PDF - arXiv.org e-Print archive · PDF fileequation,likethe n-bodyproblem in celestialm...

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05 Jacky C R ESSO N

S�ebastien D A R SES

ST O C H A ST IC EM B ED D IN G O F

D Y N A M IC A L SY ST EM S

http://arxiv.org/abs/math/0509713v1

Jacky CRESSON

Universit�e deFranche-Com t�e,�EquipedeM ath�em atiquesde Besan�con,CNRS-UM R

6623,16 routedeG ray,25030 Besan�con cedex,France..

E-m ail:[email protected]

S�ebastien DARSES

Universit�e deFranche-Com t�e,�EquipedeM ath�em atiquesde Besan�con,CNRS-UM R

6623,16 routedeG ray,25030 Besan�con cedex,France..

E-m ail:[email protected]

2000 M athem atics Subject C lassi�cation.| Prim ary 54C40,14E20;

Secondary 46E25,20C20.

K ey wordsand phrases.| Stochastic calculus, Dynamical systems, Lagrangian

systems, Hamiltonian systems.

July 2,1991

ST O C H A ST IC EM B ED D IN G O F D Y N A M IC A L

SY ST EM S

Jacky C R ESSO N ,S�ebastien D A R SES

5

A bstract.| M ostphysicalsystem sarem odelled by an ordinary ora partialdi�erential

equation,likethen-body problem in celestialm echanics.In som ecases,forexam plewhen

studying the long term behaviourofthe solarsystem orforcom plex system s,there exist

elem ents which can in uence the dynam ics ofthe system which are not wellm odelled

or even known. O ne way to take these problem s into account consists of looking at

the dynam ics ofthe system on a larger class ofobjects,that are eventually stochastic.

In this paper,we develop a theory for the stochastic em bedding ofordinary di�erential

equations. W e apply this m ethod to Lagrangian system s. In this particular case, we

extend m any resultsofclassicalm echanics nam ely,the least action principle,the Euler-

Lagrange equations,and Noether’s theorem . W e also obtain a Ham iltonian form ulation

forourstochastic Lagrangian system s.M any applicationsarediscussed attheend ofthe

paper.

C O N T EN T S

Introduction........................................................................ 11

Part I. T he stochastic derivative............................................... 19

1.A bout N elson stochastic calculus............................................. 21

1.1.Aboutm easurem entand experim ents......................................... 21

1.2.TheNelson derivatives........................................................ 23

1.3.G ood di�usion processes...................................................... 24

1.4.TheNelson derivativesforgood di�usion processes............................ 25

1.5.A rem ark aboutreversed processes............................................ 27

2.Stochastic derivative........................................................... 29

2.1.Theabstractextension problem ............................................... 29

2.2.Stochastic di�erentialcalculus................................................. 30

2.2.1.Reconstruction problem and extension.................................... 31

2.2.2.Extension to com plex processes........................................... 32

2.2.3.Stochastic derivative forfunctionsofdi�usion process.................... 36

2.2.4.Exam ples................................................................. 37

3.P roperties ofthe stochastic derivatives...................................... 41

3.1.Productrules.................................................................. 41

3.1.1.A new algebraic structure................................................. 43

3.2.Nelson di�erentiable processes................................................. 43

3.2.1.De�nition................................................................. 43

3.2.2.Exam plesofNelson-di�erentiable process................................. 44

3.2.3.Productruleand Nelson-di�erentiable processes.......................... 45

Part II. Stochastic em bedding procedures.................................... 47

8 CO N TEN TS

4.Stochastic em bedding ofdi�erentialoperators............................. 49

4.1.Stochastic em bedding ofdi�erentialoperators................................. 50

4.1.1.Abstractem bedding...................................................... 50

4.1.2.Nelson Stochastic em bedding............................................. 51

4.2.Firstexam ples................................................................ 53

4.2.1.Firstorderdi�erentialequations.......................................... 53

4.2.2.Second orderdi�erentialequations........................................ 53

5.R eversible stochastic em bedding............................................. 55

5.1.Reversible stochastic derivative................................................ 55

5.2.Iterates........................................................................ 58

5.3.Reversible stochastic em bedding............................................... 59

5.4.Reversible versusgeneralstochastic em bedding................................ 59

5.5.Stochastic m echanicsand the Stochastization procedure...................... 60

5.5.1.TheStochastic Newton Equation......................................... 60

Part III. Stochastic em bedding ofLagrangian and H am iltonian system s. 61

6.Stochastic Lagrangian system s................................................ 63

6.1.Rem inderaboutLagrangian system s.......................................... 63

6.2.Stochastic Euler-Lagrange equations.......................................... 65

6.3.Thecoherence problem ........................................................ 65

7.Stochastic calculus ofvariations.............................................. 67

7.1.Functionaland L-adapted process............................................. 67

7.2.Space ofvariations............................................................ 68

7.3.Di�erentiable functionaland stationary processes............................. 68

7.3.1.TheP = C1(I)case....................................................... 69

7.3.2.TheP = N1(I)case....................................................... 69

7.4.A technicallem m a............................................................ 70

7.5.Leastaction principles........................................................ 71

7.5.1.TheP = C1(I)case....................................................... 71

7.5.2.TheP = N1(I)case....................................................... 72

7.6.Thecoherence lem m a......................................................... 73

8.T he Stochastic N oether theorem ............................................. 75

8.1.Tangentvectorto a stochastic process......................................... 75

8.2.Canonicaltangentm ap........................................................ 75

8.3.Stochastic suspension ofone param eterfam ily ofdi�eom orphism s............ 76

CO N TEN TS 9

8.4.Lineartangentm ap........................................................... 78

8.5.Invariance..................................................................... 78

8.6.Thestochastic Noether’stheorem ............................................. 79

8.7.Stochastic �rstintegrals....................................................... 80

8.7.1.Rem inderabout�rstintegrals............................................ 80

8.7.2.Stochastic �rstintegrals................................................... 80

8.8.Exam ples...................................................................... 81

8.8.1.Translations............................................................... 81

8.8.2.Rotations................................................................. 81

8.9.About�rstintegralsand chaotic system s...................................... 82

9.N aturalLagrangian system s and the Schr�odinger equation............... 85

9.1.NaturalLagrangian system s................................................... 85

9.2.Schr�odingerequations......................................................... 85

9.2.1.Som enotationsand a rem inderofthe Nelson wave function.............. 85

9.2.2.Schr�odingerequationsasnecessary conditions............................ 86

9.2.3.Rem arksand questions................................................... 87

9.3.Aboutquantum m echanics.................................................... 89

10.Stochastic H am iltonian system s............................................. 91

10.1.Rem inderaboutHam iltonian system s........................................ 91

10.2.Them om entum process...................................................... 92

10.3.TheHam iltonian stochastic em bedding...................................... 93

10.4.TheHam iltonian leastaction principle....................................... 94

10.5.TheHam iltonian coherence lem m a........................................... 95

11.C onclusion and perspectives................................................. 97

11.1.M athem aticaldevelopm ents.................................................. 97

11.1.1.Stochastic sym plectic geom etry.......................................... 97

11.1.2.PDE’sand the stochastic em bedding.................................... 97

11.2.Applications................................................................. 98

11.2.1.Long term behaviourofchaotic Lagrangian system s..................... 98

11.2.2.Celestialm echanics......................................................101

11.2.3.Strange attractors.......................................................102

N otations...........................................................................105

B ibliography.......................................................................107

IN T R O D U C T IO N

O rdinary aswellaspartialdi�erentialequationsplay a fundam entalrolein m ostparts

ofm athem aticalphysics.Thestory beginswith Newton’sform ulation ofthelaw ofattrac-

tion and the corresponding equations which describe the m otion ofm echanicalsystem s.

Regardless the beauty and usefulness ofthese theories in the study ofm any im portant

naturalphenom ena,one m ust keep in m ind that they are based on experim entalfacts,

and asa consequenceareonly an approxim ation oftherealworld.Thebasicexam plewe

havein m ind isthem otion oftheplanetsin thesolarsystem which isusually m odelled by

the fam ousn-body problem ,i.e. n pointsofm assm i which are only subm itted to their

m utualgravitationalattraction.Ifonelooksatthebehaviourofthesolarsystem for�nite

tim e then thism odelisa very good one.Butthisisnottrue when one looksatthe long

term behaviour,which is for instance relevant when dealing with the so called chaotic

behaviourofthe solarsystem overbillionsyears,orwhen trying to predictice agesover

a very large range oftim e. Indeed,the n-body problem isa conservative system (in fact

a Lagrangian system ) and m any non-conservative e�ects, such as tidalforces between

planets,willbe ofincreasing im portance along the com putation.These non-conservative

e�ects push the m odeloutside the category ofLagrangian system s. You can go further

by considering e�ects due to the changing in the oblateness ofthe sun. In thiscase,we

do noteven know how to m odelsuch kind ofperturbations,and oneisnotsureofstaying

in the category ofdi�erentialequations(1).

(1)Note that in the context ofthe solar system we have two di� erent problem s: � rst,ifone uses only

Newton’sgravitationallaw,one m usttake into accountthe entire universe to m odelthe behaviourofthe

planets. This by itselfis a problem which can be studied by using the classicalperturbation theory of

ordinary di� erentialequations. This is di� erentifwe wantto speak ofthe \real" solar system for which

we m ust consider e� ects that we ignore. In that case,even the validation ofthe law ofgravitation as a

reallaw ofnature isnotclear.Ireferto [16]form ore detailson thispoint.

12 IN TRO D U CTIO N

As a �rst step, this paper proposes tackling this problem by introducing a natural

stochastic em bedding procedure for ordinary or partialdi�erentialequations. This con-

sistsoflooking forthebehaviourofstochastic processessubm itted to constraintsinduced

by theunderlyingdi�erentialequation(2).W epointoutthatthisstrategy isdi�erentfrom

the standard approach based on stochastic di�erentialequationsorstochastic dynam ical

system s, where one gives a m eaning to ordinary di�erentialequations perturbed by a

sm allrandom term .In ourwork,no perturbationsoftheunderlying equation arecarried

out.

A point of view that bears som e resem blance to ours is contained in V.I.Arnold’s

m aterialization ofresonances ([6],p.303-304),whose m ain underlying idea can be brie y

explained as follows: the divergence of the Taylor expansion of the arctanx function

at 0 for jx j> 1 can be proved by com puting the coe�cients ofthis series. However,

this doesnotexplain the reason for thisdivergence behaviour. O ne can obtain a better

understanding by extending the function to the com plex plane and by looking at its

singularities at � i. The sam e idea can be applied in the context ofdynam icalsystem s.

In thiscase,we look forthe obstruction to linearization ofa realsystem sin the com plex

plane.Arnold hasconjectured thatthisisduetotheaccum ulation ofperiodicorbitsin the

com plex planealong therealaxis.In ourcase,onecan try to understand som eproperties

ofthe trajectories ofdynam icalsystem s by using a suitable extension ofits dom ain of

de�nition. In ourwork,we give a precise sense to the conceptofdi�erentialand partial

di�erentialequations in the class ofstochastic processes. Thisprocedure can be viewed

asa�rststep toward thegeneral\stochasticprogram m e"asdescribed byM um ford in [51].

O urem bedding procedureisbased on a sim pleidea:in orderto writedown di�erential

or partialdi�erentialequations,one uses derivatives. An ordinary di�erentialequation

is nothing else but a di�erentialoperator oforder one(3). In order to em bed ordinary

di�erentialequations, one m ust �rst extend the notion of derivative so that it m akes

sense in the context ofstochastic processes. By extension,we m ean that our stochastic

derivative reduces to the classical derivative for determ inistic di�erentiable processes.

Having this extension,one easily de�nes in a unique way,the stochastic analogue ofa

di�erentialoperator,and asaconsequence,anaturalem beddingofan ordinary di�erential

(2)Thisstrategy ispartofa generalprogram m e called the em bedding procedure in [15]and which can be

used to em bed ordinary di� erentialequations not only on stochastic processes buton generalfunctional

spaces. A previous attem pt was m ade in [13],[14]in the context ofthe non-di� erentiable em bedding of

ordinary di� erentialequations.(3)In thiscase,we can also speak ofvector� elds.

IN TRO D U CTIO N 13

equation on stochastic processes.

O fcourse,one can think thatsuch a sim ple procedure willnotproduce anything new

forthe study ofclassicaldi�erentialequations. Thisisnotthe case. The m ain problem

that we study in this paper is the em bedding ofnaturalLagrangian system s which are

ofparticular interest for classicalm echanics. In this context,we obtain som e num erous

surprising results,from the existence ofa coherentleastaction principle with respectto

the stochastic em bedding procedure, to a derivation of a stochastic Noether theorem ,

and passing by a new derivation ofthe Schr�odinger equation. Allthese points willbe

described with detailsin thefollowing.

Two com panion papers ([18],[9]) give an application of this m ethod to derive new

results on the form ation ofplanets in a protoplanetary nebulae,in particular a proofof

theexistenceofa so called Titus-Bodelaw forthespacing ofplanetsaround a given star.

Theplane ofthe paperisasfollow:

In a �rstpart,we develop ournotion ofa stochastic derivative and study in detailsall

itsproperties.

Chapter1 givesa review ofthestochasticcalculusdeveloped by Nelson [53].In partic-

ular,we discuss the classicalde�nition ofthe backward and forward Nelson derivatives,

denoted by D and D �,with respect to dynam icalproblem s. W e also de�ne a class of

stochastic process called good di�usion processes for which one can com pute explicitly

the Nelson derivatives.

In Chapter2wede�newhatwecallan abstractextension oftheclassicalderivative.Us-

ing theNelson derivatives,wede�nean extension oftheordinary derivativeon stochastic

processes,which wecallthestochastic derivative.Aspointed outpreviously,oneim poses

thatthestochasticderivativereducestotheclassicalderivativeon di�erentiabledeterm in-

isticprocesses.ThisconstraintensuresthatthestochasticanalogueofaPDE containsthe

classicalPDE.O fcourse such a gluing constraintisnotsu�cientto de�nea rigid notion

ofstochastic derivative. W e study severalnaturalconstraintswhich allow usto obtain a

uniqueextension ofthe classicalderivative on stochastic processesas

(0.1) D � =D + D �

2+ i�

D � D �

2;� = � 1:


By extending thisoperatorto com plex valued stochastic processes,we are able to de�ne

theiterate ofD ,i.e.D 2 = D � D and so on.Them ain surpriseisthattherealpartofD2

correspond to the choice ofNelson foracceleration in hisdynam icaltheory ofBrownian

m otion. However,thisresultdependson the way we extend the stochastic derivative to

com plex valued stochastic processes. W e discuss severalalternative which covers well

known variationson theNelson acceleration.

In Chapter 3 we study the product rule satis�ed by the stochastic derivative which

is a fundam entalingredient ofour stochastic calculus ofvariation. W e also introduce

an im portant class ofstochastic processes,called Nelson di�erentiable,which have the

property to have a realvalued stochastic derivative. These processesplay a fundam ental

role in the stochastic calculus ofvariation as they de�ne the naturalspace ofvariations

forstochastic processes.

The second part of this article deals speci�cally with the de�nition of a stochastic

em bedding procedureforordinary di�erentialequations.

Chapter 4 associate to a di�erentialoperator of a given form acting on su�ciently

regularfunctionsa unique operatoracting on stochastic processesand de�ned sim ply by

replacing the classicalderivative by the stochastic derivative.Thisisthisprocedurethat

we callthe stochastic em bedding procedure. Note that the form ofthis procedure acts

on di�erentialoperatorsofa given form .Although the procedureiscanonicalfora given

form ofoperator,itisnotcanonicalfora given operator.

Thepreviousem beddingisform aland doesnottakeconstraintswhich areofdynam ical

nature,likethereversibility oftheunderlyingdi�erentialequation.Asreversibility playsa

centralrolein physics,especially in celestialm echanicswhich isonedom ain ofapplication

ofourtheory,we discussthispointin details.W e introduce an em bedding which respect

the reversibility of the underlying equation. Doing this, we see that we m ust restrict

attention to therealpartofouroperator,which istheuniqueoneto possessthisproperty

in oursetting.W ethen recoverunderdynam icaland algebraicargum entsstudiesdealing

with particularchoiceofstochasticderivativesin orderto derivequantum m echanicsfrom

classicalm echanicsunderNelson approach.

The third part is m ainly concerned with the application ofthe stochastic em bedding

to Lagrangian system s.


W e considerautonom ous(4) Lagrangian system sL(x;v),(x;v)2 U � Rd � R

d,where

U is an open set, which satisfy a num ber of conditions, one of it being that it m ust

be holom orphic with respect to the second variable which represent the derivative of

a given function. Such kind ofLagrangian functions are called adm issible. Using the

stochasticem bedding procedurewecan associateto theclassicalEuler-Lagrangeequation

a stochastic one which hastheform

@L

@x(X (t);D X (t))= D

�@L

@v(X (t);D X (t))

�

; (SE L)

whereX isa realvalued stochastic process.

At this point,our m anipulation is only form aland one can ask ifthis em bedding is

signi�cantornot.W ethen rem arkthattheLagrangian function L keep senseon stochastic

processesand can beconsidered asa functional.Asa consequence,we can search forthe

existence ofa least action principle which gives the stochastic Euler-Lagrange equation

(SEL).Theexistenceofsuch astochasticleastaction principleisfarfrom beingtrivialwith

respectto theem bedding procedure.Indeed,itm ustfollowsfrom a stochasticcalculusof

variations which is not developed apart from this procedure. O ur problem can then be

form alize asthe following diagram :

(0.2)

L(x;dx=dt)LAP��! E L

??yS

??yS

L(X ;D X )SLAP ?��! (SE L);

where LAP is the least action principle,S is the stochastic em bedding procedure,(EL)

is the classicalEuler-Lagrange equation associated to L and SLAP the at this m om ent

unknown stochastic leastaction principle. The existence ofsuch a principle iscalled the

coherence problem .

Chapter7 develop a stochastic calculusofvariationsforfunctionalsoftheform

(0.3) E

�Z b

a

L(X (t);D X (t))dt

�

;

where E denotes the classicalexpectation. Introducing the correct notion ofextrem als

and variations we obtain two di�erent stochastic analogue ofthe least action principle

depending on the regularity class we choose for the adm issible variations. The m ain

point is that for variations in the class of Nelson di�erentiable process, the extrem als

ofour functionalcoincide with the stochastic Euler-Lagrange equation obtained via the

stochastic em bedding procedure. This result is called the coherence lem m a. In the

(4)Thisrestriction isdue to technicaldi� culties.


reversiblecase,i.e.taking asa stochasticderivativeonly therealpartofouroperator,we

obtain thesam e resultbutin thiscase one can considergeneralvariations.

In chapter8 weprovidea �rststudy ofwhatdynam icaldata rem ain from theclassical

dynam ical system under the stochastic em bedding procedure. W e have focused on

sym m etriesofthe underlying equation and asa consequence on �rstintegrals. W e prove

a stochastic analogue ofthe Noethertheorem . Thisallows usto de�ne a naturalnotion

of�rst integralfor stochastic di�erentialequations. This part also put in evidence the

need for a geom etricalsetting governing Lagrangian system s which is the analogue of

sym plectic m anifolds.

Chapter 9 deals with the stochastic Euler-Lagrange equation for naturalLagrangian

system s,i.e.associated to Lagrangian functionsoftheform

(0.4) L(x;v)= T(v)� U (x);

where U is a sm ooth function and T is a quadratic form . In classical m echanics U

is the potential energy and T the kinetic energy. The m ain result of this chapter is

that by restricting our attention to good di�usion processes,and up to a a wellchosen

function ,called thewavefunction,thestochasticEuler-Lagrangeequation isequivalent

to a non linear Schr�odinger equation. M oreover,by specializing the class ofstochastic

processes,we obtain the classicalSchr�odingerequation. In thatcase,we can give a very

interesting characterization ofstochastic processes which are solution ofthe stochastic

Euler-Lagrange equation. Indeed,the square ofthe m odulusof isequalto the density

oftheassociated stochastic processsolution.

