PhYSICS REPORTS (SectionC of Physics Letters)24. No. 3 (1976) 171 --228. NORTLI-IIOLLANI) PUBLIShING COMPANY

STOCHASTIC DIFFERENTIAL EQUATIONS

N.G. VAN KAMPEN

Institute for TheoreticalI’hcsicsof the Univc’rsitv at Utrecht,Vetherlands*

ReceivedOctober 1975

Abstract;

In chapter 1 stochasticdifferential equationsare definedand classified,and their occurrencein physics is reviewed, in chapter

II it is shown for linearequationshow a differential equation for the averagedsolution is obtained by expandingin ore, where o

measuresthe sizeof the fluctuations ands-~their autocorrelation time. This result is the underlying reasonfor the existenceof

‘‘renormalired transportcoefficients’’. In chapter III the sametreatment is adaptedto nonlinearequations.In chapterIV an

alternativetreatment is described,applicableonly in a specialcase,but not confined to small ore.The emphasisis on physicalusefulnessrather than mathematicalrigor. Throughout the text applicationsarc given at the points

where they appearedto servebest asillustrations of the method. Thelist of referencesis not complete,but hopefully representative

of the literature.

Contents:

I. Generalconsiderations 14. The random harmonicoscillator - Calculations 200

• Introduction 173 15. Diffusion in a turbulent fluid 2022. Occurrenceof stochasticdifferential equations 173 16. highermoments 203

3. The connectionwith statistical mechanics 175 17. Randomly located scatterers 205

4. Formulation of the problem 177

5. First example: An assemblyof dipoles 179 III. Nonlinearequations6. Secondexample: A dipole in a fluctuating field 181 18. Nonlinear equationsreducedto the linearcase 208

19. Nonlinear equationswith rapid fluctuations 210

Ii Linear equationswith short correlation time 20. iwo applications 2127. Preliminariesabout linear equations 183 21 Use of the interactionrepresentation 215

8. Bourret’s integral equation 185

9. Wave propagationwith randomrefractive index 187 IV. A methodfor arbitrary correlation times10. lIme differential equation for the average 190 22. Markov processes 218

Ii. Keller’s expansionmethod 192 23. A method for arbitrary correlation times 22112. The cumulant expansion 194 24. Line broadening 222

13. The randomharmonic oscillator - General remark 198 References 225

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STOCHASTIC DIFFERENTIAL EQUATIONS

N.G. VAN KAMPEN

Institute for TheoreticalPhysicsof the Universityat Utrecht,Netherlands

(~

NORTH-HOLLAND PUBLISHING COMPANY — AMSTERDAM

N.G. VanKampen,Stochasticdifferentialequations 173

I. GENERAL CONSIDERATIONS

1. Introduction

A stochasticdifferential equationis adifferential equationwhosecoefficientsare random.*They may be randomconstantsor randomfunctions, but their statisticalpropertiesaresupposedto be given,just as in normalequationsthecoefficientsaregiven.Accordingly the solution of theequationwill be a randomfunction, andthe problem consistsin finding its statisticalproperties.

The importanceof stochasticdifferential equationsfor many problemsin physics,chemistry,andengineeringhasbecomeclear in the last decadeor so, andthey also find ampleuse in otherfields like biology, economicsand medicine.The literature is largeand disparate.and inevitablythe mathematiciansalso got into the game.We cannothopeto do full justice to the many andvariedfacetsof the subject.Ratherwe shall try to disentanglesomegeneralanduseful principles

and illustrate them on simple examples.We shallmainly be concernedwith ordinary differentialequationswith prescribedinitial conditions.Stochasticeigenvalueproblems(Boyce, 1 968; Porter,1965;Elliott et al., 1974)are not treated,nor stochasticintegral equations(Bharucha-Rcid,1972;TsokosandPadgett,1974), let alonemore reconditetopicslike equationsinvolving a retarded

time argument(El’sgol’ts andNorkin, 1973).The purposeof this reviewis threefold.(i) To expoundand define the problem andits relevancefor physicsandrelated fields.(ii) To critically reviewvariousapproachesandto show thata usefulexpansionexistsfor the

caseof rapid fluctuations.(iii) To enablethe readerto utilize the resultsand to makehim awareof certainpitfalls.No previousknowledgeof the subjectis required,nor of abstractmathematics,but merelya

certainfamiliarity with the methodsof mathematicalphysics.