In chapter10,wede�nea naturalnotion ofstochasticHam iltonian system .Thisresult

can be seen asa �rstattem ptto putin evidence the stochastic analogue ofa sym plectic

structure. W e de�ne a stochastic m om entum process and prove that,up to a suitable

m odi�cation ofthestochasticem beddingprocedurecalled theHam iltonian stochasticem -

bedding,and re ecting thefactthatthe\speed" ofa given stochastic processiscom plex,

we obtain a coherent picture with the classicalform alism ofHam iltonian system s. This

�rstresultiscalled theLegendrecoherencelem m a asitdealswith thecoherencebetween

the Ham iltonian stochastic em bedding procedure and the Legendre transform .Secondly,

we develop a Ham ilton leastaction principle and we prove again a coherence lem m a,i.e.


thatthe following diagram com m utes

H (x(t);p(t))

Ham ilton leastaction principle��

SH// H (X (t);P (t))

Stochastic Ham ilton leastaction principle��

(H E )SH

// (SH E )

whereSH denotesthe Ham iltonian stochastic em bedding procedure.

The last chapter discuss m any possible developm ents ofour theory from the point of

view ofm athem aticsand applications.

PA RT I

T H E ST O C H A ST IC D ER IVAT IV E

C H A PT ER 1

A B O U T N ELSO N ST O C H A ST IC C A LC U LU S

1.1. A bout m easurem ent and experim ents

In thissection,we explain whatwe think are the basisofallpossibleextensionsofthe

classicalderivative.Thesetting ofourdiscussion isthefollowing:

W e consideran experim entalset-up which producesa dynam ics.W e assum ethateach

dynam icsisobserved during a tim ewhich is�xed,forexam ple[0;T],whereT 2 R�+ .For

each experim ent i,i2 N,we denote by X i(t) the dynam icalvariable which is observed

fort2 [0;T].

Assum ethatwe wantto describethe kinem atic ofsuch a dynam icalvariable.W hatis

the strategy ?

Theusualideaistom odelthedynam icalbehaviourofavariableby ordinary di�erential

equationsorpartialdi�erentialequations. In orderto do this,we m ust�rsttry to have

access to the speed ofthe variable. In order to com pute a signi�cant quantity we can

follow atleasttwo di�erentstrategies:

{ W e do nothave access to the variable X i(t),t2 [0;T],butto a collection ofm ea-

surem entsofthisdynam icalvariable.Assum ethatwewantto com putethespeed at

tim et.W ecan only com putean approxim ation ofitfora given resolution h greater

than a given threshold h0.Assum ethatforeach experim entweareableto com pute

thequantity

(1.1) vi;h(t)=X i(t+ h)� X i(t)

h:

W e can then try to look for the behaviour ofthis quantity when h varies. Ifthe

underlying dynam icsisnottoo irregular,then wecan expecta lim itforvi;h(t)when

22 CH A PTER 1.A BO U T N ELSO N STO CH A STIC CA LCU LU S

h goesto zero thatwe denote by vi(t).

W e then com pute them ean value

(1.2) �v(t)=1

n

nX

i= 1

vi(t):

Ifthe underlying dynam icsisnottoo irregularthen �v(t)can be used to m odelthe

problem .In thecontrary the basic idea isto introducea random variable.

Rem ark thatdueto the intrinsic lim itation forh we neverhave accessto vi(t)so

thatthisprocedurecan notbeim plem ented.

{ Anotheridea isto look directly forthequantity

(1.3) �vh;n(t)=1

n

nX

i= 1

vi;h(t):

Contrary to the previouscase,ifthere existsa wellde�ned m ean value �vh(t)when

n goesto in�nity then we can have a asclose aswe wantapproxim ation. Indeed it

su�cesto do su�ciently m any experiences.W ethen look forthelim itof�v h(t)when

h goesto zero.

Forregulardynam icsthesetwo procedureslead tothesam eresultasallthesequantities

are wellde�ned and converge to the sam e quantity. This is not the case when we deal

with highly irregulardynam ics. In thatcase the second procedure iseasily im plem ented

contrary to the �rstone. The only problem isthatwe loose the geom etricalm eaning of

the resulting lim itquantity with respectto individualtrajectories asone directly take a

m ean on alltrajectoriesbeforetaking thelim itin h.

Thissecond alternative can be form alized using stochastic processes and leads to the

Nelson backward and forward derivativesthatwe de�nein thenextsection.

W e have take the opportunity to discuss these notions because the previous rem arks

provesthatonecan notjustify theform oftheNelson derivativesusingageom etricalargu-

m entlike the non di�erentiability oftrajectoriesfora Brownian m otion.Thisishowever

the argum entused by E.Nelson ([54],p.1080)in orderto justify the factthatwe need a

substitutefortheclassicalderivative when studying W ienerprocesses.Thism isleadingly

suggestthattheforward and backward derivativecapturethisnon di�erentiability in their

de�nition,which isnotthe case.

1.2.TH E N ELSO N D ER IVATIV ES 23

1.2. T he N elson derivatives

Let X (t),0 6 t6 1 be d-dim ensionalcontinuous random process de�ned on a prob-

ability space (;A ;P ), where A is the �-algebra of allm easurable events and P is a

probability m easurede�ned on A .W e denote by I the open interval(0;1).

D e�nition 1.1.| The random process X (t),a 6 t6 b,is an SO-process ifeach X (t)

belongsto L1()and the m apping t! X (t)from R to L 1()iscontinuous.

LetP = fPtg and F = fFtgbean increasingand adecreasing fam ily ofsub-�-algebras,

respectively,such thatX (t)isFt-m easurable and Pt-m easurable.In otherwords,F and

P are two �ltration to which X (t) is adapted. W e let E [� jB]denote the conditional

expectation with respectto any sub-�-algebra B � A .

D e�nition 1.2.| The random process X (t), a 6 t 6 b, is an S1-process ifit is an

SO-process such that

(1.4) D X (t)= limh! 0+

E

�X (t+ h)� X (t)

hjPt

�

;

and

(1.5) D �X (t)= limh! 0+

E

�X (t)� X (t� h)

hjFt

�

;

existin L1()and the m appingst7! D X (t)and t7! D �X (t)are both continuousfrom R

to L1().

D e�nition 1.3.| The random process X (t), a 6 t 6 b, is an S2-process ifit is an

S1-process,and

(1.6) �2X (t)= lim

h! 0+E

�(X (t+ h)� X (t))2

hjPt

�

;

and

(1.7) �2�X (t)= lim

h! 0+E

�(X (t+ h)� X (t))2

hjFt

�

;

existin L1().

D e�nition 1.4.| W edenote by C1(I)the totality ofS2-processeswith continuoussam -

ple paths,such thatX (t),D X (t)and D �X (t),a 6 t6 b,alllie in the Hilbertspace L2()

and are continuous functionsoftin L2().

A com pletion ofC1(I)in the norm

(1.8) k X k= supt2I

(k X (t)kL2() + k D X (t)kL2() + k D �X (t)kL2());

isalso denoted by C1(I),where k :kL2() denotes the norm ofHilbertspace L2().


R em ark 1.1.| The m ain pointin the previous de�nitions for a forward and backward

derivativeofa stochasticprocess,isthattheforward and backward �ltration are�xed bythe

problem .Asa consequence,wehavenotan intrinsicquantity only related to thestochastic

process. A possible alternative de�nition isthe following:

D e�nition 1.5.| LetX be a stochastic process,and �(X ) (resp. ��(X )) the forward

(resp. backward) adapted �ltration. W e de�ne

dX (t)= limh! 0+

h�1E [X (t+ h)� X (t)j�(X s;0 6 s6 t)];(1.9)

d�X (t)= limh! 0+

h�1E [X (t)� X (t� h)j�(X s;t6 s6 1)]:(1.10)

In this case, we obtain intrinsic quantities, only related to the stochastic process.

However, these new operators behave very badly from an algebraic view point. Indeed,

withoutstringentassum ptions on stochastic processes,we do nothave linearity ofd or d�.

This di�culty is not apparent as long as one restrict attention to a single stochastic

process.

1.3. G ood di�usion processes

W e introduce a specialclassofdi�usion processesforwhich we can explicitly com pute

the derivative D ,D �,D D �,D �D ,D2 and D 2

�.

D e�nition 1.6.| W e denote by �d the space of di�usion processes X satisfying the

following conditions:

i-X solvesa stochastic di�erentialequation :

dX (t)= b(t;X (t))dt+ �(t;X (t))dW (t); X (0)= X 0;(1.11)

where X 0 2 L2(),b:[0;T]� R d ! Rd and � :[0;T]� Rd ! R

d Rd areBorelm easurable

functionssatisfyingthehypothesis:thereexistsa constantK such thatforevery x;y 2 Rd

we have

supt

(j�(t;x)� �(t;y)j+ jb(t;x)� b(t;y)j)6 K jx � yj;(1.12)

supt

(j�(t;x)j+ jb(t;x)j)6 K (1+ jxj):(1.13)

ii-For any t> 0,X (t)has a density pt(x)atpointx.

1.4.TH E N ELSO N D ER IVATIV ES FO R G O O D D IFFU SIO N PRO CESSES 25

iii-Setting aij = (��)ij,forany i2 f1;� � � ;ng,forany t0 > 0,forany bounded open set

D � Rd,

(1.14)

Z1

t0

Z

D

j@j(aij(t;x)pt(x))jdxdt< + 1 :

iv-b and (t;x)!1

pt(x)@j(aij(t;x)pt(x))are continuous and bounded functions.

R em ark 1.2.| { Hypothesis iii)ensures that(1.11) hasa unique t� continuousso-

lution X (t).

{ Hypothesis i),ii)and iii)allow to apply theorem 2:3 p.217 in [49].

{ W e m ay wonder in which cases hypothesis ii) holds. Theorem 2:3:2 p:111 of[58]

givesthe existence ofa density for allt> 0 under the H�orm ander hypothesis which

is involved by the stronger condition thatthe m atrix di�usion �� is elliptic atany

pointx.A sim ple exam ple isgiven by a SDE where b isa C1 (I� Rd)function with

allitsderivativesbounded,and where the di�usion m atrix isa constantequalto cId.

In this case,pt(x) belongs to C1 (I� Rd);m oreover,ifX 0 has a di�erentiable and

everywhere positive density p0(x) with respectto Lebesgue m easure such thatp0(x)

and p0(x)�1 r p0(x)are bounded,then b(t;x)� cr log(pt(x))isbounded asnoticed in

theproofofproposition 4:1 in [64].So hypothesisii)seem snotto besuch a restrictive

condition.

{ Assum ption iv) is necessary to com pute explicitly the second order operators ofD

and D �. The existence ofD and D � is ensured under a weaker condition,the �nite

entropy condition equivalentto

(1.15) E

�Z1

0

(b(t;X (t))2 dt

�

< 1 :

W e refer to F�ollm er ([25],proposition 2.5 p.121 and lem m a 3.1 p.123) for m ore

details.

According to thetheorem 2:3 of[49]and thanksto iv),wewillseethat�d � C1([0;T])

and thatwe can com pute D X and D �X forX 2 �d (see Theorem 1.1).

1.4. T he N elson derivatives for good di�usion processes

A usefulproperty ofgood di�usionsprocessesisthattheirNelson’sderivativescan be

explicitly com puted.Precisely,we have:


T heorem 1.1.| LetX 2 �d which writesdX (t)= b(t;X (t))dt+ �(t;X (t))dW (t). Then

X isM arkov di�usion with respectto an increasing �ltration (P t)and a decreasing �ltra-

tion (Ft).M oreover,D X and D �X existsw.r.t. these �ltration and :

D X (t) = b(t;X (t))(1.16)

D �X (t) = b�(t;X (t))(1.17)

where x ! pt(x)denotesthe density ofX (t)atx and

bi�(t;x)= b

i(t;x)�1

pt(x)@j(a

ij(t;x)pt(x))

with the convention thatthe term involving 1

pt(x)is0 ifpt(x)= 0.

Proof.| The proof uses essentially theorem 2.3 of M illet-Nualart-Sanz [49] and the

techniquesofM .Thieullen fortheproofofproposition 4.1 in [64].

(1) Let X 2 �d. Then X is a M arkov di�usion w.r.t. the increasing �ltration (P t)

generated by theBrownian M otion W (t)and so :

E

�X (t+ h)� X (t)

hjPt

�

= E

�1

h

Z t+ h

t

b(s;X (s))dsjPt

�

;

and

E

��E

�X (t+ h)� X (t)

hjPt

�

� b(t;X (t))

��

�

6 E

�1

h

Z t+ h

t

jb(s;X (s))� b(t;X (t))jds

�

:

W e can apply the dom inated convergence theorem since bisbounded and

1

h

Z t+ h

t

jb(s;X (s))� b(t;X (t))jdsh! 0�! 0 a:s:

(forbiscontinuousand X hasa.s.continuouspaths).

ThereforeD X existsand D X (t)= b(t;X (t)).

(2) As X 2 �d,we can apply theorem 2:3 in [49]. So X (t)= X (1� t) is a di�usion

process w.r.t. an increasing �ltration (P t) and whose generator reads Ltf = bi@if +

1

2aij@ijf with aij(1� t;x)= aij(t;x)and b

i(1� t;x)= � bi(t;x)+

1

pt(x)@j(a

ij(t;x)pt(x)).

Setting Ft= P 1�t,X isa M arkov di�usion w.r.t.the decreasing �ltration (F t).W e have

:

E

�X (t)� X (t� h)

hjFt

�

= E

�X (1� t)� X (1� t+ h)

h

��P1�t

�

= � E

�1

h

Z 1�t+ h

1�t

b(s;X (s))ds��P 1�t

�

:(1.18)

1.5.A R EM A R K A BO U T R EV ER SED PRO CESSES 27

Using thesam ecalculationsand argum entsasabove(sincehypothesisiv)in thede�nition

ofclass�d im pliesthatb iscontinuousand bounded),we obtain thatD �X (t)existsand

isequalto � b(1� t;X (1� t)).

In the case offractionalBrownian m otion oforderH 6= 1=2,the Nelson derivativesdo

not exist. However,one can de�ne new operators using the so-called quasiconditional

expectation introduced by [1].W ereferto thework ofDarsesand Sausserau [19]form ore

details.

1.5. A rem ark about reversed processes

This part reviews basic results about reversed processes,with a specialem phasis to

di�usion processes.W e useNelson’sstochastic calculus.

Let X be a process in the class C1([0;1]). W e denote by eX the reversed process :

eX (t)= X (1� t),with his"past" ePt and his"future" eFt.Asa consequence,wealso have

ex 2 C1([0;1]! H ).

Using the operators d and d� de�ned in de�nition 1.5,we have:

Lem m a 1.1.| d�x(t)= � dex(1� t)= � edex(t):

Proof.| Thede�nition ofd� givesim m ediately:

d�x(t)= lim�! 0+

E

�ex(1� t)� ex(1� t+ �)

�

��Ft

�

:

ButFt= �fx(s);t6 s6 1g = �fex(u);0 6 u 6 1� tg = eP1�t.

Thus:

d�x(t)= lim�! 0+

� E

�ex(1� t+ �)� ex(1� t)

�

��eP1�t

�

= � dex(1� t)= � edex(t):

The sam e com putation is not at allpossible when dealing with the operators D and

D �.

C H A PT ER 2

ST O C H A ST IC D ER IVAT IV E

In this part, we construct a natural extension(1) of the classical derivative on real

stochastic processesasa uniquesolution to an algebraic problem .Thisstochastic deriva-

tive turns out to be necessarily com plex valued. O ur construction relies on Nelson’s

stochastic calculus [53]. W e then study properties ofour stochastic derivative and es-

tablish a num beroftechnicalresults,including a generalization ofNelson’sproductrule

[53]aswellasthestochasticderivativeforfunctionsofdi�usion processes.W ealso com -

pute the stochastic derivative in som e classicalexam ples. The m ain point is that,after

a naturalextension to com plex processes,the realpartofthe second derivative ofa real

stochastic processcoincide with Nelson’sm ean acceleration. W e de�ne a specialclassof

processescalled Nelson di�erentiable,which willbe ofim portance forthe stochastic cal-

culusofvariationsdeveloped in chapter7.Thispartisselfcontained and allbasicresults

aboutNelson’sstochastic calculusare rem inded.

2.1. T he abstract extension problem

In thissection,we discussin a generalabstract setting,whatkind ofanalogue ofthe

classicalderivative we are waiting foron stochastic processes.

W e �rstrem ark thatreal(2) valued functionsnaturally em bed in stochastic processes.

Indeed,letf :R ! R beagiven function.W edenoteby X f thedeterm inisticstochastic

processde�ned by

(2.1) X f(!)= f 8! 2 :

(1)A precisem eaning to thisword willbegiven in thefollowing.Itshould benoted thatM alliavin calculus

isnotan extension ofthe ordinary di� erentialcalculus(see below).(2)O uraim was� rstto study dynam icalsystem soverR

n.However,aswewillseewewillneed to consider

com plex valued objects.

30 CH A PTER 2.STO CH A STIC D ER IVATIV E

W e denoteby � :RR ! P them ap associating to f 2 RR the stochastic processX f.

W e denote by Pdet the subsetofP consisting ofdeterm inistic processes,and by P kdet

the set�(Ck),k > 1.

As a consequence,we have a naturalaction ofthe classicalderivative on the set of

di�erentiable determ inistic processes,thatwe denote again d=dt.

LetK = R orC.In thesequel,wedenoteby PK � SK a subsetofthesetofK -valued

stochastic processes(3).

LetK = R orC.

D e�nition 2.1.| LetK = R or C. An extension ofd=dton PK is an operator �,i.e.

a m ap � :PK ! SK such that:

i)� coincideswith d=dton P1det,

ii)� is R-linear.

Condition i), which is a gluing condition on the classical derivative is necessary as

longasonewantstorelateclassicaldi�erentialequationswith theirstochasticcounterpart.

Condition ii)ism oredelicate.O fcourse,onehaslinearity of� on Di�.A naturalideais

then to preserve fundam entalalgebraic propertiesofd=dt,R-linearity being one ofthem .

Thiscondition isnotso stringent,ifforexam ple we considerK = C.But,following this

pointofview,onecan ask form oreprecise propertieslike theLeibniz rule

(2.2) d=dt(X � Y )= d=dt(X )� Y + X � d=dt(Y ); 8X ;Y 2 P1det:

In whatfollows,we constructa stochastic di�erentialcalculus based on Nelson’s deriva-

tives.

2.2. Stochastic di�erentialcalculus

In thispart,weextend theclassicaldi�erentialcalculusto stochastic processesusing a

previouswork ofNelson [53]on the dynam icaltheory ofBrownian m otion. W e de�ne a

stochastic derivative and review itsproperties.

(3)W e do not give m ore precisions on this set for the m om ent,the set P can be the whole set ofrealor

com plex valued stochastic processes,ora particularclasslike di� usion processes,...etc.

2.2.STO CH A STIC D IFFER EN TIA L CA LCU LU S 31

2.2.1. R econstruction problem and extension.| Letusbegin with som eheuristic

rem arkssupporting ourde�nition and construction ofa stochastic derivative.

O ur aim is to construct a "natural" operator on C1(I) which reduces to the classical

derivative d=dt over di�erentiable determ inistic processes(4). The basic idea underlying

the whole construction is that, for exam ple in the case of the Brownian m otion, the

trajectories are non-di�erentiable. At least, this is the reason why Nelson [53]intro-

duces the left and right derivatives D X and D �X for a given process X . Ifwe refer to

geom etry,forgetting for a m om ent processes for trajectories,the fundam entalproperty

ofthe classicalderivative dx=dt(t0) ofa trajectory x(t) at point t0,is to provide a �rst

order (geom etric) approxim ation ofthe curve in a neighbourhood oft0. O ne wants to

construct an operator,that we denote by D ,such that the data ofD X (t0)allows usto

give an approxim ation ofX in a neighbourhood oft0. The di�erence is that we m ust

know two quantities,nam ely D X and D �X ,in order to obtain the inform ation(5). For

com putationalreasons,one wants an operator with values in a �eld F . This �eld m ust

bea naturalextension ofR (aswewantto recovertheclassicalderivative)and atleastof

dim ension 2.Thenaturalcandidateto such a �eld isC.O necan also recoverC by saying

thatwe m ustconsidernotonly R butthe doubling algebra which correspondsto C.