2. Occurrenceof stochasticdifferential equations

In principle every differential equationthat purportsto describea physicalsystemshould be re-placedwith a stochasticone,in order to takeinto accountthe inevitableperturbationsdueto in-teractionswith the surroundings.In practice,of course,this is only donewhenthe effect of theseperturbationsis actuallyof interest.We list a numberof representativeapplicationsin order of in-

creasingsophistication.(i) The Langevin equation for the velocity of a Brownian particle

m~—---,3v+E(t), (2.1)

wherethe Langevinforce ~(t) is a randomfunction of time, whosestochasticpropertiesaresupposedto be known. Thisequationhasbeenthe subjectof extensivestudiesand we shallnotdealwith it here(UhlenbeckandOrnstein,1930;Chandrasekhar,1943;Wangand Uhlenbeck,1945).Note, however,that the classicaltreatmentis basedon the assumptionthat ~(t) is rapidlyfluctuating,which will also be the leadingideaof our laterdevelopments.

*We do not makethe somewhatsubtledistinction of Soong(1973)between“stochastic”and “random”equations,but useboth

termsinterchangeably.

174 N.G. Van Kampen,Stochasticdifferential equations

The problemof randomcurrentsin anelectricalnetwork with noise sourcesis of the sametype,but often involvesseveralcoupleddifferential equationswith severalunknowns(Middleton, 1960;Stratonovich,1963. 1967;VaIl derZiel, 1970;Van Vliet. 1971). However, aslong as the equationsare linear andmerely tile driving force is random, the problem is of a simpletype. It canbe solvedeasily owing to the fact that one can decomposethe noise in small parts(e.g. pulses,or Fourier componeilts),and addtile effectsof theseseparateparts.

This is no longertrue when the parametersof the circuit. i.e., the coefficientsin the equation,are fluctuating, nor if the equationsthemselvesare nonlinear.This is tile type of problem that weshall be concernedwith. Mechanicalsystemsandelectromechanicalcontrol systemsare often ofthis type (Wonham.1970;SrinivasanandVasudevan,1971;Soong, 1973),andso are the follow-ing examples.

(ii) Propagationof’ radio wavesthroughan atmospherewith snlall density fluctuations hasbeenextensivelystudied*. In the simplestcaseof planewavesthey are describedby tile equation

d2~. w2(2.2)dx c

wherethe refractiveindex n(x) hasan averagevaluen0 plus a small term n1(x),which is a random

function of x. A moregeneraltreatment,however,starts from the three-dimensionalwaveequa-

tion, seeSec.9.Light wavesin the atmosphereareaffectedby the fluctuationsmamly in that their pathsare

bent, as is demonstratedby the twinkling of stars.In principle this is coveredby the three-dimen-

sional wave equation,but in practicethe geometricalopticsapproximationis used,which leadstostochasticnonlinearequationsfor the pathsof the light rays (Chernov. 1 96 1; Keller. 1962).

(iii) Soundpropagationin tile atmosphereor in the ocean(Cllernov, 1961;Horton, 1969) isalso of practicalinterest.Sonar,however,becauseof its short wavelength, is lessaffectedbydensityfluctuationsthanby scatteringon randomly locatedobjects,like fish. One is interestedin the resulting “reverberation” (backscatteringinto the receiver)and the attenuationof thesignal(Sec. 17). Reflection by the bottom andby the waveson the surfacegives rise to stochasticboundaryconditionsand is outsidethe scopeof this review(Kohler, 1975;Kravtsov et al., 1975).Similar problemsoccur in the study of thermoelasticwaves(Chow, 1 973) andgravity waves(Lange, 1973),and the propagationof sonicboons(Pierceand Maglieri, 1972;Wenzel, 1975).