Thisinform aldiscussion leadsustobuild acom plex valued operatorD :C1(I)! C1

C(I),

with the following constraints:

i)(Gluing property)ForX 2 P 1det,D X (t)= dX =dt,

ii)TheoperatorD isR-linear,

iii)(Reconstruction property)ForX 2 C1(I),letusdenote by

D X = A(D X ;D �X )+ iB (D X ;D �X );

whereA and B are linearR-valued m appingsby ii).W e assum ethatthem apping

(D X ;D �X )7! (A(D X ;D �X );B (D X ;D �X ))

isinvertible.

(4)A rigourousm eaning to thissentence willbe given in the sequel.

(5)Thisrem ark isonly valid forgeneralstochastic processes.Indeed,aswewillsee,fordi� usion processes,

there isa close connection between D X and D �X ,which allowsto sim plify the de� nition ofD .


Lem m a 2.1.| The operator D has the form

D �X = [aD X + (1� a)D �X ]+ i�b[D X � D �X ]; � = � 1;

where a;b2 R and b6= 0.

Proof.| W edenoteby A(X )= aD X + bD �X and B (X )= cD X + dD �X .IfX 2 C 1(I),

we have D X = D �X = dX =dt,and i)im plies

a+ b= 1; c+ d = 0:

W ethen obtain thedesired form .By iii),wem usthaveb6= 0 in orderto haveinvertibility.

In orderto rigidify thisoperator,weim posea constraintcom ing from theanalogy with

the construction ofthe scale-derivative fornon-di�erentiable functionsin [13].

iv)IfD � = � D ,then A(X )= 0,B (X )= D .

W e then obtain the following result:

Lem m a 2.2.| An operator D satisfying conditions i),ii),iii)and iv)isofthe form

(2.3) D � =D + D �

2+ i�

D � D �

2; � = � 1:

Proof.| Using lem m a 2.1,iii)im pliesthe relations: 2a � 1 = 0 and 2b= 1,so a = b=

1=2.

W e then introduce thefollowing notion ofstochastic derivative:

D e�nition 2.2.| W e denote by D � the operators de�ned by

D � =D + D �

2+ i�

D � D �

2; � = � 1:

2.2.2. Extension to com plex processes.| In order to em bed second order di�er-

entialequations,we need to de�ne the m eaning ofD 2,and m ore generally ofD n,n 2 N.

Thebasicproblem isthat,contrary to whathappensfortheordinary di�erentialoperator

d=dt,even ifwe consider realvalued processes X ,the derivative D X is a com plex one.

Asa consequence,one m ustextend D to com plex processes.

For the m om ent,let us denoted by DCthe extension to be de�ne ofD ,to com plex

processes. Let F be a �eld containing C to be de�ned,and D C :C1C(I) ! F . There

are essentially two possibilities to extend the stochastic derivative leading to the sam e

de�nition:an algebraic and an analytic one.


2.2.2.1. Algebraic extension.| Letusassum ethat:

i)theoperatorDCisR-linear.

Let Z = X + iY be a com plex process,where X and Y are two realprocesses. By

R-linearity,we have

DC(Z)= D

CX + D

C(iY ):

AsDCreduceto D on realprocesses,we obtain

DC(Z)= D X + D

C(iY );

which reduce the problem ofthe extension to �nd a suitable de�nition ofD on purely

im aginary processes.

W e now m ake an assum ption aboutthe im age ofDC:

ii)TheoperatorDCisC-valued.

Thisassum ption isfarfrom being trivial,and hasm any consequences.O neofthem is

that,whateverthede�nition ofDC(iY )is,wewillobtain a com plex quantity which m ixes

with the quantity D X in a non trivialway.

R em ark 2.1.| One can wonder if another choice is possible, as for exam ple, using

quaternions in order to avoid this m ixing problem . However,a heuristic idea behind the

com plex nature ofD isthatitcorresponds to a fundam entalproperty ofNelson processes,

the (in general) non-di�erentiable character of trajectories. Then, the doubling of the

underlying algebra is related to a sym m etry breaking(6). The com putation ofD 2 is not

related to such phenom enon.

In the following, we give two di�erent extensions of D to com plex processes under

hypothesisi)and ii).Thebasic problem isthe following:

LetY bea realprocess.W e denote

(2.4) D Y = S(Y )� iA(Y );

where

(2.5) S(Y )=

�D + D �

2

�

(Y ); and A(Y )=

�D � D �

2

�

(Y );

(6)This reduces to D X = D �X for determ inistic di� erentiable processes, nam ely the invariance under

h ! � h.


and the letters S and A stand for the sym m etric and antisym m etric operators with

respectto the exchange ofD with D �.

W e denote

(2.6) DC(iY )= R(Y )+ iI(Y );

whereR(Y )and I(Y )are two realprocesses.

O necan ask ifwe expectforspecialrelationsbetween R(Y ),I(Y )and S(Y ),A(Y ).

2.2.2.1.1. C-linearity.| Ifno relations are expected for,the naturalhypothesis is to

assum e C-linearity ofDC,i.e.

(2.7) DC(iY )= iD Y:

Asa consequence,we obtain the following de�nition fortheoperatorDC:

W e denote by C1

C(I) the set ofstochastic processes ofthe form Z = X + iY ,with

X ;Y 2 C1(I).

D e�nition 2.3.| The operator DC:C1

C! C

1

Cisde�ned by

DC;�(X + iY )= D �X + i�D �Y; � = � 1;

where X ;Y 2 C1.

In the sequel,we denote DCforD

C;�.

Thefollowing lem m agivesastrongreason tochoosesuch ade�nition ofDC.W edenote

by

D n

C= D

C� � � � � D

C:

Lem m a 2.3.| W e have

D 2

C=

�D D � + D �D

2

�

+ i

�D 2 � D 2

�

2

�

:(2.8)

Proof.| O neusethe C-linearity ofoperatorD .

W e note thatthe realpartofD 2 isthe m ean acceleration asde�ned by Nelson [53].

R em ark 2.2.| In ([53],p.81-82),Nelson discussesnaturalcandidatesforthestochastic

analogue ofacceleration. M ore or less,the idea is to consider quadratic com binations of

D and D �,respecting a gluing property with the classicalderivative:


LetQ a;b;c;d(x;y) = ax2 + bxy + cyx + dy2 be a realnon-com m utative quadratic form

such thata+ b+ c+ d = 1.A possible de�nition fora stochastic acceleration isQ (D ;D �).

W e rem ark thatthe condition a + b+ c+ d = 1 im plies thatwhen D = D �,we have

Q (D ;D �)= D = D �.

The sim plestexam ples ofthiskind are: D 2,D 2�,D D � and D �D .

W e can also im pose a sym m etry condition in order to take into accountthatwe do not

wantto give a specialim portance to the m ean-forward or m ean-backward derivative,by

assum ing thatQ (x;y)= Q (y;x),so thatQ isofthe form

Q a(x;y)= a(x2 + y2)+ (1� 2a)

xy+ yx

2;a 2 R:

The sim plestexam ple in thiscase isobtained by taking a = 0,i.e.

Q 0(D ;D �)=D D � + D �D

2:

Thislastone corresponds to Nelson’sm ean acceleration and coincide with the realpartof

our stochastic derivative.

Itm ustbe pointed outthatNelson discuss only �ve possible candidates where atleast

a three param eters fam ily can be de�ned by Q a;b;c;1�a�b�c (D ;D �). His �ve candidates

correspond to the sim plestcaseswe have described.

The choice ofQ 0(D ;D �)asa m ean acceleration isjusti�ed by Nelson using a Gaussian

M arkov process X (t)in equilibrium ,satisfying the stochastic di�erentialequation

dX (t)= � !X (t)dt+ dW (t):

W e willreturn to thisproblem below.

2.2.2.2. Analyticextension.| W e�rstrem arkthatD and D � possessanaturalextension

to com plex processes.Indeed,letX = X 1 + iX 2,with X i2 C1(I)then

D (X 1 + iX 2)= D (X 1)+ iD (X 2)and D �(X 1 + iX 2)= D �(X 1)+ iD �(X 2):

As a consequence,the quantities S(Y ) and A(Y ) introduced in the previous section for

realvalued processesm ake sense forcom plex processes,and the quantity A(X )+ iS(X )

is wellde�ned for the com plex process X 2 C1C(I). As a consequence,we can naturally

extend D (X )to com plex processesby sim ply posing

D (X )=D + D �

2+ �i

D � D �

2;


with the naturalextension ofD and D �.

2.2.2.3. Sym m etry.| A possible way to extend D is to assum e that the regular part

ofDC(iY ) is equalthe im aginary part ofD (Y ),i.e. that the geom etric m eaning ofthe

com plex and realpartofD Y isexchanged.W e then im posethe following relation:

R(Y )= �A(Y ):

Thisleadsto the following extension:

D e�nition 2.4.| The operator DC:C1

C! C

1

Cisde�ned by

DC;�(X + iY )= D �X � i�D �Y; � = � 1;

where X ;Y 2 C1.

2.2.3. Stochastic derivative forfunctionsofdi�usion process.| In thefollowing,

we need to com pute the stochastic derivative off(t;X t)where X t is a di�usion process

and f isa sm ooth function.O urm ain resultisthe following lem m a:

Lem m a 2.4.| Let X 2 �d and f 2 C 1;2(I � Rd) such that @tf, r f and @ijf are

bounded. Then,we have:

D f(t;X (t)) =

�

@tf + D X (t)� r f +1

2aij@ijf

�

(t;X (t));(2.9)

D �f(t;X (t)) =

�

@tf + D �X (t)� r f �1

2aij@ijf

�

(t;X (t)):(2.10)

Proof.| Let X 2 �d and f 2 C 1;2(I � Rd) such that @tf,r f and @ijf are bounded.

Thusf belongsto thedom ain ofthegeneratorsLtand Ltofthedi�usionsX (t)and X (t).

M oreovertheseregularity assum ptionsallow usto usethesam eargum entsasin theproof

oftheorem 1.1 in orderto write:

D f(t;X (t)) = @tf(t;X (t))+ Lt(f(t;� ))(X (t))

=

�

@tf + bi@if +

1

2aij@ijf

�

(t;X (t))

=

�

@tf + D X (t)� r f +1

2aij@ijf

�

(t;X (t))

and

D �f(t;X (t)) = @tf(t;X (t))� L1�t(f(t;� ))(X (t))

=

�

@tf + D �X (t)� r f �1

2aij@ijf

�

(t;X (t))

W e deduceim m ediately thefollowing corollary :


C orollary 2.1.| LetX 2 �d and f 2 C 1;2(I � Rd) such that @tf,r f and @ijf are

bounded. Then,we have:

D �f(t;X (t)) =

�

@tf + D �X (t)� r f +i�

2aij@ijf

�

(t;X (t)):(2.11)

and

C orollary 2.2.| LetX 2 �d with aconstantdi�usion coe�cient� andf 2 C 1;2(I� Rd)

such that@tf,r f and @ijf are bounded. Then,we have:

D �f(t;X (t)) =

�

@tf + D �X (t)� r f +i��2

2�f

�

(t;X (t)):(2.12)

2.2.4. Exam ples.| W e com pute the stochastic derivative in som e fam ousexam ples,

like theO rnstein-Uhlenbeck processand a Brownian m ation in an externalforce.

2.2.4.1. The Ornstein-Uhlenbeck process.| A good m odelofthe Brownian m otion ofa

particle with friction isprovided by the O rnstein-Uhlenbeck equation:

(2.13)

�X 00(t)= � �X 0(t)+ ��(t)

X (0)= X 0; X0(0)= V0;

where X(t) is the position ofthe particle at tim e,� is the friction coe�cient,� is the

di�usion coe�cient,X 0 and V0 aregiven G aussian variables,� is"whitenoise".Theterm

� �X 0(t)representsa frictionaldam ping term .

The stochastic di�erentialequation satis�ed by the velocity process V (t) := Y 0(t) is

given by:

(2.14)

�dV (t)= � �V (t)dt+ �dW (t)

V (0)= V0;

W e can explicitly com puteD V and D 2V :

Lem m a 2.5.| LetV (� )be a solution of

(2.15)

�dV (t)= � �V (t)dt+ �dW (t)

V (0)= V0;

where V0 hasa norm aldistribution with m ean zero and variance �2

2�.

Then V 2 C2(]0;+ 1 ))and:

D V (t) = � i�V (t)(2.16)

D 2V (t) = � �2V (t):(2.17)

Proof.| Thesolution isa G aussian processexplicitly given by:

(2.18) 8t> 0; V (t)= V0e��t + �

Zt

0

e��(t�s)

dW (s):


Therefore,we can com pute the expectation and the variance ofthe norm alvariable

V (t):

(2.19)

(E [V (t)]= E [V0]e

��t

Var(V (t))= �2

2�+

�

Var(V0))��2

2�

�

e�2�t;

W e notice,as in [30],that ifV0 has a norm aldistribution with m ean zero and variance

�2

2�,then X isa stationary gaussian processwhich distribution pt(x)ateach tim e treads

(2.20) pt(x)=

p�

p��

e�

�x2

�2 :

Asa consequence,we have

(2.21) 8t> 0;ln(pt(x))= ln(

p�

p��

)��x2

�2;

and

(2.22) �2@x ln(pt(x))= �

2� 2�x

�2= � 2�x:

M oreover,we have

(2.23) D V (t)= � �V (t);

and according to theorem 1.1,we obtain

(2.24) D �V (t)= � �V (t)� �2@x ln(pt(V (t)))= �V (t):

Therefore D V (t) = � i�V (t), and using the C� linearity of D , we obtain D 2V (t) =

� �2V (t),which concludestheproof.

2.2.4.2. Brownian particlesubm itted to an externalforce.| In som eexam plesofrandom

m echanics,onehasto considerthestochastic di�erentialsystem :

(2.25)

8<

:

dX (t)= V (t)dt

dV (t)= � �V (t)dt+ K (X (t))dt+ �dW (t)

X (0)= X 0; V (0)= V0;

X and V m ay representtheposition and thevelocity ofa particleofm assm being under

the in uence ofan externalforce F = � r U where U isa potential. SetK = F=m . The

"free" case K = 0 istheabove exam ple.

W hen K (x)= � !2x (a linearrestoring force),the system can also be seen asthe ran-

dom harm onicoscillator.In thiscase,itcan beshown thatif(X 0;V0)hasan appropriate

gaussian distribution then (X (t);V (t))isa stationary gaussian processin the sam e way

asbefore.

Letuscom e back to thegeneralcase.


First,werem ark thatX isNelson-di�erentiableand wehaveD X (t)= D �X (t)= V (t).

M oreover,Nelson claim sin ([53],p.83-84)that,when theparticle isin equilibrium with a

specialstationary density,

D V (t) = � �V (t)+ K (X (t));(2.26)

D �V (t) = �V (t)+ K (X (t)):(2.27)

W e can sum m arize theseresultswith thecom putation ofD :

D X (t) = V (t);(2.28)

D 2X (t) = K (X (t))� i�V (t):(2.29)

C H A PT ER 3

PR O PERT IES O F T H E ST O C H A ST IC D ER IVAT IV ES

3.1. P roduct rules

In chapter7,we develop a stochastic calculusofvariations. In m any problem s,we will

need the analogue ofthe classicalform ula ofintegration by parts,based on the following

identity,called the productor Leibnizrule

d

dt(fg)=

df

dtg+ f

dg

dt; (P )

wheref;g are two given functions.

Using a previous work of Nelson [53], we generalize this form ula for our stochastic

derivative. W e begin by recalling the fundam entalresult ofNelson on a product rule

form ula forbackward and forward derivatives:

T heorem 3.1.| LetX ;Y 2 C1(I),then we have:

d

dtE [X (t)� Y (t)]= E [D X (t)� Y (t)+ X (t)� D�Y (t)](3.1)

W e referto ([53],p.80-81)fora proof.

R em ark 3.1.| It m ust be pointed out that this form ula m ixes the backward and for-

ward derivatives. As a consequence,even withoutour de�nition ofthe stochastic deriva-

tive,which takes into accountthese two quantities,the previous productrule suggests the

construction ofan operator which m ixesthese two term sin a "sym m etrical" way.

W e now take up the variousconsequencesofthisform ula regarding ouroperatorD .A

straightforward calculation gives:

Lem m a 3.1.| LetX ;Y 2 C1(I),we then have:

d

dtE [X (t)� Y (t)] = E [Re(D X (t))� Y (t)+ X (t)� Re(D Y (t))](3.2)

E [Im (D X (t))� Y (t)] = E [X (t)� Im (D Y (t))](3.3)

42 CH A PTER 3.PRO PERTIES O F TH E STO CH A STIC D ER IVATIV ES

Lem m a 3.2.| LetX ;Y 2 C1C(I). W e write X = X 1 + iX 2 and Y = Y1 + iY2 where

X i;Yi2 C1(I). Therefore :

(3.4) E [D �X � Y + X � D�Y ]=d

dtg(X (t);Y (t))+ r(X (t);Y (t));

where

(3.5) g(X ;Y )= E [X � Y ];

and

(3.6)r(X ;Y ) = � 2E [Y1 � Im (D�X 2)]� 2E [Y2 � Im (D�X 1)]

+ i(2E [Y1 � Im (D�X 1)]� 2E [Y2 � Im (D�X 2)]):

Proof.| W e have

(3.7)

Y D �X = Y1Re(D �X 1)� Y1Im (D �X 2)

� Y2Im (D �X 1)� Y2Re(D �X 2)

+ i(Y1Im (D �X 1)+ Y1Re(D �X 2)+ Y2Re(D �X 1)� Y2Im (D �X 2)):

In a sym m etricalway,we obtain

(3.8)

X D �Y = X 1Re(D �Y1)� X 1Im (D �Y2)

� X 2Im (D �Y1)� X 2Re(D �Y2)

+ i(X 1Im (D �Y1)+ X 1Re(D �Y2)+ X 2Re(D �Y1)� X 2Im (D �Y2)):

Form ing the sum oftheseexpressionsand using lem m a 3.1,we obtain (3.4).

Thenextlem m a willbeofim portance in chapter7 forthe derivation ofthestochastic

analogue ofthe Euler-Lagrange equations:

Lem m a 3.3.| LetX ;Y 2 C1C(I). W e write X = X 1 + iX 2 and Y = Y1 + iY2 where

X i;Yi2 C1(I). Therefore,we have:

(3.9) E [D �X � Y + X � D�� Y ]=d

dtg(X (t);Y (t))

where g(X ;Y )= E [X 1 � Y1 � X 2 � Y2]+ iE [Y1 � X2 + Y2 � X1]= E [X � Y ]

Proof.| W e have

(3.10)

Y D �X = Y1<(D �X 1)� Y1=(D �X 2)

� Y2=(D �X 1)� Y2<(D �X 2)

+ i(Y1=(D �X 1)+ Y1<(D �X 2)+ Y2<(D �X 1)� Y2=(D �X 2));

and in a sym m etricalway

(3.11)

X D �� Y = (X 1 + iX 2)(D �Y1 + iD �Y2)

= X 1<(D �Y1)+ X 1=(D �Y2)

+ X 2=(D �Y1)� X 2<(D �Y2)

+ i(� X 1=(D �Y1)+ X 1<(D �Y2)+ X 2<(D �Y1)+ X 2=(D �Y2)):

W e form the sum oftheseexpressionsand we usethe lem m a 3.1 to obtain (3.4).

3.2.N ELSO N D IFFER EN TIA BLE PRO CESSES 43

3.1.1. A new algebraic structure.| A convenientway to writeequation (3.9)isto

usethe following Herm itian product:

ForallX ;Y 2 PC,we denote by ? theproduct

(3.12) X ?Y = X � Y ;

where:denotesthe usualscalarproduct.

Form ula (3.9)isthen equivalentto:

(3.13) D E [X ?Y ]= E [D X ?Y + X ?D Y ];

where we have im plicitly used the factthatD reducesto d=dtwhen thisquantity hasa

sense.

Thisnew form leadsusto the introduction ofthe following algebraic structure,which

is,asfaraswe know,new.Let� bethe canonicalm apping

(3.14) � :PC P

C! P

C

X Y 7! X ?Y :

W e de�ne for D the quantity �(D ) = D 1+ 1 D ,which we willcallthe coproduct

ofD . Then,denoting by E the classicalm apping which takesthe expectation ofa given

stochastic process,we obtain the following diagram :

(3.15)

PC P

C

�(D )��! P

C P

C

X Y ��! D X Y + X D Y??y�

??y�

X ?Y ��! D X ?Y + X ?D Y??yE

??yE

E [X ?Y ]D

��! E [D X ?Y + X ?D Y ]

Thisstructureissim ilarto theclassicalalgebraicstructureofHopfalgebra.Thedi�erence

isthatwe perturb theclassicalrelationsby a linearm apping,here given by E .Itwillbe

interesting to study thiskind ofstructurein fullgenerality.

3.2. N elson di�erentiable processes

3.2.1. D e�nition.| W ede�nea specialclassofprocesses,called Nelson-di�erentiable

processes,which willplay an im portant role in the stochastic calculus ofvariations of

chapter7.

44 CH A PTER 3.PRO PERTIES O F TH E STO CH A STIC D ER IVATIV ES

D e�nition 3.1.| A process X 2 C1(I)iscalled Nelson di�erentiable ifD X = D �X .

N otation 3.1.| W e denote by N 1(I)the setofNelson di�erentiable processes.

A better de�nition is perhaps to use D instead of D and D � saying that Nelson

di�erentiable processeshave a realstochastic derivative.

Them ain idea behind thisde�nition isthatwewantto de�nea classP ofprocessesin

C1(I)such thatifX 2 C

1(I)then forallY 2 P,we have

Im (D (X + Y ))= Im (D X ):

Thiscondition im posesthatIm (D Y )= 0.

Thiscondition willappearm oreclearly in chapter7 concerning thestochastic calculus

ofvariations.

R em ark 3.2.| W e m ustkeep in m ind thatour de�nition ofthe stochastic derivative

followstheidea ofthescalecalculusdeveloped in [13]to study non-di�erentiablefunctions.

In thatcontext,the existence ofan im aginary partfor the scale derivative ofa function

is seen as a resurgence of its non-di�erentiability. In particular, when the underlying

function is di�erentiable then the scale derivative is real. Thatis why we have chosen to

callprocesses such thatD = D � Nelson di�erentiable.

Thede�nition ofNelson di�erentiable processesisonly given forprocessesin C 1(I).It

isnotatallclear to know whatisthe correctextension to C1C(I). Aswe have no use of

such kind ofnotion on C1C(I)we don’tdiscussthispointhere.

O fcourse a di�cultproblem is to characterize these processes. The nextsection dis-

cussessom e exam ples.

3.2.2. Exam ples ofN elson-di�erentiable process. | W egiveexam plesofNelson-

di�erentiable processes.

3.2.2.1. Di�erentiable determ inistic process.| Itisprobably the �rstand the sim plest

exam ple.Letx(� )bea di�erentiable determ inistic processde�ned on I� .The pastP

and thefutureF are trivial:

8t2 I; Pt= Ft= f;;g:

Asa consequence,we have

8t2 I; D x(t)= D �x(t)= x0(t);

wherex0istheusualderivative ofx.

3.2.N ELSO N D IFFER EN TIA BLE PRO CESSES 45

3.2.2.2. A very specialrandom exam ple.| Let X 2 C1(I). In [53],Nelson shows that

X is a constant (i.e. X(t) is the sam e random variable for allt) ifand only if: 8t 2

I; D X (t)= D �X (t)= 0.So itprovidesusa random exam ple ofN 1(I)� process.

3.2.2.3. Nelson-di�erentiable di�usion processes.| Using theorem 1.1, we can �nd a

su�cient and necessary condition for a di�usion process to be a Nelson-di�erentiable

process:

Lem m a 3.4.| LetX 2 �d with � = const,then X 2 N 1(I)ifand only if

(3.16) r (�2p)(t;X (t))= 0:

W hen the di�usion equation istim e hom ogeneousand thesolutionshave a density,we

notethatthisdensity m ustbeastationary density.M oreover,theFokker-Planck equation

(K olm ogorov forward equation)allowsustogiveanecessary condition (arelation between

thedriftand thedi�usion coe�cient)foradi�usion equation togiveaNelson-di�erentiable

solution.

3.2.2.4. Therandom harm onicoscillator.| Therandom harm onicoscillatorsatis�esthe

stochastic di�erentialequation:

(3.17)

8<

:

dX (t)= V (t)dt

dV (t)= � �V (t)dt� !2X (t)dt+ �dW (t)

X (0)= X 0; V (0)= V0;

Asa consequence,wehaveX (t)=

Z t

0

V (s)dswith E

�Z b

0

jV (s)j2ds

�

< 1 (b> 0),and

X hasa strong derivative in L2. W e then obtain D X (t)= D �X (t)= V (t). Finally,we

have X 2 N 1([0;b])and D X (t)= V (t).

3.2.3. P roduct rule and N elson-di�erentiable processes.|

C orollary 3.1.| LetX ;Y 2 C1C(I).IfX isNelson-di�erentiable then :

(3.18) E [D �X (t)� Y (t)+ X (t)� D�Y (t)]=d

dtE (X (t);Y (t))

Proof.| This is a sim ple consequence of the fact that if X = X 1 + iX 2 is Nelson-

di�erentiable then Im (D �X 1)= Im (D �X 2)= 0.

PA RT II

ST O C H A ST IC EM B ED D IN G

PR O C ED U R ES

C H A PT ER 4

ST O C H A ST IC EM B ED D IN G O F D IFFER EN T IA L

O PER AT O R S

A naturalquestion concerning ordinary and partialdi�erentialequationsconcernstheir

behaviourundersm allrandom perturbations. Thisproblem isparticularly im portantin

naturalphenom ena where we know that m odels are only an approxim ation ofthe real

setting.Forexam ple,the study ofthe long term behaviourofthe solarsystem isusually

done by running num ericalcom putationson the n-body problem .However,m any e�ects

in thesolarsystem sarenotincluded in thism odeland can beofim portanceifone looks

foralongterm integration,asnon conservativee�ects(duetotidalforcesbetween planets)

and theoblatnessofthesun which isnotyetm odelled by a di�erentialequation.

The m ain problem isthen to �nd the correctanalogue ofa given di�erentialequation

taking into accountthefollowing facts:

i)Theclassicalequation isa good m odelatleastin �rstapproxim ation,

ii)O nem ustextend thisequation to stochastic processes.

Using the stochastic derivative introduced in the previouspart,we give a naturalem -

bedding ofpartialor ordinary di�erentialequations into stochastic partialor ordinary

di�erentialequations. Itm ustbe pointed outthatwe do notperturb the classicalequa-

tion by a random noise oranything else.In thisrespectwe are farfrom the usualway of

thinking underlying the �eldsofstochastic di�erentialequationsorstochastic dynam ical

system s.

O fcourse,having this naturalem bedding,we can naturally de�ne what a stochastic

perturbation ofa di�erentialequation is. Thisissim ply a stochastic perturbation ofthe

stochastic em bedding ofthe given equation. The m ain pointisthatwe stay in the sam e

classofobjectsdealing with perturbations,which isnotthe case in the stochastic theory

ofdi�erentialequations,wherewejum p from classicalsolutionsto stochasticprocessesin

50 CH A PTER 4.STO CH A STIC EM BED D IN G O F D IFFER EN TIA L O PER ATO R S

one step using forexam ple Ito’sstochastic calculus(1).

In thispartwe�rstgiveageneralem beddingprocedureforpartialdi�erentialequations.

W e discussclassicalexam ples,in particular�rstand second orderdi�erentialequations.

The case ofLagrangian system s is studied in details in chapter 7. An im portant part

ofclassicaldi�erentialequations com ing from m echanics are reversible. This property

is not conserved by the previous stochastic em bedding procedure. W e de�ne a special

em beddingcalled reversible,which preservesthisproperty,m eaning thatifX isa solution

ofthestochastic em bedded equation,then ~X ,the reversed process,isagain a solution.

4.1. Stochastic em bedding ofdi�erentialoperators

In this part,we �rstgive an abstract em bedding procedure based on an extension of

the classicalderivative de�ned in the previous part. W e then specialize our em bedding

procedureusing the stochastic derivative.

4.1.1. A bstract em bedding.| LetA bea ring,we denote by A [x]thering ofpoly-

nom ialswith coe�cientsin A .LetA = C 1(Rd � R).

D e�nition 4.1.| A di�erentialoperator isan elem entsofA [d=dt].

LetO 2 A [d=dt],the di�erentialoperatorO isofthe form

(4.1) O = a0(� ;t)+ a1(� ;t)d

dt+ � � � + an(� ;t)

dn

dtn; ai2 A ; = 0;:::;n;

fora given n 2 N,called the degree ofO .

Theaction ofO on a given function x :R ! Rd,t7! x(t)isdenoted O � x and de�ned

by

(4.2) O � x =

nX

i= 0

ai(x(t);t)dx

dt:

D e�nition 4.2 (A bstract stochastization).| LetO 2 A [d=dt]be a di�erentialop-

erator,ofthe form

(4.3) O = a0(� ;t)+ a1(� ;t)d

dt+ � � � + an(� ;t)

dn

dtn; ai2 A ; = 0;:::;n;

where n 2 N isgiven.

(1)This rem ark is also valid for allthe theories ofthis kind,using yourfavourite stochastic calculus,like

M alliavin calculusforexam ple.

4.1.STO CH A STIC EM BED D IN G O F D IFFER EN TIA L O PER ATO R S 51

The stochastic em bedding ofO with respectto the extension � :P ! P is an elem ent

O � ofP[�]de�ned by

(4.4) O � = a0(� ;t)+ a1(� ;t)� + � � � + an(� ;t)�n; ai2 P; i= 0;:::;n;

where �n = � � � � � � �.

The action ofO � on a given stochastic process X ,denoted by O � � X isde�ned by

(4.5) O � � X =

nX

i= 0

ai(X ;t)�iX ;

where the notation ai(X ;t)stands for the stochastic process de�ned for all! 2 by

(4.6) ai(X ;y)(!)= ai(X (!;t);t):

Them ain property ofthisem bedding isthe factthat

(4.7) O � jP n

det= O ;

so thattheclassicaldi�erentialequation associated to O ,and given by

O � x = 0; (E )

iscontained in the stochastic di�erentialequation

O � � X = 0: (SE ):

4.1.2. N elson Stochastic em bedding.| Using the stochastic derivative,we have a

particularstochastic em bedding procedure.

D e�nition 4.3 (Stochastization).| LetO 2 A [d=dt]bea di�erentialoperator,ofthe

form

(4.8) O = a0(� ;t)+ a1(� ;t)d

dt+ � � � + an(� ;t)

dn

dtn; ai2 A ; = 0;:::;n;


The stochastic em bedding ofO with respectto the stochastic extension D � isan elem ent

O stoc ofC1(I)[D �]de�ned by

(4.9) O stoc = a0(� ;t)+ a1(� ;t)D + � � � + an(� ;t)Dn; ai2 C

1(I); i= 0;:::;n:

W edenoteby S theoperatorassociating to an operatorO oftheform 4.8 theoperator

O stoc.Asa consequence,wewillfrequently usethenotation S(O )forO stoc.


In som e occasions,in particularforthe Euler-Lagrange equation,we willneed to con-

siderdi�erentialoperatorsin anon-standard form .Precisely,weneed toconsideroperators

like

(4.10) B a =d

dt� a(� ;t):

Thisnotation m eansthatB a actson a given function as

(4.11) B a � x =d

dt(a(x(t);t))):

Thebasic idea isto de�nethestochastic em bedding ofB a asfollow:

D e�nition 4.4.| The stochastic em bedding ofthe basic brick B a isgiven by

(4.12) Ba = D � a(� ;t):

However,classicalpropertiesofthedi�erentialcalculusallow ustowriteB a equivalently

as

(4.13) B a � x = a0(x)

dx

dt:

Thestochastic em bedding ofthisnew form ofB a isgiven by

(4.14) Ba:X = a0(X )D X :

Them ain problem isthatin general,wedo nothave

(4.15) Ba = Ba;

asin theclassicalcase.

Thisre ectsthe factthatS actson operatorsofa given form and noton operatorsas

an abstractelem entofa given algebra.In particular,thisisnota m apping.

Nevertheless,thereexistsa classoffunctionsa such thatequation (4.15)isvalid:

Lem m a 4.1.| Equation (4.15) issatis�ed on the set� d with constantdi�usion ifa is

an harm onic function.

Proof.| Thisfollowseasily from corollary 2.2.

In thesequelwestudy som ebasicpropertiesofthisem beddingprocedureon di�erential

equations.

4.2.FIR ST EX A M PLES 53

4.2. First exam ples

4.2.1. First order di�erentialequations.| Letusconsidera �rstorderdi�erential

equation

dx

dt= f(x;t); 1� (O D E )

wherex 2 R and f :R � R ! R isa given function.Thestochasticem beddingof(1-O DE)

leadsto

D X = F (X ;t); 1� (SO D E )

whereF isrealvalued.

The reality ofF im posesim portantconstraints on solutions of1-(SO DE).Indeed,we

m usthave

D X = D �X ;

so thatX belongsto theclassofNelson-di�erentiable processes.

In our generalphilosophy,ordinary di�erentialequations are only coarse approxim a-

tions to reality which m ustinclude stochastic behaviour in its foundation. A stochastic

perturbation ofa �rst order di�erentialequation is then highly non-trivial. Indeed,we

m ustconsiderSO DE’softhe form

D X = F (X ;t)+ �G (X ;t);

where G (X ;t)isnow com plex valued. Asa consequence,we allow solutionsto leave the

Nelson-di�erentiable class.

4.2.2. Second order di�erentialequations.| Letus considera second orderdif-

ferentialequation

d2x

dt2+ a(x)

dx

dt+ b(x)= 0; (2� (O D E )

where x 2 R,and a;b :R ! R are given functions. The stochastic em bedding of(2 �

(O D E ))leadsto

D 2X + a(X )D X + b(X )= 0:

In this case,contrary to what happensfor �rstorder di�erentialequations,we have no

reality condition which constrainsourstochastic process.

In order to study such kind of equations, one can try to reduce it to a �rst order

equation,using standard ideas. W e denote by Y = D X ,then the second orderequation


isequivalentto the following system of�rstorderstochastic di�erentialequations:

(4.16)

�D X = Y;

D Y = � a(X )Y � b(X ):

O nem ustbecarefulto takeY 2 C1C(I)asY isa prioria com plex stochasticprocess.This

rem ark is ofim portance since ifwe apply the stochastic em bedding procedure(2) to the

classicalsystem of�rstorderdi�erentialequations

(4.17)

8><

>:

dx

dt= y;

dy

dt= � a(x)y� b(x);

by saying that we apply separately the stochastic em bedding on each di�erentialequa-

tions,we obtain thestochastic equation (4.16)butwith Y 2 C1(I),which im posesstrong

constraintson thesolutionsofourequations.

Thisexam ple provesthatthe stochastic em bedding procedure isnotso easy to de�ne

ifonewantsto dealwith system sofdi�erentialequations.W ewillreturn on thisproblem

concerning thestochastic em bedding ofHam iltonian system s.

(2)Notethatwe havenotde� ned thestochastic em bedding procedureon system sofdi� erentialequations.

C H A PT ER 5

R EV ER SIB LE ST O C H A ST IC EM B ED D IN G

5.1. R eversible stochastic derivative

In ourconstruction ofthestochasticderivative,wehaveim posed som econstraintsasfor

exam plethegluing to theclassicalderivativeon di�erentiabledeterm inisticprocesses.W e

have m oreoverkeptsom epropertiesoftheclassicalderivative such aslinearity.However,

wehavenotconserved m oreim portantpropertiesoftheclassicalderivativewhich areused

in the study ofclassicaldi�erentialequations.Forexam ple,letusconsider

d2x

dt2= f(x); (E )

which is the basic equation ofNewton’s m echanics. An im portantproperty ofthiskind

ofequationsisitsreversibility:

Lett! x(t)bea solution of(E).W e denote by ~x(t)= x(� t).Then,wehave

d2~x

dt2=

d

dt(�

dx

dt(� t))=

d2x

dt2(� t)= f(x(� t))= f(~x(t));

proving thatthe reversed solution ~x(t)is again a solution ofthe sam e equation. In this

case,we say thatthedi�erentialequation isreversible.

Thereversibility argum entused thefollowing im portantproperty:

d

dt(x(� t))= �

dx

dt(� t): (R)

Thenaturalway to introducea notion ofreversibility isthen to look forthestochastic

di�erentialequation satis�ed by ~X (t)= X (� t)2 C1(I)the reversed processes.However,

in general,we do nothave access to D ~X or D �~X . As a consequence,a de�nition using

thischaracterization isnote�ective.In the following,wefollow a di�erentstrategy.

56 CH A PTER 5.R EV ER SIBLE STO CH A STIC EM BED D IN G

A convenient way to characterize the reversibility ofa given di�erentialequation,de-

scribed by a di�erentialoperator

(5.1) O =X

i

aidi

dti2 R[d=dt]

isto prove thatthisoperatorisinvariantunderthe substitution

(5.2) r:R[d=dt]�! R[d=dt]

which isR linearand de�ned by

(5.3) r(d=dt)= � d=dt:

W e then introducein oursetting,thefollowing analogoussubstitution:

D e�nition 5.1.| The reversibility operator R :C[D ;D �]! C[D ;D �]isa C m orphism

de�ned by

(5.4) R(D )= � D �; R(D �)= � D :

W e have thefollowing im m ediate consequence ofthe de�nition:

Lem m a 5.1.| The reversibility operator isan involution ofC[D ;D �].

Thisoperatoractsnon trivially on ourstochastic derivative.Precisely,we have:

Lem m a 5.2.|

(5.5) R(D )= � D :

The com plex nature ofthe stochastic derivative induces new phenom enon which are

di�erentfrom theclassicalcase.Forexam ple,we have

(5.6) R(D 2)= D2;

contrary to whathappensforr.

W e now de�neournotion ofa reversible stochastic equation.

D e�nition 5.2.| [Reversibility]LetO 2 R[D ;D �],then thestochastic equation O � X =

0 isreversible ifand only ifR(O )� X = 0.

A naturalproblem isthefollowing:

R eversibility problem : Find an operator such that the stochastic em bedding of a

reversible equation isagain a reversible equation in the sense ofde�nition 5.2.

5.1.R EV ER SIBLE STO CH A STIC D ER IVATIV E 57

Letusconsiderthe fam ily ofstochastic derivatives D �,� = 0;� 1. W ithoutassum ing

a particularform fortheunderlying equation,thepreservation ofthereversiblecharacter

reducesto prove thatthe operator� which ischosen satis�es

(5.7) R(�)= � �:

In thefam ily ofstochastic derivativesD �,� = 0;� 1,only one case ispossible:

Lem m a 5.3.| A reversibility of a di�erential equation is always preserved under a

stochastic em bedding ifand only ifthisem bedding isassociated to the stochastic derivative

D 0.

Proof.| Essentially thisfollowsfrom equation (5.5).Ifwewantto preservereversibility

then theoperatorD � m ustsatis�ed R(D �)= � D �.Thisisonly possibleifD � isreal,i.e.

� = 0.

Itm ustbepointed outthatthe operator

D 0 =D + D �

2;

hasbeen obtained by di�erentauthorsusing thefollowing argum ent:

Ifwe use only D (orD �)then,we give a specialim portance to the future (orpast)of

the process,which has no physicaljusti�cation. As a consequence,one m ust construct

an operator which com bines these two quantities in a m ore or less sym m etric way. The

sim plestcom bination isa linearone aD + bD � with equalcoe�cients a = b. The gluing

to the classicalderivative leadsto a = b= 1=2.