(iv) Diffusion in a moving fluid** is describedby

—D~72n--V(nu). (2.3)

wheren(r, t) is the density of’ the testparticlesando(r, t) the velocity of the fluid. If the fluid isturbulent,u is only known as a stochasticfunction of r andt. The problem is then to find theaverage(n(r, t)) for given initial n(r, 0) = ~(r), seeSec. 15.

*Lighthill (1953);Tatarski (1961, 1969);Bourret(1962a,I 962b);Keller (1964);Karal andKeller (1964); Iloffmann (1964);

Frisch (1968); Barabanenkovetal. (1971);Howe (1971);Chow (1972, 1973);DenceandSpence(1973); Lax (1973);Lee(1974); Klyatskin and Tatarskii(1974);Kohler andPapanicolaou(1974);Prokhorovet al. (1974).

**Flemshman(1956); Bourret (I 962a, I 962b); Lo Dato (1973);Chow (1974); BedeauxandMazur (1974);MazurandBedeaux(1974, 1975);Klyatskin andTatarskii (1974).

N.G. VanKampen,Stochasticdifferentialequations 175

(v) The numberof chargecarriersin the conductionbandof a photoconductoris determined

by the incident beamof photons.If the arrival timesof the photonsareuncorrelated(shot noise)the probability distribution of that numberobeysa linearequation(Chapman—Kolmogorovormasterequation,seeSec.22). If, however,the arrival timesarecorrelatedthe coefficientsin thatlinear equationhaveto be treatedasstochasticfunctionsof time (Ubbink, 1971).

(vi) Broadeningof the spectrallinesemitted andabsorbedby an atom in an ionized gasrequiresthe studyof theSchrodingerequationof the atom

(2.4)

HereP is the operatorof the electric dipole of the atom andE is the electric field dueto thechargedparticles.ThusE(t) is a stochasticfunctionof time whosepropertiesaresupposedto beknown from kinetic theory. The difficulty hereis thatE(t) involves both fast andslowly moving

components(Sec. 24).(vii) In magneticresonancetheory onestudiesthemotion of a spin in a solid,governedby the

linearequationof motion(Bloch, 1 946)

S=—gBXS. (2.5)

The magneticfield B consistsof an externallyimposedpart(not necessarilyconstantin time), afixed part dueto the lattice,anda fluctuatingpart causedby the latticevibrations(WangsnessandBloch, 1953;Redfield, 1965;Slichter, 1963).In analogywith the Brownian motion thestochasticpropertiesof thesefluctuationsare obtainedfrom a reasonableconjecture.Ratherthanasinglespin also moregeneralquantummechanicalsystemsin interactionwith a heatbathoflattice vibrations havebeenstudied(Redfield, 1957, 1965).

(viii) Analogousmethodsareusedin the theoryof the laser*. In order to describethe energyloss of an electromagneticmodeoneintroducesa couplingwith a somewhatabstract“heat bath”.The equationfor the densitymatrix of the combinedsystemis reducedto an equationfor thedensitymatrix of the electromagneticmodealoneby taking the tracein the Hilbert spaceof thebath.This canonly be doneat the expenseof certainassumptionsconcerningthe propertiesof thebath,which areof the samenatureas Langevin’sassumptions.The result is a Langevinequation(or the equivalentFokker—Planckequation)for the densitymatrix of that mode. In the samewaythe atomsare coupledto fictitious heatbathsto describedampingeffects(other thanthroughtheir couplingwith the lasermodes),the effect of pumping,andthe concomitantfluctuations.Similar ideashavebeenappliedto Josephsonjunctionsin interactionwith the radiationfield (LeeandScully, 1971).

3. The connectionwith statistical mechanics

It appearsfrom theseexamplesthatstochasticdifferentialequationsoccurwhenthe totalphysicalworld, or atleasta largesystem,is subdividedin a subsystemandits environment.Theinfluenceof the environmenton the subsystemis treatedin a way similar to thatof a heatbath,

*Wilhjs (1966);HakenandWeidlich (1969);Haken(i970); Haake(1973);Louisell (1973);Agarwal (1973).