Theproblem with thisconstruction isthatthisargum entisused on di�usion processes,

where D and D � are not free. As a consequence,working with D is the sam e (even if

the connection with D � is not trivial) than working with D �. W e can not really justify

then the use ofD 0. It m ust be pointed out that E.Nelson [53]does not use D 0 in his

derivation oftheSchr�odingerequation,butsim ply D .

Here,thisoperatorisobtained by specialization ofD �,which form isim posed by our

construction (linearity,gluing to the classicalderivative,reconstruction property). The

reconstruction property im posesthat� 6= 0 unlesswe work with di�usion processes.

Im posing a new constrainton the reversibility on thisoperatorleadsusto � = 0.The

operatorD 0 isofcourse de�ned on C1(I),butin orderto satisfy thewhole constraintsof


ourconstruction,we m ustrestrictitsdom ain to di�usion processes.

W e can ofcourse �nd reversible equations without using D 0 but D �. W e keep the

notationsand conventionsofchapter4.W e�rstde�netheaction ofR on a given operator

oftheform

(5.8) O =

nX

i= 0

ai(� ;t)(� 1)iD

i:

D e�nition 5.3.| The action ofR on (5.8)isdenoted R(O )and de�ned by

(5.9) R(O )=

nX

i= 0

ai(� ;t)Di:

The de�nition 5.2 ofa reversible equation can then be extended to cover operatorsof

the form 5.8.

Using thisde�nition,we can prove thatthe stochastic equation

D 2�X = � r U (X ); (E )

isreversible.

Indeed,we have:

Lem m a 5.4.| Equation (E)isreversible.

Proof.| W e have

(5.10)R(D 2

�X + r U (X )) = D2X + r U (X );

= D 2�X + r U (X ):

AsU isrealvalued and X are realstochastic processes,we deducefrom (E)that

(5.11) D 2�X = � r U (X )= � r U (X ):

W e deducethat

(5.12) R(D 2�X + r U (X ))= 0;

which concludestheproof.

5.2. Iterates

There exists a fundam entaldi�erence between D 0 and D �,� 6= 0. The operator D 0

send realstochastic processes to realstochastic processes in the contrary ofD �,� 6= 0,

5.4.R EV ER SIBLE V ER SU S G EN ER A L STO CH A STIC EM BED D IN G 59

which leadsto com plex stochastic processes.Asa consequence,then-i�em e iteratesofD 0

issim ply de�ned by

(5.13) D n0 = D 0 � � � � � D0;

withoutproblem ,where a specialextension ofD �,� 6= 0 to com plex stochastic processes

m ustbediscussed.

5.3. R eversible stochastic em bedding

Using D 0,wecan de�nea stochasticem bedding which conservesthefundam entalprop-

erty ofreversibility ofa given equation.W e keep notationsfrom chapter4.

D e�nition 5.4 (R eversible stochastization).| LetO 2 A [d=dt]beadi�erentialop-

erator,ofthe form

O = a0(� ;t)+ a1(� ;t)d

dt+ � � � + an(� ;t)

dn

dtn; ai2 A ; = 0;:::;n;


The reversible stochastic em bedding ofO isan elem entO rev ofC1(I)[D 0]de�ned by

(5.14) O rev = a0(� ;t)+ a1(� ;t)D0 + � � � + an(� ;t)Dn0; ai2 C

1(I); i= 0;:::;n:

A di�erentialequation (E) is de�ned by a di�erentialoperator O 2 A [d=dt],i.e. an

equation oftheform

O � x = 0; (E )

wherex isa function.

Using stochastization,thereversible stochastic analogue of(E)isde�ned by

O rev � X = 0; (RSE )

whereX isa stochastic process.

5.4. R eversible versus generalstochastic em bedding

The reversible stochastic em bedding leads to very di�erent results than the general

stochastic em bedding. W e can already see thisdi�erence on �rstorderdi�erentialequa-

tions.Letusconsiderdx

dt= f(x);

wherex 2 R and f isa realvalued function.Thereversible stochastic em bedding gives

D 0X = f(X ):

Contrary to what happensfor the stochastic em bedding,this equation does not im pose

forthe solution to bea Nelson di�erentiable processes.


5.5. Stochastic m echanics and the Stochastization procedure

5.5.1. T he Stochastic N ew ton Equation.| Thestochastized version oftheclassical

system :

_x(t) = v(t)

_v(t) = K (x(t))(5.15)

isgiven by:

D X (t) = V (t)

D V (t) = K (X (t))(5.16)

whereV 2 C1C(I)and K isa force:K (x)= � r U (x)and U a potential.

W ecan giveatleasttwo di�erentkind ofsolutionsofthisequation,and so two relevant

m odels.

In the �rstone,the com ponentX isthe position in the O rnstein-Uhlenbeck theory of

Brownian M otion and isnotsubm itted to a random noise.Thesystem writes:

(5.17)

8<

:

dX (t)= V (t)dt

dV (t)= � �V (t)dt+ K (X (t))dt+ �dW (t)

X (0)= X 0; V (0)= V0;

W e have noticed in a previoussection that,atan equilibrium (i.e. X hasa stationary

density)and ife�U isintegrable,then:

D X (t) = V (t);(5.18)

D 2X (t) = K (X (t))� i�V (t):(5.19)

Therefore (X ;V ) solves the Newton stochastized system (5.16) if and only if � = 0.

M oreoverwe note in thisparticuliarcase thatX isa Nelson-di�erentiable process.

Thesecond one isdescribed by

(5.20) dX (t)= b(t;X (t))dt+ �dW (t);

where the function b m ustbe determ ined. In thiscase,we proved thatthe density pt(x)

ofa solution X of(5.16) writes pt(x) = (t;x)(t;x) where solves the Schr�odinger

equation:i�2@t+�4

2@xx= U .In thiscase,X isdriven by a Brownian m otion and is

notNelson-di�erentiable.

PA RT III

ST O C H A ST IC EM B ED D IN G O F

LA G R A N G IA N A N D H A M ILT O N IA N

SY ST EM S

C H A PT ER 6

ST O C H A ST IC LA G R A N G IA N SY ST EM S

M ost ofclassicalm echanics can be form ulated using Lagrangian form alism ([5],[2]).

Lagrangian m echanicscontainsim portantproblem s,like the n-body problem .Using our

fram ework,we study Lagrangian dynam icalsystem sunderstochastic perturbations(1).

O urapproach is�rsttoem bed classicalLagrangian system s,in particulartheassociated

Euler-Lagrangeequation (EL)in orderto obtain an idea ofwhatkind ofequation govern

stochastic Lagrangian system s. W e then develop a stochastic calculus ofvariations. W e

obtain an analogue ofthe least-action principle(2) which givesa second stochastic Euler-

Lagrangeequation,denoted by(SEL)in thesequel.W ethen provethefollowingsurprising

result,called the coherence lem m a:we have S(E L)= (SE L).

TheprincipalinterestofLagrangian system sisthattheaction ofa group ofsym m etries

leads to �rst integrals ofm otion,i.e. functions which are constants on solutions ofthe

equationsofm otion.Thecelebrated theorem ofE.Noethergivesapreciserelation between

sym m etriesand �rstintegrals.W e prove a stochastic analogue ofE.Nothertheorem .

Finally,we prove thatthe stochastic em bedding ofNewton’sLagrangian system slead

to a non linearSchr�odinger’sequation fora given wave function whose m odulusisequal

to the probability density oftheunderlying stochastic process.

6.1. R em inder about Lagrangian system s

W e referto [5]form oredetails,aswellas[2].

(1)For the n-body problem ,which is usually used to study the long term behavior ofthe solar system

[47],thisproblem is ofcrucialim portance. Indeed,the n-body problem isonly an approxim ation ofthe

realproblem ,and even ifsom e num ericalsim ulationstake into accountrelativistic e� ects[40],thisisnot

su� cient[50].(2)In ourcase,the word least-action ism isleading and a betterterm inology isstationary (see below).

64 CH A PTER 6.STO CH A STIC LAG R A N G IA N SY STEM S

Lagrangian system splay a centralrolein dynam icalsystem sand physics,in particular

form echanicalsystem s. A Lagrangian system isde�ned by a Lagrangian function,com -

m only denoted by L,and depending on three variables:x,v,and twhich belongsin the

sequelto R.AsLagrangian system scom efrom m echanics,theletterx standsforposition,

theletterv forspeed and thelettertfortim e.In whatfollows,weconsidera specialtype

ofLagrangian function called adm issible in thefollowing.

D e�nition 6.1.| An adm issible Lagrangian function isa function L such that:

i)The function L(x;v;t)isde�ned on Rd � C

d � R,holom orphic in the second variable

and realfor v 2 R.

ii)L isautonom ous,i.e.L does notdepend on tim e.

Condition i)isfundam ental.Thiscondition isnecessary in orderto apply thestochas-

tization procedure (see below). The fact that we only consider autonom ous Lagrangian

function isdueto technicaldi�cultiesin orderto takeinto accountbackward and forward

�ltrationsin the com putation ofthe stochastic Euler-Lagrange equation (see below).

R em ark 6.1.| In applications,adm issible Lagrangian functions L are analytic exten-

sions to the com plex dom ain of real analytic Lagrangian functions. For exam ple, the

classicalNewtonian Lagrangian L(x;v)= (1=2)v2 � U (x),de�ned on an open(3) subsetof

R � R,with an analytic potentialisan adm issible Lagrangian function.

A Lagrangian function L being given,the equation

d

dt

�@L

@v

�

=@L

@x: (E L)

iscalled the Euler-Lagrange equations.

An im portant property ofthe Euler-Lagrange equation is that it derives from a vari-

ational principle, nam ely the least action principle (see [5],p.59). Precisely, a curve

:t7! x(t)isan extrem al(4) ofthe functional

Ja;b( )=

Z b

a

L(x(t);_x(t);t)dt;

on the space ofcurvespassing through the pointsx(a)= xa and x(b)= xb,ifand only if

itsatis�estheEuler-Lagrange equation along the curve x(t).

(3)This Lagrangian function is notalways de� ned on R � R. An exam ple is given by Newton’s potential

U (x)= 1=x,x 2 R�.

(4)W e referto [5],chapter3,x.12 foran introduction to the calculus ofvariations.

6.3.TH E CO H ER EN CE PRO BLEM 65

6.2. Stochastic Euler-Lagrange equations

W e now apply ourstochastic procedureS to an adm issibleLagrangian.

Lem m a 6.1.| LetL(x;v) :Rd � Cd ! C be an adm issible Lagrangian function. The

stochastic Euler-Lagrangeequation obtained from (EL)by thestochastic procedure isgiven

by

D �

�@L

@v(X (t);D �X (t)

�

=@L

@x(X (t);D �X (t)): S(E L)

Proof.| TheEuler-Lagrangeequation associated to L(x;v)can beseen asthefollowing

di�erentialoperator

O E L =d

dt�@L

@v�@L

@x;

acting on (x(t);_x(t)).Theem bedding ofO E L gives

O E L = D � �@L

@v�@L

@x:

AsO E L actson (x(t);_x(t)),theoperatorO E L actson (X (t);D �X (t)).Thisconcludesthe

proof.

Thefreeparam eter� 2 f� 1;0;1g can be�xed dependingon thenatureoftheextension

used.

Itm ustbepointed outthatthere existcrucialdi�erencesbetween allthese extensions

due to the factthatD � iscom plex valued for� = � 1 and realfor� = 0. Indeed,letus

considerthe following adm issibleLagrangian function:

L(x;v)=1

2v2 � U (x);

whereU isa sm ooth realvalued function.Then,equation S(EL)gives

D �V = U (X );

where V = D �X . W hen � = � 1,thisequation im posesstrong constraints on X due to

the realnatureofU (X ),nam ely thatD 2�X 2 N

1(I).

O n the contrary,when � = 0,i.e. in the reversible case,these intrinsic conditions

disappear.

6.3. T he coherence problem

Up tonow,thestochasticem beddingprocedurecan beviewed asaform alm anipulation

ofdi�erentialequations. M oreover,asm ostclassicalm anipulations on equations do not

66 CH A PTER 6.STO CH A STIC LAG R A N G IA N SY STEM S

com m ute with the stochastic em bedding,this procedure is not canonical (5). In order

to rigidify this construction and to m ake precise the role ofthis stochastic em bedding

procedure,we study thefollowing problem ,called the coherence problem :

W e know that the Euler-Lagrange equations are obtained via a least-action principle

on a functional.Them ain problem isthe existence ofa stochastic analogue ofthisleast-

action principle,thatwe can calla stochastic leastaction principle,com patible with the

stochastic em bedding procedure.

L(x(t);_x(t))

Leastaction principle��

S// L(X (t);D X (t))

Stochastic leastaction principle?��

(EL)S

// (SEL)

(6.1)

In the next chapter,we develop the necessary tools to answer to this problem ,i.e. a

stochastic calculusofvariations.Notethatdueto thefactthatthestochasticLagrangian

aswellasthestochasticEuler-Lagrangeequation are�xed,thisproblem isfarfrom being

trivial. The m ain result ofthe next chapter is the Lagrangian coherence lem m a which

says precisely that the stochastic Euler-Lagrange equation obtained via the stochastic

em bedding procedure coincide with the characterization ofextrem als for the functional

associated to the stochastic Lagrangian function using the stochastic calculus ofvaria-

tions. As a consequence,we obtain a rigid picture involving the stochastic em bedding

procedureand a �rstprinciplevia the stochastic leastaction principle.

Thispicture willbethen extended in anotherchapterwhen dealing with the Ham ilto-

nian partofthistheory.

(5)W e return to thisproblem in ourdiscussion ofa stochastic sym plectic geom etry which can be used to

bypassthiskind ofproblem .

C H A PT ER 7

ST O C H A ST IC C A LC U LU S O F VA R IAT IO N S

The em bedding procedure allowsusto associate a stochastic Euler-Lagrange equation

to a stochasticLagrangian function.A basicquestion isthen theexistenceofan analogue

oftheleastaction principle.In thissection,wedevelop a stochasticcalculusofvariations

forourLagrangian function following a previouswork ofK .Yasue [71].O urm ain result,

called the coherence lem m a,states that the stochastic Euler-Lagrange equation can be

obtained asan application ofa stochasticleastaction principle.M oreover,thisderivation

isconsistentwith the stochastic em bedding procedure.

7.1. Functionaland L-adapted process

In the sequelwedenote by I a given open interval(a;b),a < b.

W e �rstde�nethe stochastic analogue oftheclassicalfunctional.

D e�nition 7.1.| LetL be an adm issible Lagrangian function. The functionalassoci-

ated to L isde�ned by

Ja;b(X )= E

�Z b

a

L(X (t);D �X (t))dt

�

;(7.1)

for allX 2 C1(I).

In what follows, we need a specialnotion introduced by Yasue [71], and called L-

adaptation:

D e�nition 7.2.| Let X 2 C1(I) be a stochastic process. W e denote by P and F

the past and the future of X . Let L be an adm issible Lagrangian function. A process

X 2 C1(I)iscalled L-adapted if:

i)@L

@v(X (t);D �X (t))isadapted to P and F .

68 CH A PTER 7.STO CH A STIC CA LCU LU S O F VA R IATIO N S

ii)@L

@v(X (t);D �X (t))2 C

1(I).

Di�usion processesare L-adapted.

7.2. Space ofvariations

Calculusofvariationsisconcerned with the behaviouroffunctionalsundervariations

ofthe underlying functionalspace,i.e.objectsofthe form + h,where belongsto the

functionalspace and h is a given functionalspace ofvariations. A specialcare m ustbe

taken in ourcase to de�ne whatisthe classofvariations we are considering. In general,

thisproblem isnotreally pointed outasboth variationsand curvescan be taken in the

sam efunctionalspace (see[5],p.56,footnote 26).W e introducethefollowing term inology:

D e�nition 7.3.| LetP be a subspace ofC1(I)and X 2 C1(I). A P -variation ofX is

a stochastic process ofthe form X + Z,where Z 2 P .

In the sequel,we considertwo subspacesofvariations:N 1(I)and C1(I).

Thechoice ofC1(I)isnatural.However,doing thiswe can obtain stochastic processes

with com pletely di�erentbehaviourthan X (1).

W hatisthe speci�cproperty ofX 2 C1(I)thatwe wantto keep ?

Ifwereferto theconstruction ofthestochasticderivative,then a m ain pointistheexis-

tenceofan im aginary partin D �X(2).Thisproperty isrelated to thenon-di�erentiability

oftheunderlyingstochasticprocess.W earethen lead tosearch forvariationsZ which con-

servethisim aginary part.Asa consequence,wem ustconsiderNelson di�erenceprocesses

introduced in thepreviouspart(3),and denoted by N 1(I).

7.3. D i�erentiable functionaland stationary processes

W e now de�neournotion ofdi�erentiable functional.LetP bea subspaceofC 1(I).

(1)O fcourse,thisisnotthe case in the classicalcase: one consider x 2 C1 (I)and z 2 C

1 (I)such that

x + h 2 C1(I)isvery sim ilarto x.Forexam ple,we don’tchoose z 2 C

0(I)which leadsto radically new

behaviourofx + z with respectto x.(2)O fcourse,as long as � = � 1. This is ofim portance since we willbe able to choose a m ore general

variationsspace in thiscase.(3)An analogousproblem isconsidered in [14],where a non di� erentiable variationalprinciple isde� ned.

7.3.D IFFER EN TIA BLE FU N CTIO N A L A N D STATIO N A RY PRO CESSES 69

D e�nition 7.4.| LetL be an adm issible Lagrangian function and Ja;b the associated

functional.ThefunctionalJa;b iscalled P -di�erentiableatan L-adapted processX 2 C1(I)

if

(7.2) Ja;b(X + Z)� Ja;b(X )= dJa;b(X ;Z)+ R(X ;Z);

where dJa;b(X ;Z)isa linear functionalofZ 2 P and R(X ;Z)= o(k Z k).

Thestochastic analogue ofa stationary pointisthen de�ned by:

D e�nition 7.5.| A P -stationary process for the functionalJa;b isa stochastic process

X 2 C1(I)such thatdJ(X ;Z)= 0 for allZ 2 P .

7.3.1. T he P = C1(I) case.| O urm ain resultis:

Lem m a 7.1.| The functional Jab de�ned by (7.1) is C1(I)-di�erentiable at any L-

adapted process X 2 C1(I),and for allZ 2 C

1(I),the di�erentialisgiven by:

(7.3)

dJab(X ;Z) = E

�Zb

a

�@L

@x(X (u);D �X (u))� D ��

�@L

@v(X (u);D �X (u))

��

Z(u)du

�

+ g(Z;@vL)(b)� g(Z;@vL)(a);

where

(7.4) g(Z;@vL)(s)= E [Z(u)@vL(X (u);D �X (u))]:

Proof.| LetX and Z betwo L-adapted processes.TheTaylorexpansion ofL gives:

(7.5)

L(X + Z;D �(X + Z))� L(X ;D �(X )) = @xL(X ;D �(X ))Z

+ @vL(X ;D �(X ))D �(Z)

+ o(kZk);

which yields(7.6)by integration and (3.9).

7.3.2. T he P = N1(I) case.| O urm ain resultis:

Lem m a 7.2.| The functionalJab de�ned by (7.1) is N1(I)-di�erentiable at any L-

adapted process X 2 C1(I),and for allZ 2 N

1(I)the di�erentialisgiven by:

(7.6)dJab(X ;Z) = E

�Zb

a

(@xL � D �@vL)(X (u);D �X (u))Z(u)du

�

+ g(Z;@vL)(b)� g(Z;@vL)(a);

where

(7.7) g(Z;@vL)(s)= E [Z(u)@vL(X (u);D �X (u))]:


Proof.| LetX a L� adapted processand H a Nelson-di�erentiable process.TheTaylor

expansion ofL gives

(7.8)

L(X + H ;D �(X + H ))� L(X ;D �(X )) = @xL(X ;D �(X ))H

+ @V L(X ;D �(X ))D �(H )

+ o(kH k);

which yields(7.6)by integration and (3.18).

7.4. A technicallem m a

The classicalderivation ofthe least action principle used a wellknown result about

bum p functions (see [5],p.57). In the stochastic fram ework,we willneed the following

result:

Lem m a 7.3.| LetY 2 PC be a com plex stochastic process. IfY satis�es

(7.9)

Z 1

0

E [Y (u)D �Z(u)]du = 0;

for allZ 2 N 1([0;1])then Y isa constantprocess.

Proof.| W e denote Y = Y1 + iY2,where Yi 2 PR and D �Z = A,where A 2 PR. The

equation (7.9)isequivalentto

(7.10)

R10E [Y1(u)A(u)]du = 0;

R10E [Y2(u)A(u)]du = 0;

forallA 2 PR such thatthere existsZ 2 C1([0;1])satisfying D �Z = A.

LetZY1 bethe processde�ned by

(7.11) ZY1(u)=

Z u

0

Y1(s)ds� u

Z 1

0

Y1(s)ds:

W e have ZY1 2 N 1(I)with Z(0)= Z(1)= 0.Indeed,we have

(7.12) D �Z(u)= Y1(u)�

Z 1

0

Y1(s)ds:

Asa consequence,wehavein ournotationsB = 0 and the�rstequation of(7.10)reduces

to

(7.13)

Z 1

0

E [Y1(u)A(u)]du = E

"Z 1

0

�

Y1(u)�

Z 1

0

Y1(s)ds

� 2

du

#

:

W e deduce thatY1 isa constantprocess,thatisforallu 2 I,Y1(u)= C a.s.,where C is

a random variable.

Thesam e argum entwith thesecond equation of(7.10)and ZY2 concludestheproofof

the lem m a.

7.5.LEA ST ACTIO N PR IN CIPLES 71

7.5. Least action principles

As for the com putation ofthe di�erentialoffunctionals,we m ust consider two cases:

P = C1(I)and P = N

1(I).

7.5.1. T he P = C1(I)case.| Them ain resultofthissection isthefollowing analogue

oftheleast-action principleforLagrangian m echanics.

T heorem 7.1 (G lobalLeast action principle).| A necessary and su�cientcondi-

tion for an L-adapted process to be a C1(I)-stationary process ofthe functionalJab with

�xed end points X (a):= X a 2 H etX (b):= X b 2 H isthatitsatis�es

@L

@x(X (t);D �X (t))� D ��

�@L

@v(X (t);D �X (t))

�

= 0:(7.14)

W e callthis equation the GlobalStochastic Euler-Lagrange equation (GSEL).

W e have conserved the term inology ofleast-action principle even ifwe have no notion

ofextrem alsforourcom plex valued functional.

Proof.| W e denote by I = ]0;1[.LetX 2 C1(I)bea solution of

(7.15) (@xL � D �@vL)(X (u);D �X (u))= 0;

then X isa N 1(I)-stationary processforthe functionalJI.

Conversely,letX isa C1(I)-stationary processforthefunctionalJI,i.e.dJI(X ;Z)= 0.

W riting

(@xL � D �@vL)(X (u);D �X (u))= D �Y (u);

where

(7.16) Y (u)=

Z u

0

@xL(X (s);D �X (s))ds� @vL(X (u);D �X (u));

we obtain forany Z 2 C1(I)with Z(0)= Z(1)= 0:

dJI(X ;Z) = E

�Z1

0

D �Y (u)Z(u)du

�

(7.17)

=

Z1

0

E [D �Y (u)Z(u)]du:

Using theC1(I)-productrule(see equation 3.9),we obtain

(7.18) dJI(X ;Z)= �

Z 1

0

E [Y (u)D �Z(u)]du:

Using lem m a 7.3 weobtain thatY isa constantprocess.


Hence,we have D �Y (u)= 0 and

(7.19) (@xL � D �@vL)(X (u);D �X (u))= 0;


7.5.2. T he P = N1(I) case.| O urm ain resultis:

T heorem 7.2 (least action principle).| A necessary and su�cientcondition foran

L-adapted process to be a N1(I)-stationary process of the functionalJab with �xed end

points X (a):= X a 2 H etX (b):= X b 2 H isthatitsatis�es

@L

@x(X (t);D �X (t))� D �

�@L

@v(X (t);D �X (t))

�

= 0:(7.20)

W e callthis equation the weak stochastic Euler-Lagrange equation (SEL).

Proof.| W e denote by I = ]0;1[.LetX 2 C1(I)bea solution of

(7.21) (@xL � D �@vL)(X (u);D �X (u))= 0;

then X isa N 1(I)-stationary processforthe functionalJI.

Conversely,letX isaN 1(I)-stationary processforthefunctionalJI,i.e.dJI(X ;Z)= 0.

W riting

(@xL � D �@vL)(X (u);D �X (u))= D �Y (u);

where

(7.22) Y (u)=

Zu

0

@xL(X (s);D �X (s))ds� @vL(X (u);D �X (u));

we obtain forany Z 2 N 1(I)with Z(0)= Z(1)= 0:

dJI(X ;Z) = E

�Z1

0

D �Y (u)Z(u)du

�

(7.23)

=

Z 1

0

E [D �Y (u)Z(u)]du:

Using theN 1(I)-productrule(see equation 3.18),we obtain

(7.24) dJI(X ;Z)= �

Z 1

0

E [Y (u)D �Z(u)]du:

Using lem m a 7.3,we deduce thatY isa constantprocess,thatisforallu 2 I,Y (u)= C

a.s.whereC isa random variable.

Hence,we obtain D �Y (u)= 0 and

(7.25) (@xL � D �@vL)(X (u);D �X (u))= 0;

7.6.TH E CO H ER EN CE LEM M A 73


7.6. T he coherence lem m a

Itisnotclearthatthe stochastic Euler-lagrange equation obtained by the stochastiza-

tion procedureand theN 1(I)orC1(I)least-action principlecoincide.O neeasily seesthat

thisisnotthe case forP = C1(I). In the contrary,we have the following lem m a,called

the coherence lem m a,which ensurethatforP = N1(I)weobtain the sam e equations.

Lem m a 7.4 (coherence lem m a).| The following diagram com m utes :

L(x(t);x0(t))


S// L(X (t);D X (t))

Stochastic Leastaction principle��

(E L)S

// (SE L)

(7.26)

Proof.| Thisisan im m ediate consequence ofthepreviousresults.

R em ark 7.1.| W hen � = 0,i.e. in the reversible case,the previous lem m as and the-

orem s are true under C1(I) variations. Note thatwhen � = 0,our stochastic derivatives

coincideswith the M isawa-Yasue [52]canonicalform alism for stochastic m echanics.

C H A PT ER 8

T H E ST O C H A ST IC N O ET H ER T H EO R EM

A naturalquestion arising from the stochastization procedure ofclassicaldynam ical

system s,in particular,Lagrangian system s,isto understand whatrem ainsfrom classical

�rstintegrals ofm otion. Firstintegrals play a centralrole in m any problem slike the n-

bodyproblem .In thissection,weobtain astochasticanalogueoftheNoethertheorem .W e

then de�ned thenotion of�rstintegralsforstochasticdynam icalsystem s.W ealso discuss

the consequences ofthe existence of�rst integrals in the context ofchaotic dynam ical

system s.

8.1. Tangent vector to a stochastic process

LetX 2 C1(I)be a stochastic process. W e de�ne the analogue ofa tangentvectorto

X atpointt.

D e�nition 8.1.| LetX 2 C1(I),I � R. The tangent vector to X at point t is the

random variable D X (t).

R em ark 8.1.| Ofcourse,in order to de�ne stochastic Lagrangian system s in an in-

trinsic way,one m ust de�ne the stochastic analogue ofthe tangentbundle to a sm ooth

m anifold. In our case, it is not clear what is the adequate geom etric object underlying

stochastic Lagrangian dynam ics. For exam ple,we can think ofm ultidim ensionalBrow-

nian surfaces ([23],x.16.4). Allthese questions willbe developed in a forthcom ing paper

[17].

8.2. C anonicaltangent m ap

In the sequel,we willneed the following m apping called the canonicaltangentm ap:

76 CH A PTER 8.TH E STO CH A STIC N O ETH ER TH EO R EM

D e�nition 8.2.| For allX 2 C1(I),we de�ne the canonicaltangentm ap as

(8.1) T :C1(I) �! C

1(I)� PC;

X 7�! (X ;D X ):

Them apping T willbeused in thefollowing section to de�netheanalogueofthelinear

tangentm ap fora stochastic suspension ofa oneparam etergroup ofdi�eom orphism s.

8.3. Stochastic suspension ofone param eter fam ily ofdi�eom orphism s

W e begin by introducing a usefulnotion ofstochastic suspension ofa di�eom orphism .

D e�nition 8.3.| Let� :Rn ! Rn be a di�eom orphism . The stochastic suspension of

� isthe m apping �:P ! P de�ned by

(8.2) 8X 2 P; �(X )t(!)= �(X t(!)):

In whatfollows,we willfrequently usethe sam e notation forthe suspension ofa given

di�eom orphism and the di�eom orphism .

R em ark 8.2.| Itseem s strange thatwe have notde�ned directly the notion ofdi�eo-

m orphism on a subsetE ofthe stochastic processes,i.e. m apping � :E ! E which are

Fr�echetdi�erentiable with an inverse which isalso Fr�echetdi�erentiable. However,these

objectsdo notalways exist.

Usingthestochasticsuspension,weareabletode�nethenotion ofstochasticsuspension

fora one-param etergroup ofdi�eom orphism s.

D e�nition 8.4.| A one-param etergroup oftransform ations�s :A ! A,s2 R,where

A � P,iscalled a �-suspension group acting on A ifthere exista one param eter group of

di�eom orphism s �s :Rn ! R

n,s2 R,such thatfor alls2 R,we have:

i)�s isthe stochastic suspension of�s,

ii)for allX 2 A,�s(X )2 A.

This notion of suspension group com es from our fram ework. It relies on the fact

that we want to understand how sym m etries ofthe underlying Lagrangian system s are

transported via the stochastic em bedding. The non-trivialcondition on the stochastic

suspension ofaone-param etergroup ofdi�eom orphism sacting on A com esfrom condition

ii). However,im posing som e conditions on the underlying one param eter group,we can

obtain astochasticoneparam etergroup which actson thesetE ofgood di�usionprocesses.

Precisely,letusintroducethefollowing classofone-param etergroups:

8.3.STO CH A STIC SU SPEN SIO N O F O N E PA R A M ETER FA M ILY O F D IFFEO M O R PH ISM S 77

Lem m a 8.1.| An adm issible one param eter group ofdi�eom orphism s � = f� sgs2R is

a one param eter group ofC 2-di�eom orphism s on Rn such that

(8.3) (s;x)7!@

@x�s(x)isC

2:

The m ain property ofadm issible one param eter groupsis the factthatthey are well-

behaved on thesetofgood di�usions.

Lem m a 8.2.| Let�= (� s)s2R be a stochastic suspension ofan adm issible one param -

eter group ofdi�eom orphism s. Then,for allX 2 E ,we have for allt2 I,and alls2 R:

i)The m apping s7! D ��sX (t)2 C 1(R),(a.s.),

ii)W e have@

@s[D �(�s(X ))]= D �

�@�s(X )

@s

�

(a:s:):

This lem m a is trivialin the classicalcase where X is a sm ooth function and D � is

the classicalderivative with respect to tim e. Indeed,it reduces to the Schwarz lem m a.

However,thisinequality playsan essentialrolein thederivation oftheclassicalNoether’s

theorem (see [5],p.89).

Proof.| According to (2.4),

D ��s(X )(t)= D �X (t)�@x�s

@xX (t)+ i�

�(t;X t)2

2

@2x�s

@x2X (t) (a:s:):

So :

@

@sD ��s(X )(t)= D �X (t)�

@

@s

@x�s

@xX (t)+ i�

�(t;X t)2

2

@

@s

@2x�s

@x2X (t) (a:s:):

Since (s;x)7! �s(x)isC2,we have

@

@s

@

@x�s(x)=

@

@x

@

@s�s(x)by the Schwarz lem m a.In

the sam eway,

@

@s

@2

@x2�s(x)=

@

@x

@

@s

@

@x�s(x)

because(s;x)7! @@x�s(x)isC

2.

Therefore:@

@s

@2

@x2�s(x)=

@2

@x2

@

@s�s(x):

Applying (2.4)to@

@s�s,wecan conclude that:

@

@s[D �(�s(X ))]= D �

�@�s(X )

@s

�

(a:s:):

It m ust be pointed out that every extension ofthis lem m a willlead to a substantial

im provem entofthe following stochastic Noethertheorem .


8.4. Linear tangent m ap

Let X 2 C1(I) and � :Rn ! R

n be a di�eom orphism . The im age ofX under the

stochastic suspension of�,denoted by �,inducesa naturalm ap fortangentvectors de-

noted by ��,called thelineartangentm ap,and de�ned asin classicaldi�erentialgeom etry

by:

D e�nition 8.5.| Let � be a stochastic suspension of a di�eom orphism . The linear

tangentm ap associated to �,and denoted by � �,isde�ned for allX 2 C1(I)by

(8.4) ��(X )= T(�(X ))= (�(X );D (�(X ))):

Allthequantitiesare wellde�ned asdi�eom orphism ssend C 1(I)on C1(I).

8.5. Invariance

W e then obtain the following notion of invariance under a one param eter group of

di�eom orphism s.

D e�nition 8.6.| Let� = f� sgs2R be a one-param eter group ofdi�eom orphism s and

letL be a functionalL :C1(I) ! C1C(I). The functionalL is invariantunder the one-

param eter group ofdi�eom orphism s � if

L(��X )= L(X ); for all� 2 �:

Asa consequence,ifL isinvariantunder�,wehave

L(�s(X );D (�s(X )))= L(X ;D X );

foralls2 R and X 2 C1(I).

R em ark 8.3.| W e note that this notion ofinvariance under a one param eter group

ofdi�eom orphism s does notcoincide with the sam e notion as de�ned by K.Yasue ([71],

p.332,form ula (3.1)) which in our notation isgiven by:

L(�s(X );�s(D X ))= L(X ;D X ); for alls2 R and X 2 C1(I):

In fact,K.Yasue de�nition ofinvariance does notreduce to the classicalnotion (see for

exam ple [5],p.88) for di�erentiable determ inistic stochastic processes.

M oreover,Yasue’sde�nition isnotcoherentwith theinvariancenotion used in hisproof

ofthe stochastic Noether’s theorem ([71],theorem 4,p.332). See the com m entbelow.

8.6.TH E STO CH A STIC N O ETH ER ’S TH EO R EM 79

8.6. T he stochastic N oether’s theorem

Noether’s theorem has already been generalized a great num ber oftim es and covers

som etim esdi�erentstatem ents[32].Here,we follow V.I.Arnold’s([5],p.88)presentation

ofthe Noethertheorem forLagrangian system s.W e correcta previouswork ofK .Yasue

([71],Theorem 4,p.332-333).

T heorem 8.1.| LetJa;b be a functionalon C1(I)given by

Ja;b(X )= E

�Z b

a

L(X (t);D X (t))dt

�

:

with L invariantunder the one-param eter group �= f� sgs2R.

LetX 2 C1(I)be a C1(I)-stationary pointofJa;b with �xed end points condition

X (a)= X a; and X (b)= X b:

Then,we have

d

dtE

�

gradvL@Y

@s

��s= 0

�

= 0;

where

(8.5) Ys = �s(X ):

Proof.| LetY (s;t)= �sX (t)fors2 R and a 6 t6 b.

AsL isinvariantunder�= f� sgs2R,we have

@

@sL(Y (s;t);D �Y (s;t))= 0 (a:s:):

AsY (:;t)and D �Y (:;t)belong to C1(R)forallt2 [a;b]by de�nition 8.4,iii),we obtain

(8.6) gradxL �@Y

@s+ gradvL

@D �Y

@s= 0 (a:s:):

Using (Lem m a 8.2,ii),thisequation isequivalentto

(8.7) gradxL �@Y

@s+ gradvLD �

�@Y

@s

�

= 0 (a:s:):

AsX = Y js= 0 isa stationary processforJa;b,we have

(8.8) gradxL = D �� gradvL:

Asa consequence,we deducethat�

[D �gradvL]�@Y

@s+ gradvLD �

�@Y

@s

��s= 0

= 0 (a:s:):

Taking the absolute expectation,we obtain

(8.9) E

� �

[D �gradvL]�@Y

@s+ gradvLD �

�@Y

@s

��s= 0

�

= 0:


Using theproductrule,weobtain

d

dtE

�

gradvL@Y

@s

��s= 0

�

= 0;


8.7. Stochastic �rst integrals

The previous theorem leads us to the introduction ofthe notion of�rst integralfor

stochastic Lagrangian system s(1).

8.7.1. R em inder about �rst integrals.| LetX be a C k vector�eld orR n,k > 1

(k could be 1 or !,i.e. analytic). W e denote by �x(t) the solution ofthe associated

di�erentialequation,such that�x(0)= x and by S the setofallthesesolutions.

A �rstintegralofX isa realvalued function f :R n ! R such thatforall�x(t)2 S,

we have

(8.10) f(�x(t))= cx;

wherecx isa constant.

W e have not im posed any kind ofregularity on the function f,so thatf can be just

C 0.In thiscase,the existence ofa �rstintegraldoesnotim pose m any constrainton the

dynam ics.

Iff isatleastC 1,then we can characterize �rstintegralsby the following constraint:

(8.11) X � f = 0:

8.7.2. Stochastic �rst integrals.| Thepreviousparagraph leadsusto searching for

an analogue ofthe classicalnotion of�rstintegrals asa functionalde�ned on the setof

solutions ofa given stochastic Euler-Lagrange equation(2) and realvalued. Looking for

the stochastic Noethertheorem ,wechoose thefollowing de�nition:

D e�nition 8.7.| LetL be an adm issible Lagrangian system .A functionalI :C1(I)!

R isa �rstintegralfor the Euler-Lagrange equation associated to L if

(8.12)d

dt[I(X )]= 0;

for allX satisfying the Euler-Lagrange equation.

(1)O fcourse,one can extend thisde� nition to generalstochastic dynam icalsystem s.

(2)O fcourse,thisde� nition willextend to arbitrary stochastic dynam icalsystem s.

8.8.EX A M PLES 81

W e can now interpret the stochastic Noether theorem in term of�rst integrals, i.e.

the fact that the invariance of the Lagrangian L under of a one param eter group of

di�eom orphism s � = (� s)s2R induces the existence ofa �rst integralfor the associated

Euler-Lagrange equation,de�ned by

(8.13) I(X )= E

�

gradvL@�sX (t)

@s

��s= 0

�

:

8.8. Exam ples

8.8.1. Translations.| W e follow the�rstexam plegiven by V.I.Arnold ([5],p.89)for

Noethertheorem .LetL betheLagrangian de�ned by

(8.14) L(X ;V )=V 2

2� U (X ); whereX 2 R

3;

V = (V1;V2;V3)2 C3,V 2 := V 2

1 + V 22 + V 2

3 and U istaken to beinvariantunderthe one

param etergroup oftranslations:

(8.15) �s(x)= x + se1;

wherefe1;e2;e3g isthecanonicalbasisofR3.

Then,by theStochastic Noether’stheorem ,thequantity

(8.16) E [D X 1]

isa �rstintegralsince @V L = V and @s�s(X 1(!))= e1.

8.8.2. R otations.| W e keep the notations ofthe previous paragraph. W e consider

the Lagrangian ofthe two-body problem in R3,i.e.

(8.17) L(X ;V )= q(V )�1

jX jwhere q(V )=

V 2

2;

where j:jdenotesthe classicalnorm on R3 de�ned forallX 2 R

3,X = (X 1;X 2;X 3)by

jX j2= X 21 + X 2

2 + X 23.

W e already know that the classicalLagrangian L is invariant under rotations when

X 2 R3 and V 2 R

3.Here,we m ustprove thatthe sam e istrue forthe extended object,

i.e.forL de�ned overR 3 nf0g� C3.Thisextension,aslong asitisde�ned,iscanonical.