1 76 N. G. Van Kampen,Stochasticdifferential equations

viz., as a randomforce whosestochasticpropertiesare supposedgiven. In this way the equatIons

of motion for the total systemare reducedto equationsof motion for the subsystemalone, atthe expenseof introducing randomcoefficients.

More serious,however, is that the actualstocilasticpropertiesof thesecoefficients can strictly

speakingbe found Dilly by solving tile microscopic equationsof nlotloll of tile total system.hut

this cannot be doneexce~itfor somesimple niodels.~Insteadone conjectures thesepropertiesott

the basisof physical intuition, aswasfirst donewith greatsuccessin Langevin’s treatment ofBrownian motion. Yet when dealingwith lcss sitllple and coticreteequations,physical intuition is

less reliableand often borders011 wishful thlitlking. Itt partIcular for SOlllC notihnearequations

this metilod can be Silo\Vll to lead to wronv results(Vati Kampeit. I 965).Yet in somecasesit is clear tilat tile stochasticpropertIesale determinedby the environnlent

alone, i.e., the reaction of tile subsystemon them may sal’elv he neglected.For example. tiledensity fluctuations ill tile atnlospllereaffect tile propagationof electromagneticwaves.but arenot influenced by them. In other cases.the reaction catlnOt be igllored. but Somesinlpie fornl is

assumedfor it. For example.in the Langevinequation (2. II tile whole rigilt-hand Illeniber is tile

f’orce exerted by the environment on tile Brownlan particle: this force will not be itldependentof’tile velocity of tile particle, but one conjecturesl’irst that its aVerageIS --~V.and tllen that after

subtractingthis tile rernainillg force is itldepetldent of v. and hastile lamihar propertiesof a

Langevin force. (Note that this averagedanipitlg is indispensablefor making the velocity and itsmeansquaretend to their equilibrium valuesasdetermInedby equilibrium statistical hllechallicS.)

Statistical mccilanics of irreversible processesalso t’its into this scheme.it is basedon a subdivi-

sion of tile total system of V -~ 1023 particles into a “subsystem” consistingof,i ~ N macroscopicvariables.,and an ‘‘environment’’ (also ‘‘reservoir or “bath’’) consistingof tile relllailulig A —- n

variables.The macroscopicvariablesobey a seif-contaitled setof t’qtlatioliS of fllOtioll by tileni-

selves(e.g. tile hydrod namic equatioils), itt analogy with tile equationmv = -~vfor theBrownian particle. The influence of the remainlllg degreesof Ireedontis then taken into account

by addingfluctuating terms.which are assumedto havesimilar propertiesas the Latlgevin force.That is, they musthavea negligibly short auto-correlationtime, and theremust be n~feedbackof the macroscopicstateon tileir stochasticproperties.

The crucial questlonis wilether or not this asscirnptionlS true. it should be emphasizedthat tiliS

question cannot be answeredby formal transformationsof the microscopicequationsof’ nlotion,sucil as tile cumulation of N-- n variableswith the use of Zwan~ig’sprojection operatortcchiliiqtle(Zwanzig, 1960;Nakajima. 1 958; Mon. 1965), It is not sufficient to rewrite the equationsill aform tilat looks like a Langevinequation and to bestow tile Ilaille “Langevin force’’ on one of the

terms.unlessit is at least piausible that it has the stochasticpropertiesthat go with it. An appeal

to tile large size of the heat bath,the weaknessof’ the coupling and ~VCll the rapidity of the fluctua-tions is not enough. It is necessarythat all long correlation tinies in the system arefully accountedf’or by the macroscopicequations.1-lencetile proper choice of macroscopicvariablesis crucial. Sofar no systematicmethod is known for makitlg this ciloice or even for decidingwhetiler any pro-

Posedchoice is correct.Similar stocllastification by adding fluctuating termshasbeenapplied to linear relaxationequa-

tions (OnsagerandMachiup, 1953;Fox atld Uhlenbeck, I 970a), to the hydrodynamicequations

~Ruhin(1958); hemmer(1959);Turner (1960); Kogure(1962): Marur and Braun (1964)~Uilersma (1966);Lopez (1966):Nakazawa (1966); Braun (1967);\Vada (1973);Kinm (1974): Adclmanand Doll (1974);Van Kanipen (1974e).

N G. VanKampen,Stochasticdifferentialequations 177

(LandauandLifshitz, 1959;Zwanzig, 1972;HaugeandMartin-Löf, 1973;Klyatskin and

Tatarskii, 1974;Kuramoto, 1975), to the diff’usion equation(Bedeauxand Mazur, 1974). toMaxwell’s equationsin a medium(Landauand Lifshitz, 1960), to theBoitzmannequation(Abrikosov andKhahatnikov, 1958;Bixon andZwanzig, 1969;Fox and Uhlenbeck,1970b),andto the equationsfor tile gravitational field in the universe(Nariai, 1974, 1975).

4. Formulation of the problem

A stochasticor randoml’ariable ~ is determinedby the set ~2of valuesit cantake (called in

different coilnectionsthe “range”, the “set of states”,or the “phasespace”).and by its probabilitydistribution defined on ~‘l. This distribution may be specified by its ‘density’ D(~)obeying

D(~)~ 0, fD(~)d~= 1. (4.1)

It may happenthat D involvesdelta functions or evenconsistsof delta functions alone(discretecase).The set~2andthe distribution D(~)will alwaysbe consideredasgiven.*

Any mathematicalobject that dependson a stochasticvariable~ is itself stochastic.In particular.a function ~(t; ~) of t and~ is astochasticfunction of time or a ‘random process’. It may beviewedas an ensembleof functionsof t, eachindividual function or “sample function” being iden-

tified by ~. Averagesandhighermomentsare integralsover f2, t’or instance

~(t1) ~(t2)) = f~(t1~) ~(t2 ~)D(~) d~.

The variable~ is often not written whenit is not necessaryto emphasizethat the object in ques-tion is stochastic.An object that doesnot dependon a stochasticvariableis called non-stochastic

or sure.A stochasticprocesswith the property that all moments

dependon the time differencesaloneis calledstationary. One definesthe auto-correlationfunctionof a stationaryprocessby

F(r) = K ~(t) -- K~(t))}{~(t+ r) — (~(t+ r))}) = K~(t)~(t + r)) (4.2)

Of courseF is a surefunction.F(0) is thevarianceof ~(t). In manycasesa correlation time r~canbe definedsuchthat F(r) 0 for r> r~.The Wiener—Khintchinetheoremstatesthat the Fouriertransformof’ F,

S(w)=ifF(r)e~wT dr~f F(r)coswrdr (4.3)

is the spectraldensityof the fluctuationsin ~(t; ~).

*More exactly we shouldhavepostulateda set t2, a u-algebra E of subsetsof 12, anda probability measureon E, not necessarilyhaving a density D (Kolmogorov, 1950).

178 NC. VanKampenStochasticdifferential equations

A stochasticequation (Bharllcha-Reid, 1 964, 1970) is ai~equationof’ tile form F(x; ~) = 0.whereF is given and.v 1S tile unknown. Its solution is a well-defined stocilasticvariablex(~),providedthat for each~ in &2 there existsoneand only onex obeyingF(x. ~) 0. Tilis condi-

tion can be nlademore explicit for linear equations:

L(~)xJ~ (4.4)

wheref-is a given surevector.L(~)a randomniatrix~5.andx tile unknown vector. Thisequation

determinesa stochasticvector.v(~)provided that L(~)is nonsingular f’or all ~. For instance

/1 ~\ /x1\ /f’m\

I II 1=1 I. f2=(--~,oo).\0 I / \x~! ~1~J

If L(~)is singularf’or certain ~ in ~2, then in generalno .v existsatld the equationdoesnotdeterminea stochasticvector.v(~).For instailce