Indeed,we de�neq(z)forz 2 C3 as

(8.18) q(z)=1

2(z21 + z

22 + z

23); z = (z1;z2;z3)2 C

3:

Note that our problem is not to discuss an analytic extension ofthe realvalued kinetic

energy but only to look for the sam e function on C3 sim ply replacing realvariables by

com plex one.Aslong asthenew objectiswellde�ned thisprocedureiscanonical,which


isnotthe case ifwe search foran analytic extension ofq over C3 which reducesto q on

R3.

O urm ain resultisthen thatthisgroup ofsym m etry ispreserved understochastization,

which isin facta generalphenom enon thatwillbediscusselsewhere.

Lem m a 8.3.| The lagrangian L de�ned overR 3nf0g� C3 isinvariantunderrotations

��;k around the ek axisby the angle �,k = 1;2;3.

Theproofisbased on thetwo following facts:

{ As��;k isa linearm ap whosem atrix coe�cientsdo notdepend on t,we have

(8.19) D � [��;k(X )]= ��;k[D �X ];

where��;k istrivially extended to C3.

{ A sim ple calculation gives

(8.20) 8 z 2 C3; q(��;k(z))= q(z):

W e easily deducethe��;k invariance ofL,i.e.that

(8.21) L(��;kX ;D (��;kX ))= L(X ;D X ):

W e now com pute:@��;k(X )j�= 0 = ek ^ X and

@V L(X ;D X )� @��;k(X )j�= 0 = (X ^ D X )k:

Thereforetheexpectation ofthe"com plex angularm om entum " X ^ D X isa conserved

vector (̂ isextended in a naturalway to com plex vectors).

8.9. A bout �rst integrals and chaotic system s

In this section,we discuss som e consequences ofthe stochastic Noether’s theorem in

the contextofchaotic dynam icalsystem s. The study ofdeterm inistic chaotic dynam ical

system sisdi�cult.

Hereagain,wereturn to theclassicaln-body problem ,n > 3.In thiscase,in particular

forlargen,thedynam icsofthesystem isvery com plicated and only num ericalresultsgive

a globalpicture ofthe phase space. Despite the existence ofa chaotic behaviour,there

existseveralwellknown �rstintegralsofthe system .

Theseintegralsareused asconstraintson thedynam icsand can giveinteresting results,

asforexam ple J.Laskar’s[41]approach to the Titus-Bode law forthe repartition ofthe

8.9.A BO U T FIR ST IN TEG R A LS A N D CH AO TIC SY STEM S 83

planetsin thesolarsystem sand extra-solarsystem s.

Usingourapproach,wecan gofurtherbyclaim ingthatsuch kind ofintegralscontinueto

existeven ifoneconsidera m oregeneralclassofperturbationsincludingstochasticity.W e

notethatthisresultisfundam entalaslong asonewantsto relatenum ericalcom putations

on the n-body problem with the realdynam icalbehaviour ofthe solar system s,and in

thisparticularexam ple,the dynam icsofthe protoplanetary nebulae.

C H A PT ER 9

N AT U R A L LA G R A N G IA N SY ST EM S A N D T H E

SC H R �O D IN G ER EQ U AT IO N

In this section, we explore in details the stochastization procedure for natural La-

grangian system s. In particular, by introducing a suitable analogue of the action

functional,we prove that the stochastic Euler-Lagrange equation leads to a non-linear

Schr�odingerequation,dependingon afreeparam eterrelated toanorm alization constraint.

For a suitable choice ofthis param eter we then obtain the classicallinear Schr�odinger

equation.

9.1. N aturalLagrangian system s

In ([5],p.84),V.I.Arnold introducesthefollowing notion ofnaturalLagrangian system s:

D e�nition 9.1.| A Lagrangian system is called naturalifthe Lagrangian function is

equalto the di�erence between kinetic and potentialenergy:

L(x;v)= T(v)� U (x):

Asan exam ple,we have the naturalLagrangian function associated to Newtonian m e-

chanics:

L(x;v)=1

2v2 � U (x);

whereU isofclassC 1 .

9.2. Schr�odinger equations

9.2.1. Som e notationsand a rem inderofthe N elson w ave function.| W erecall

that�d is the space of"good" di�usion processes. Let�g

dbe the subspace of�d whose

elem entshave a sm ooth gradientdrift.W e then set:

S = fX 2 �d jD 2X (t)= � r U (X (t))g:

86 CH A PTER 9.N ATU R A L LAG R A N G IA N SY STEM S A N D TH E SCH R �O D IN G ER EQ U ATIO N

Fora di�usion X in � d with driftband density function pt(x),weset:

(9.1) �= (R + � Rd)nf(t;x);jpt(x)= 0g:

IfX 2 �g

dthen there existrealvalued functionsR and S sm ooth on � such that

(9.2) D X (t)=

�

b��2

2r log(pt)+ i

�2

2r log(pt)

�

(X (t))= (r S + ir R)(X (t));

since bisa gradient.O bviously:

(9.3) R(t;x)=�2

2log(pt(x)):

In thiscase,weintroduce thefunction:

(9.4) (t;x)= e

(R + iS)(t;x)

K

(whereK isa positive constant)called the wave function.

The wave function hasthe sam e form than thatofNelson one (see [53]). W e then set

A = S � iR. So = eiA

K and r A(t;X (t)) = D X (t). For a suitable K ,Nelson shows

thatifX satis�esitsstochastized Newton equation (which isthe realpartofours)then

satis�esa Schr�odingerequation. W e show,by using ouroperatorD ,the sam e kind of

resultin the nextsection.

9.2.2. Schr�odinger equations as necessary conditions. |

T heorem 9.1.| IfX 2 S \ �g

d, then the wave function (9.4) satis�es the following

non-linear Schr�odinger equation on the set�:

(9.5) iK @t+K (K � �2)

2

(@x)2

+�2

2�= U ;

Proof.| AsU isa realvalued function,X 2 S im plies

D2X (t)= � r U (X (t)):

Thede�nition of im pliesthaton �

r A = � iKr

:

Sincer A(t;X (t))= (D X )(t),we obtain

iK D@x

(t;X (t))= r U (t;X (t)):

Therefore,considering the k-th com ponentofthe lastequation and using lem m a 2.4,we

deduce

iK

�

@t@k

+ D X (t)� r

@k

� i

�2

2�@k

�

(t;X (t))= @kU (X (t)):

9.2.SCH R �O D IN G ER EQ U ATIO N S 87

Now D X (t)= � iKr

(t;X (t)).Thus,by Schwarz lem m a,weobtain

D X (t)� r@k

= � iK

dX

j= 1

@j

@j@k

= �

iK

2@k

dX

j= 1

�@j

� 2

;

and

�@k

=

dX

j= 1

@2j

@k

= @k

dX

j= 1

@j

�@j

�

= @k

dX

j= 1

@2j

�

�@j

� 2

:

Therefore

iK @k

0

@@t

+ i

�2 � K

2@k

dX

j= 1

�@j

� 2

� i�2

2

�

1

A (t;X (t))= @kU (X (t)):

By addingan appropriatefunction oftin S,wecan arrangetheconstantin x ofintegration

in equation to bezero,and form ula (9.5)followsasclaim ed.

In orderto recover the classicallinear Schr�odingerequation,we m ustchoose the nor-

m alization constant K . The m ain point is that in this case,we obtain a clear relation

between the m odulus ofthe wave function and the density ofthe underlying di�usion

process.Precisely,we have:

C orollary 9.1.| W e keep the notations and assum ptions oftheorem (9.2.3). W e as-

sum e that

K = �2:

Then the wave functional satis�esthe linear Schr�odinger equation

i�2@t+

�4

2�= U ;(9.6)

M oreover,ifpt(x)isthe density ofthe process X (t)atpointx,then we have

()(t;x)= p t(x):

Proof.| K = �2 killsthe non-linearity in equation (9.5)and furtherm ore

log()=2

KR =

2

�2R = log(p):


9.2.3. R em arks and questions.|

{ O bviously �1 � �g

1 since biscontinuous.

{ A naturalquestion is to know ifthe converse ofthe corollary of() is true. M ore

precisely,if satis�esa linearSchr�odingerequation,can we constructa processX

which belongsto S \ �g

dand whosedensity issuch thatpt(x)= j(t;x)j2 ?

88 CH A PTER 9.N ATU R A L LAG R A N G IA N SY STEM S A N D TH E SCH R �O D IN G ER EQ U ATIO N

R.Carm ona tackled theproblem aticoftheso-called Nelson processesand proved

in [11]undersom econditionstheexistenceofa processX with gradientdriftrelated

to and whose density is such that p t(x) = j(t;x)j2. However we do not know

ifthis process belongs to our space ofgood di�usions processes (which m ay turn

to be a little restrictive class in thiscase),butwe can prove form ally,i.e. even so

assum ingthattheform ulaeofthestochastized derivativeto afunction oftheprocess

holds,thatX satis�esthe Newton stochastized equation. Therefore,thisleadsone

to question the extension ofthe derivative operator and the way itacts on a large

classofprocesses.Thisproblem willbetreated in a forthcom ing paper(See [18]).

{ ThefactthataprocessX satis�esthestochastized Newton equation ofNelson im plies

(D 2� D 2�)X = 0 (forthepotentialU isreal).Thisisa generalfactfordi�usion with

gradientdrift.Indeed,we can prove:

Lem m a 9.1.| LetX 2 �d,b its driftand p its density function. LetG i be the

i-th colum n ofthe m atrix (G ij):= (@jbi� @ibj). Then (D 2 � D 2�)X = 0 ifand only

iffor allt> 0,div(ptG i)= 0.

Thus,ifX 2 �g

ditisclearthat(D 2 � D 2

�)X = 0 since the form

Pbk@k isclosed

and so G = 0.An interesting question isthen to know iftheconverse istrue.So we

m ay wonderourselvesifS � �g

d.

The di�culty relies on the fact that p and b are related via the Fokker-Planck

equation,sothecondition div(ptG i)= 0 m ay notbethegood form ulation.However,

onecould usethework ofS.Roelly and M .Thieullen in [61]who usean integration

by partsvia M alliavin Calculusto characterize gradientdi�usion,in orderto give a

positive ornegative answerto ourquestion.

{ A basic notion in m echanics is that ofaction (see [5],p.60). The action associated

to a Lagrangian system is in generalobtained via the action functional. In our

fram ework,a naturalde�nition forsuch an action functionalisgiven by:

D e�nition 9.2.| LetA be the functionalde�ned on [a;b]� C1([a;b])by:

(9.7) 8t2 [a;b]; 8X 2 C1(I); A (t;X )= E

�Z t

a

L(X s;(D X )s)dsjX t

�

:

Thisfunctionaliscalled the action functional.

9.3.A BO U T Q U A N TU M M ECH A N ICS 89

Using this action functional,we have som e freedom to de�ne the corresponding

\action".Thenaturalone isde�ned by

(9.8) A X (t;x)= E

�Z t

a

L(X s;(D X )s)dsjX t= x

�

:

Usually,the wave function associated to A X an denoted by ~ isthen de�ned as

(9.9) ~ X (t;x)= expiA X (t;x):

However,itisnotatallclearthatsuch kind offunction satis�esthegradientcondi-

tion,i.e.that

(9.10) r A(t;X (t))= D X (t);

which isfundam entalin ourderivation ofthe Schr�odingerequation.

However,the condition 9.10 is equivalentto prove thatthe realpartofD X isa

gradient,which isnotatalltrivialin dim ension greaterthan two.

9.3. A bout quantum m echanics

Even ifwe look for dynam icalsystem s,our work can be used in the context ofthe

so-called Stochastic m echanics,developed by Nelson [53]. The basic idea isto reexpress

quantum m echanicsin term sofrandom trajectories.W e referto [12]fora review.

Thestochasticem bedding theory can beseen asa quantization procedure,i.e.a form al

way to go from classicalto quantum m echanics. Thisapproach isalready di�erentfrom

Nelson’sapproach,which do notde�nea rigid procedureto associate to a given equation

a stochastic analogue.M oreover,theacceleration de�ned by Nelson as

(9.11) a(X )=D D �(X )+ D �D (X )

2;

isonly a particularchoice. M any authorshave tried to justify thisform ([59],[60])orto

try anotherone.In ourcontext,the form ofthe acceleration is�xed and corresponds,as

in theusualcase,to thesecond (stochastic)derivativeofX .Asa consequence,stochastic

em beddings can be used to provide a conceptual fram ework to stochastic m echanics.

W e refer to [59]where a com plex valued velocity for a stochastic process is introduced

corresponding to the stochastic derivative ofX .

However,stochasticm echanicsaswellasitsvariantshavem any drawbackswith respect

to the initialwish to describequantum m echanicalbehaviours.W e referto [55]and [12]

fordetails.Thisisthereason why we willnotdevelop furtherthistopic.

C H A PT ER 10

ST O C H A ST IC H A M ILT O N IA N SY ST EM S

In thispart,we introduce the stochastic pendantofHam iltonian system sforclassical

Lagrangian system s.Thestrategy is�rstto de�nethestochasticanalogueoftheclassical

m om entum . W e then de�ne a stochastic Ham iltonian. However,thisHam iltonian isnot

obtained by theclassicalstochasticem bedding procedure.Thisisdueto thefactthatthe

m om entum processiscom plex valued.Asaconsequence,wem ustm odifytheprocedurein

orderto obtain a coherentpicturebetween theclassicalform alism and thestochasticone.

Thisleadsusto de�nethestochastic Ham iltonian em bedding procedurewhich re ectsin

factthenon trivialcharacteroftheunderlying stochasticsym plecticgeom etry to develop.

Having the stochastic Ham iltonian we prove a Ham ilton least action principle using our

stochasticcalculusofvariations.W ethen obtain an analogueoftheLagrangian coherence

lem m a in thiscase up to the factthatthe underlying stochastic em bedding procedure is

now the Ham iltonian one.

10.1. R em inder about H am iltonian system s

W e denote by I an open interval(a;b),a < b.

Let L :Rd � Rd � R ! R be a convex Lagrangian. The Lagrangian functionalover

C 1(R)isde�ned by

(10.1) L :C 1(R) �! C 1(R);

x 7�! L(x;_x;t):

W e can associate to L a Ham iltonian function using the Legendre transform ation

([5],p.65). From the functionalside,thisinducesa change ofpointofview,asthe func-

tionalisnotseen asacting on x(t),which isthe so-called con�guration space ofclassical

m echanics, but on (x(t);_x(t)) which is associated to the phase-space. This dichotom y

between position and velocitieshasofcourse m any consequences,one ofthem being that

the system ism ore sym m etric (the sym plectic structure).

92 CH A PTER 10.STO CH A STIC H A M ILTO N IA N SY STEM S

D e�nition 10.1.| LetL(x;v)be an adm issible Lagrangian system .Forallx 2 C 1,we

denote by

(10.2) p(x)=@L

@v(x;_x);

the m om entum variable.

W e now introducean im portantclassofLagrangian system s.

D e�nition 10.2.| LetL(x;v) be an adm issible lagrangian system . The Lagrangian L

is said to possess the Legendre property ifthere exists a function f :Rd ! Rd,called the

Legendre transform ,such that

(10.3) _x = f(x;p);

for allx 2 C 1.

M ostclassicalexam plesin m echanicspossesstheLegendreproperty.Thisfollowsfrom

the convexity ofL in the second variable (see [5],p.61-62).

W e can introducethe fundam entalobjectofthissection:

D e�nition 10.3.| LetL be an adm issible Lagrangian system which possesses the Leg-

endre property. The Ham iltonian function associated to L isde�ned by

(10.4) H (p;x)= pf(x;p)� L(x;f(x;p));

where f isthe Legendre transform .

TheHam iltonian function playsafundam entalrolein classicalm echanics.W eintroduce

the stochastic analogue in thenextsection.

10.2. T he m om entum process

A naturalstochastic analogue ofthe m om entum variable isde�ned asfollow:

D e�nition 10.4.| LetL(x;v)be an adm issible Lagrangian system .ForallX 2 C1(I),

we de�ne the stochastic process P (t),called the canonicalm om entum process,by

(10.5) P (t)=@L

@v(X (t);D X (t)):

Thisde�nition can bem adem orenaturalusing theem bedding � de�ned from C0(I)on

Pdet and thelineartangentm ap introduced in chapter8.Indeed,them om entum process

can be viewed asa functionalon X 2 C1(I),P :C1(I)! PC de�ned by (10.5). W e have

forallX 2 P 1det

= �(C1(I)),

(10.6) P (X )= �(p(x));

10.3.TH E H A M ILTO N IA N STO CH A STIC EM BED D IN G 93

wherex 2 C 1(I)issuch thatX = �(x).Asby de�nition,we have

(10.7) �(p(x))= p(�(x))= p(X ):

Asp keepsa senseforX 2 C1(I),we extend form ula (10.7)to C1(I)leading to de�nition

10.4.

Ifwe assum e thatthe Lagrangian possessesthe Legendre property,then there existsa

Legendretransform f such thatforallx 2 C 1,_x = f(x;p).W ecan ask ifsuch a property

isconserved forthe m om entum process.W e have:

Lem m a 10.1.| LetL(x;v)bean adm issibleLagrangian system possessingtheLegendre

property. Letf be the Legendre transform associated to L.W e have

(10.8) D X (t)= f(X ;P );

for allX 2 C1(I).

W e can now de�nethestochastic Ham iltonian associated to L:

D e�nition 10.5.| LetL(x;v)be an adm issible Lagrangian system possessing the Leg-

endre property. The stochastic Ham iltonian system associated to L isde�ned by

(10.9) H :PC � C

1(I) �! PC(P;X ) 7�! P f(X ;P )� L(X ;f(X ;P )):

10.3. T he H am iltonian stochastic em bedding

As in the previous chapter, we want to use the stochastic em bedding procedure to

associate a naturalstochastic analogue ofthe Ham iltonian equations. However,we m ust

be carefulwith such a procedure,asalready discussed in chapter 4,x.4.2.2. Indeed,the

em bedding procedure does not allow us to �x the notion ofem bedding for system s of

di�erentialequations.M oreover,wem ustkeep in m ind thattheprincipalidea behind the

Ham iltonian form alism istowork notin thecon�guration space,i.e.thespaceofpositions,

butin the phase space,i.e.the space ofpositionsand m om enta.Asthe stochastic speed

isby de�nition com plex,thisinducesa particularchoice forthe em bedding procedure in

the case ofHam iltonian di�erentialequations.

D e�nition 10.6.| Let F :Rd � Cd 7! C be a holom orphic function, realvalued on

realargum ents. Thisfunction de�nesa realvalued functionaloverC 1(I)� C 1(I),forI a

given open intervalofR.TheHam iltonian em bedding ofthe functionalF isthefunctional

denoted by FS,de�ned on C1(I)� PC(I)by H ,i.e.

(10.10) FS(X ;P )(t)= F (X (t);P (t)):


W e denote by SH the procedure associating the stochastic functionalFS to F . This

procedurereducestochangethefunctionalspacesforF from C 1(I)� C 1(I)to C1(I)� PC.

The m ain property ofthe Ham iltonian stochastic em bedding procedure (and in fact

it can be used as a de�nition) is to lead to a coherent de�nition with respect to the

m om entum process.Precisely,wehave:

Lem m a 10.2 (Legendre coherence lem m a).| Let L(x;v) be an adm issible La-

grangian system possessing the Legendre property. The following diagram com m utes

(x;p)

SH

��

H// H (x;p)

SH

��

(X ;P )H S

// H (X ;P )

(10.11)

Theprooffollowsessentially from the factthatthe stochastic Ham iltonian em bedding

ofthe functionalH ,denoted by H S coincide with the de�nition 10.5 ofthe stochastic

Ham iltonian system associated to H via theLegendretransform and thede�nition ofthe

m om entum process.

10.4. T he H am iltonian least action principle

Using the stochastic Ham iltonian function,we can use the stochastic calculusofvari-

ationsin orderto obtain the setofequationswhich characterize the stationary processes

ofthefollowing functional:

(10.12) Ia;b(X ;P )= E

�Zb

a

(P (t)D X � H (X (t);P (t)))dt

�

;

de�ned on C1(I)� PC.