/1 ~\ 1x1\ Ifi\H )( I. ~( ~,oo),

\~ I I \x21 \f~/

For specialchoiceoff, however,tile singularmatrix doeshavea solution,but in that casethere

are infinitely many, sothatagain the solution .v(~)is not well-defined. For instance

/1 &~\ /.\‘~\ if\I II 11’ I, ~2=(0.=).\~ I / \.x’2! \f/

Yet in that caseit is possibleto define a stochasticvariablex(~)by prescribingwhich solution is

to be selected.In the aboveexampleonehasx = x2 for all ~ ~ I ; it would be natural to prescribethat for ~ = I also the solution with .v1 = x2 is to be selected.It mustbe bornein mind, however.that the resultingstochasticvariablex(~)dependson this additional prescription.

A stochasticdifferentialequation of the n-th order may be written in the form

i’m = F(u, t; ~). (4.5)

whereu and F arevectorswith n components.It is obviousthat the solution is not unique for any

~, unlessoneprescribesappropriateboundaryconditions. If onesetsu(0, ~) = a, wherea is a surevector, the resultingsolution u(t; ~) is a well-def’ined stochasticprocess.It dependson the choiceof initial condition, however,not merely throughthe choiceof a,but also throughthe choiceoftile initial time. Tilis is a differencewith surediff’erential equationsandneedssonic further elabora-tion.

On selectingdifferent vectorsa one obtainsan n-dimensionallinear spaceof solutions.In thecaseof a suredifferential equationthis spacecontainsall solutions.More precisely,if one im-posesat some t0 ~ 0 a boundarycoilditiOn u(t0) = b oneobtainsoneof the solutionsfrom thesamespace.In the caseof a stochasticdifferential equation,however,this is no longertrue: thestochasticprocessdefinedby (4.5) and u(t0 ~) = b is not the sameas anyoneof thosedefinedby (4.5) andu(O; ~) = afbr anychoiceofa. This is evidentfrom the fact that the varianceof

*The tcrnm “stochasticmatrix” is also usedfor a special typeof surematricesoccurring in the theoryof Markov chains(Gantmnacher,

1959).

N.G. VanKampen,Stochasticdifferentialequations 1 79

u(t; ~) vanishesat the time at which the initial condition is imposedandnot at any other time.As a consequence,evenif F doesnotdependexplicitly on t, the solutionu(t; ~) is not a stationaryprocess,owing to the choiceof initial time.

The following tablesummarizesthevariouscategoriesof stochasticdifferential equationsmen-

tioned so far.

Table I

Category0. Ordinarydifferentialequationswith surecoefficientsbut randominitial valuesat t = 0. In contrastto someauthors(e.g.,Syski, 1967) we shallhenceforthnot includethis in theterm “stochasticdifferentialequations”.Thesolution simply amountsto solving theequationfor arbitraryinitial value andsubsequentlytaking appropriateaveragesover theinitial values.

CategoryI. Lineardifferentialequationsin which only theinhomogeneousterm is random,asin the Langevinequation.Thiscategoryis sometimesreferredto as“linear” or “additive” (Fox, 1972)andis too elementaryfor the presentreview,

CategoryII. Lineardifferentialequalionswith randomcoefficients,sometimesreferredto as “multiplicative” or somewhatconfusingly“nonlinear” (Kraichnan,1961;BedeauxandMazur, 1974). They arethe main substanceof this review, As initial con-ditionswe prescribesurevaluesat t = 0, but they may afterwardsberandomizedasin category0.

CategoryIII. Nonlineardifferentialequationswith randomcoefficients.It is shownin sec. 18 that theycanbe reducedtocategoryII. It will appearin thecourseof that reductionthat thedistinction betweensureandrandominitial valuesbecomesmoot.

CategoryIV. Eigenvalueproblemsof differentialequationswith randomcoefficientsarenot covered.