In orderto apply ourstochasticcalculusofvariations,werestrictourattention to I on

C1(I)� C

1(I).Thefundam entalresultofthissection isthe following:

T heorem 10.1.| A necessary and su�cientcondition foran L-adapted process(X ;P )

to be N 1(I)-stationary process ofthe functionalIa;b with �xed end points (X (a);P (a))=

(X a;Pa)2 H ,(X (b);P (b))= (X b;Pb)2 H is thatitsatis�es the stochastic Ham iltonian

equations

(10.13)D X =

@H

@P(X (t);P (t));

D P = �@H

@X(X (t);P (t)):

10.5.TH E H A M ILTO N IA N CO H ER EN CE LEM M A 95

Proof.| W e m ust use the weak least action principle using the process Z = (X ;P ) 2

C1(I)� PC and theLagrangian denoted by L de�ned on R d � C

d � Cd � C

d by

(10.14) L(x;p;v;w)= pv� H (x;p):

As L(x;p;v;w) = L(x;v) form ally via the Legendre transform ,and L is assum ed to be

adm issible,wededuce thatL isagain adm issible.

Let�Z bea N 1(I)variation oftheform Z + �Z = (X + X1;P + P1),whereX 1 and P1

are N 1 processes.

TheEuler-Lagrange equation associated to L isgiven by

(10.15)

@L

@x(Z(t);D Z(t))� D �

�@L

@v(Z(t);D Z(t))

�

= 0;

@L

@p(Z(t);D Z(t))� D �

�@L

@w(Z(t);D Z(t))

�

= 0:

An easy com putation leadsto

(10.16)�@H

@x(Z(t);D Z(t))� D �P (t) = 0;

D X (t)�@H

@p(Z(t);D Z(t)) = 0:

Thisconcludesthe proof.

R em ark 10.1.| In thisproofwe do notneed a uniform assum ption on the setofvari-

ations as the Lagrangian does notdepend on the variable w. In fact,we can assum e a

variation in the direction P which belongsto C1(I).

10.5. T he H am iltonian coherence lem m a

In thissection,wederivetheHam iltonian analogueoftheLagrangian coherencelem m a.

Lem m a 10.3 (T he H am iltonian cohrence lem m a).| LetH :Rd � Rd ! R be an

adm issible Ham iltonian system . Then,the following diagram com m utes

H (x(t);p(t))


SH// H (X (t);P (t))

Stochastic leastaction principle��

(H E )SH

// (SH E )

(10.17)

Them ain pointisthatthisresultisnotvalid ifonereplacestheHam iltonian stochastic

em bedding by the naturalstochastic em bedding that we have used up to now. W e can

keep the classicalem bedding procedure only when dealing with realvalued versions of

the stochastic derivative.Forexam ple,ifone dealswith the reversible stochastic em bed-

ding procedure,we obtain a uni�ed stochastic em bedding procedureforboth Lagrangian


an Ham iltonian system s. W e think however that as wellas the com plex nature ofthe

stochasticderivativehasa fundam entalin uenceon theform ofthestochasticLagrangian

equations,i.e. that we obtain the Nelson acceleration,the fact to m ove from S to SH

re ectsa basic propertiesofthe underlying stochastic sym plectic geom etry we m usttake

into accountthiscom plex characterofthespeed.Thisproblem willbestudied in another

paper.

C H A PT ER 11

C O N C LU SIO N A N D PER SPEC T IV ES

Thispartaim s at discussing possible developm ents and applications ofthe stochastic

em bedding procedure.

11.1. M athem aticaldevelopm ents

11.1.1. Stochastic sym plectic geom etry.| The Ham iltonian form alism developed

in the last part suggest the introduction ofwhat can be called a stochastic sym plectic

geom etry.An interesting construction ofsym plectic structureson Hilbertspacesisgiven

in [34].

The m ain point here is to construct an analogue ofthe geom etricalstructure which

putsin evidence the very particularsym m etries ofthe Lagrangian equations in classical

m echanics.There existsalready m any attem ptto constructa given notion ofsym plectic

geom etry or at least a given geom etry for stochastic processes,but they are as far as

we know ofa di�erentnature. W e referto the book ofElworthy,LeJan and Li[44]for

an overview. These geom etries are only associated to stochastic processes and translate

into data ofgeom etricalnaturepropertiesoftheunderlying stochastic processes(like the

Riem annian orsub-Riem annian structureassociated to Brownian m otionsand di�usions).

A recent work ofJ-C.Zam briniand P.Lescot ([37]and [38]) deals speci�cally with

sym plectic geom etry and a notion ofintegrability by quadratures.

Fora discussion ofintegrability in ourcontextsee section 11.1.2.

11.1.2. P D E’s and the stochastic em bedding.| Thestochasticem bedding ofLa-

grangian system s over di�usion processes lead to a PDE governing the density of the

98 CH A PTER 11.CO N CLU SIO N A N D PER SPECTIV ES

solutionsofthestochasticEuler-Lagrangeequation.M oreover,wehavede�ned a stochas-

tic Ham iltonian system naturally associated to the Lagrangian. However,som e classical

PDEs,asforexam pletheSchr�odingerequation,possessan Ham iltonian form ulation.This

rem ark,which goes back to the work ofZakharov V.E.and Faddeev D.[72]is now an

im portant subject in PDEs known as Ham iltonian PDEs (see for exam ple [34]). As a

consequence,we have thefollowing situation:

(11.1)

H S

#

P D E �! H

O fcoursetherelation between thePDE and H S isnotofthesam enatureastherelation

with H .

In the sequel,we list a num berofproblem s and questions which naturally arise from

the previousdiagram :

{ There exists a notion ofcom pletely integrable Ham iltonian PDE (see [34]). W hat

aboutoutstochastic Ham iltonian system s?

Assum ing that we have a good notion ofintegrability for H S,we have the following

questions:

{ Arethere any relationsbetween the integrability ofH and H S?

{ Istherea stochastic analogue oftheArnold-Liouville theorem ?

{ Istherea specialsetof\coordinates" sim ilarto the action/angle variables?

W e note thatthere already existssuch a notion forHam iltonian PDEs(see [72]).

{ Istherea notion ofintegrability by \quadratures"?

In thatrespect,we think aboutLax work [36]on the integrability ofPDEs.

11.2. A pplications

11.2.1. Long term behaviour ofchaotic Lagrangian system s.| Thedynam ical

behaviourofunstable orchaotic dynam icalsystem sisfarfrom being understood,unless

we restrict to a very particular class ofsystem s like hyperbolic system s or weak version

ofhyperbolicity. This question arises naturally for sm allperturbations ofHam iltonian

system sforwhich there exists a large fam ily ofresultsdealing with thisproblem ,asfor

11.2.A PPLICATIO N S 99

exam ple the K AM (K olm ogorov-Arnold-M oser)theorem ,Nekhoroshev theorem and spe-

cialphenom ena liketheArnold di�usion related to theso-called quasi-ergodichypothesis.

Unfortunately,these results are di�cult to use in concrete situations and only direct

num ericalsim ulationsprovide som eunderstanding ofthedynam ics[22].

There exists ofcourse ergodic theory which tries to look for weaker inform ation on

the dynam icsthan a directqualitative approach.However,thistheory leadsalso to very

di�cult problem s when one tries to im plem ent it, as for exam ple in the case ofSina�i

billiard.M oreover,thereisa widely opinion in theapplied com m unity thatthelong term

behaviour ofa chaotic system s is m ore or less equivalent to a stochastic process. O ne

exam ple ofsuch opinion is wellexpressed in the article ofJ.Laskar [41]in the context

of the chaotic behaviour of the Solar system : \Since the characteristic tim e scale for

the divergence ofnearby orbits in the Solar system is approxim ately 5 M yr,the orbital

evolution ofthe planet becom es practically unpredictable after 100 M yr. Thus in the

long term ,the m otion ofthe Solarsystem m ay be described by a random process,where

orbitswandererratically in a chaotic zone."

W hatarethe argum entsleading to thisidea ?

The�rstpointisthatchaoticdynam icalsystem sarein generalcharacterized by theso-

called sensitivity to initialconditions,m eaning thata sm allerroron the initialcondition

leadsto very di�erentsolutions. O fcourse,one m ustquantify thiskind ofsentence,and

we can do that,with m ore or less canonicity,by introducing Lyapounov exponents and

Lyapounov tim e.W hateverwedo,thereisa non canonicaldata in this,which isprecisely

to what extent we consider that two solutions are di�erent. This m ust be a m atter of

choice fora given system ,and cannotbe �xed by any m athem aticaltool. In the sequel,

we assum e thata system is sensitive to initialconditions in som e region R ofthe phase

space,and fora given m etric,ifforallx0 2 R and all� > 0,thedistanceattim etbetween

a trajectory starting atx0 and x0 + �,denoted by d(t)is(1) approxim ately given by

(11.2) d(t)= �et=T

;

(1)Aswealready stress,wecan in som esituationsgivesa precisem eaning to allthispoint,likeforexam ple

in the Sm ale Horseshoes,butthisis farto coverthe wide variety ofchaotic behaviourwhich are studied

in the applied literature.


whereT > 0 istheso-called Lyapounov tim eorhorizon ofpredictability forthesystem (2).

Foran exam ple ofsuch an estim ate,we referto J.Laskar[42]where he gives num erical

evidencesforthe chaotic behaviourofthesolarsystem .

Asa consequence,fortsu�ciently large with respectto T,we have no prediction any

m ore,orin otherwords,wecan notassign to a given prediction a preciseinitialcondition.

W e then have lost the determ inistic character ofthe equations ofm otions. An idea is

then to say that one m usts then consider not a �xed initialcondition x0,but a given

random variable representing allthepossiblebehaviours(kind oftrajectories)one islead

to aftera �xed tim e t:forexam ple,� > 0 being �xed,we considerallthe intersectionsof

trajectoriesstarting in thedisk D (x0;�)with theballB (x0;�).W ethen obtain a fam ily of

directions.Assum ing thatwe can com pute an average overthe fam ily ofsuch a quantity

which obtain an averaged direction which select a given point ofthe ballB (x0;�). W e

then follow the selected trajectory during the tim e t,and continue again thisprocedure.

Such a construction is rem iniscent ofthe classicalconstruction ofthe Brownian m otion

(see [30],p.66).O fcourse,thisprogram m ecan only becarried in som especi�cexam ples.

W ereferto thearticleofY.Sina�i[62]foran heuristicintroduction to alltheseproblem s.

Ifwe agree with the previousheuristicidea,one can then ask forthe following:how is

the underlying stochastic processgoverned by thedynam icalsystem ?

W e return again to the Ham iltonian/Lagrangian case. The stochastic em bedding

procedureanswersprecisely thisquestion.Thestochastic Euler-Lagrange equation isthe

track ofthe underlying Lagrangian system on stochastic processes. As a consequence,

we can think that we are able to capture even the desired long term behaviour ofthe

Lagrangian system using thisprocedure.

In orderto supportourpointofview,we suggestthefollowing strategy:

Considera perturbation ofa com pletely integrableHam iltonian system H �(x)= h(x)+

�f(x),with x 2 R2n for exam ple. Let us assum e that h(x) leads to a particular PDE

under stochastic em bedding,which can be wellunderstood and solved. The long term

behaviour ofthe com pletely integrable Ham iltonian system is trivial. This notthe case

forthe stochastic analogue.W hataboutthe long term behaviourofH � ? W e think that

(2)In concrete system s,one m ustinvolve a m acroscopic scale (see [21],p.17),which bound the adm issible

size ofan erroron a prediction.Here,thisquantity isarbitrary replaced by e.


itiscontrolled by thestochastic analogue oftheunperturbed Ham iltonian.Thisresultis

related to a kind ofstochasticstability which wem ustde�ne.However,thisapproach can

betested on a widevariety ofexam ples,in particularcelestialm echanicalproblem s.

11.2.2. C elestialm echanics. | There existm any theoriesdealing with the problem

oftheform ation ofgravitationalstructures.Forplanetary system sthisquestion isrelated

to a long standingproblem related to the\regular" spacing ofplanetsin theSolarsystem .

Thisproblem which goesback to K epler(1595),K ant(1755),von W olf(1726),Lam bert

(1761),takes a m athem aticalform under the Titius (1766) form ulation ofthe so called

Titius-Bode law giving a geom etric progression ofthe distance ofthe planets from the

sun.W ereferto thebook ofNieto [56]form oredetails.Even ifthisem piricallaw failsto

predictcorrectly the realdistance forthe PlanetPluto forexam ple,itsinterestisthatit

suggests thatthe repartition ofexoplanetorbitalsem i-m ajor axescould satisfy a sim ple

law. Asa consequence,one searchsfora possible physical/dynam icaltheory supporting

theexistenceofsuch kind oflaw.M oreover,thediscovery ofm any exo-planetary system s

can be used to testifthe theory isbased on universalphenom ena and notrelated to our

knowledge oftheSolarsystem .

Allthe actualtheories aboutthe origin ofthe solar system presuppose the form ation

ofa protoplanetary nebula,form ed by som e m aterial(gas,dust,etc ...) with a central

body (a starora big planet).W e referto Lissauer[43]form oredetails.

Instead,weusea sim pli�ed m odelconsisting ofa largecentralbody ofm assm 0 with a

largenum berofsm allbodies(m j)j= 1;:::;n,whosem assisassum ed to besm allwith respect

to m 0.Them ain problem isto understand the long term dynam icsofthism odel.

Following the work ofAlbeverio S.,Blanchard Ph. and R.Hoegh-K rohn ([3],see also

[4]), we can m odelize the m otion of a given grain in the protoplanetary nebula by a

stochastic process(see [3],p.366-367),m ore precisely a di�usion process. The problem is

then to �nd what is the equation governing the dynam ics ofsuch a stochastic process.

Using our stochastic em bedding theory,we can use the classicalform ulation in orderto

obtain thedesired equation.Thisquestion willbedetailed in a forthcom ing article.

Them ain idea behind stochastic m odelisation isthe following:

The m otion ofa given sm allbody in a protoplanetary nebula is given by the K epler

m odeland a perturbation due to the large num ber ofnum ber ofsm allbodies. In [3],


this perturbation is replaced by a white noise. As a consequence,the m ovem ent ofa

sm allbody isassum ed to bedescribed by a di�usion process.Itm ustbe noted thatthis

assum ption isrelated to a num berofargum ents,one ofthem being thatthe dynam icsof

theunderlying classicalsystem isunstable.W ethen return to ourpreviousdescription of

the chaotic behaviour ofa dynam icalsystem . However,using the stochastic em bedding

theory, we can try to justify the passage from a classical m otion to a stochastic one

looking atthe following problem :

LetL� = LK epler+ P�,betheLagrangian system describing thedynam icsofourm odel.

The Lagrangian LK epler isthe classicalLagrangian ofthe K eplerproblem ,and P� isthe

perturbation. Using the stochastic em bedding theory,we can deduce two stochastic dy-

nam icalsystem s,one associated to L� and denoted by S� and one associated to LK epler

denoted by SK epler.Ifthe previousstrategy to replace theperturbativee�ectby a W hite

noise is valid,then we m ust have a kind ofstochastic stability between SK epler and S�.

The notion ofstochastic stability m ustbe de�ned rigorously and be consistent with the

stochastic em bedding theory(3). W hy such a stability result is reasonable ? The m ain

thing isthatwe already look in SK epler forstatisticalpropertiesofthe setoftrajectories

ofstochastic (di�usion)processesunderthe K eplerLagrangian. There isno reason that

the statistic ofthis trajectories really di�ers when adding a sm allperturbation. Thisis

ofcoursedi�erentifonelook fortheunderlying determ inisticsystem .Allthesequestions

willbestudied in a forthcom ing paper.

11.2.3. Strange attractors.| Strange attractors play a fundam entalrole in turbu-

lence and lead to m any di�cultproblem s. M ost ofthe tim e,one is currently interested

in the geom etrical properties of attractors (Hausdorf dim ension,...), specialdynam ical

properties (existence of an SRB (Siba�i-Ruelle-Bowen) m easure [68], stability under

perturbations....).However,focusing on a given attractorhidesthe factthatm ostofthe

tim e we can not predict from the equation the existence ofsuch an attractor. This is

in particular the case for the Lorenz attractor or the Henon attractor. These attractors

are obtained num erically. In som e m odels, we can construct a geom etric m odelfrom

which we can prove the existence ofsuch a structure (this is the case for the geom etric

Lorenz m odel) [27]. For exam ple, S. Sm ale [63] asks for an existence proof for the

Lorenz equation ofthe attractor.Thishasbeen donerecently by W .Tucker([66],[67]).

However,no generalstrategy existsin orderto predictsuch an attractor.

(3)Itm ustbe noted thatthere existsalready severalnotion ofstochastic stability in the literature,asfor

exam ple Has’inskii[29],K ushner[35]and m ore recently Handel[28].


O ur idea is to use the stochastic em bedding theory in order to predict the existence

of such an object. Let us consider the Lorenz equations. These equations are not a

Lagrangian system . However,there exitsa canonicalem bedding in a Lagrangian system

(see the reportofM .Audin [7]). Thislagrangian can then be studied via the stochastic

em bedding procedure. The solutionsare stochastic processeswhose density iscontrolled

by a PDE.Aswe already explain,we expectthatthe long term behaviourofthe system

is coded by this PDE. As the long term dynam ics of the Lorenz system if precisely

supported by the Lorenz attractor,we think that this structure can be detected in the

PDE (asa stationary state forexam ple).

W e can also take this problem as a �rst step towards understanding the existence

ofcoherent structures in chaotic dynam icalsystem s. M oreover,the Lorenz attractor is

widely studied and there exists a great am ount ofresults like the existence ofa unique

SRB m easure (see [67]). W e can then take this exam ple as a good system to com pare

classicalm ethodsofergodic theory and ourapproach. Form ore problem srelated to the

Lorenzattractor,SRB m easure :::,see ([69],[70]).

N O TAT IO N S

d:dim ension

(;A ;P )a probability space

-Stochastic processes

{ W e denote by

dX = b(t;X )dt+ �(t;X )dW ; (� )

the stochastic di�erentialequation where b is the drift,� the di�usion m atrix and

W isa d-dim ensionalW ienerprocessde�ned on (;A ;P ).

{ W e denote by X (t) the solution of(*) and by pt(x) its density (when it exists) at

pointx.

{ �(X s;a 6 s6 b):the �-algebra generated by X between a and b

{ Ft:an increasing � algebra

{ Pt:an decreasing � algebra

{ E [� jB]:the conditionalexpectation.

{ k :k:norm on stochastic processes.

-Functionalspaces

{ PR:realvalued stochastic processes

{ PC:com plex valued stochastic processes

{ Pdet:thesetofdeterm inistic stochastic processes

{ P kdet:thesetofdeterm inistic stochastic processessuch thatX (!)isofclassC k

{ �d:good di�usion processes

{ �g

d:good di�usion processeswith a gradientdrift

106 N O TATIO N S

{ Lp():setofrandom variableswhich belongsto L p

{ L2: the set ofrealvalued processes which are Pt and Ft adapted and such that

E

�Z 1

0

X2t dt

�

< 1 .

{ C 1;2((0;1)� Rd)thesetoffunction which are C 1 in the�rstvariable and C 2 in the

second one.

{ N 1:thesetofNelson di�erentiable processes.

-O perators

{ r :the gradient

{ �:the Laplacian

{ Letf(x1;:::;xn)bea given function.W e denoteby @xif thepartialderivative off

with respectto xi

{ Letf(x1;:::;xn;y1;:::;ym )beagiven function.W edenoteby @xf,x = (x1;:::;xn)

thepartialdi�erentialoff in thedirection x.

{ D :Nelson forward derivative

{ D �:Nelson backward derivative

{ D :the stochastic derivative

{ D n,D n�,D n:then-th iterate ofD ,D � orD

{ d and d�:adapted forward and backward derivative

k > 1

{ Ck: the setofrealvalued processeswhich are Pt and Ft adapted and such thatD i

exists,1 6 i6 k.

{ CkC:the setofcom plex valued processeswhich arePt and Ft adapted and such that

D i exists,1 6 i6 k.

{ Re(z):realpartofz 2 C.

{ Im (z):im aginary partofz 2 C.

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