5. First example: An assemblyof dipoles

Considerthe first orderdifferentialequationfor a complexsinglecomponentquantityu(t),

tm—i&5u, u(0)a. (5.1)

This equationis a specialcaseof(2.5) obtainedby takinggB= (0,0. ~) andsettingS~-- iS,1, = u.If one has an assemblyof suchdipoles,all with slightly differentvaluesof the magneticfield, ~is a randomvariablewith somedistributionD(~).The boundaryconditionu(0) = a signifiesthatat t = 0 all dipolesarealigned.They thenall precessat slightly different ratesand(u(t)) tells themagnetizationin the x andy directionsof the wholeassembly(the magnetizationalong thez-axisis constant).The sameequation(5.1) hasalso beenemployedto illustrate certainpoints in thetheory of turbulence(Kraichnan, 1 961; Leslie, 1973).

The equation(5.1) canbe solvedexplicitly

u(t; L~)= e~1(~ta.

It follows that

(u(t)) fetD(ez,)deza.

Thus theresult is given by the Fouriertransformof D, that is, the characteristicfunction of thedistribution.Table II lists (u(t)) for a few different choicesof D; c andy are fixed parameters,therangeis 12 = (~-_oo,oo).

First observethat in all threeitemsthe averagedecaysowing to gradualloss of alignmentbe-tween the dipoles(“phasemixing”). The decayis morerapidwheny is larger, that is, whenthe dis-tribution D is broader.However, the way in which the amplitudedecreaseswith time dependsonthe specialform of D. In the first item (u(t)) is simply a dampedwave,and obeysa first-order

I 8)) N.G. Van Kaompt’n StochasticdmJJcremmtialequation.s

Table II

:11(t):

in(i) — —-—- :‘xp(- ct —yt)a

+ ~ C)-

r n ((212 exp~-- I cxp( -ict -- ~yt2)a

L -~ I

(iii) (7y)i exp) I{7 c7’y) —a1 +

differential equation

—(11(t)) = ---i(c — iy)(u(t)). (5.2)cit

Thus the effect of the randomness,as far as the averageis concerned,is to addan imaginary part

to thefrequency. But this is an artifact of the special f’orm of D: the same is Ilot true for the othertwo items.

Let usseewhat happensif onesolves tile sameequationwith initial valueh at t0. In item ( i) the

situation is the sameas ill suredifferential equations.By a suitablechoice of b one can reproduce

the same(u(t)). viz.,

b = exp(-- ict0 ~ -yt0)a.

In (ii), however,the solutionwith initial value b is

(u(t)) exp~-ic(t — t0) --- ~(t — t0)2 ~b.

Thereis no choice of b that makesthis coincidewith the solution in table II. The stochasticprocess

obtained in this way is different from all thoseObtainedby imposinga boundarycondition at

t = 0. By observingthe processat somelater period one is able to tell at what initial time tile SpIllshavebeenaligned.The santeremarkappliesto the third item.

It also follows from this discussionthat thereis no real lossof memory.That fact is physicallydemonstratedin the spin echoexperiment(Hahn, 1950;WalgraefandBorckmans,1972).One firstalignsthe nuclearspinsby astrong magneticfield B

0 in the z direction,then appliesan rf pulseB1along the x axis that turnsthem into the y direction.Theyare then allowed to precessfreely inthe field B0. Owing to local inhomogeneitiesof B0 their precessionis describedby the stochasticequation(5.1). The resulting magneticmomentrotatesin thex---y planeand is given by (u(t)). Itgradually disappearsdue to phasemixing. At sometime r asecondpulse,twice as strongas thepreviousone is appliedwith theeffect that all y-componentschangesign: u(r + 0) = u(r ~— 0)*.The result is that at t = 2r all spinsareagainaligned,

u(2r; ~) = e’°27~e ~ u(Ofl* = u(0)* = --u(0).

The basic reasonfor this effect -- which hascausedmisgivingsaboutthe foundationsof irrevers-

N.G. VanKampen, Stochasticdifferentialequations 181 :ibie statisticalmechanics(Blatt, 1959;Mayer, 1961) --- is that tile randomnessis due to phase :mixing ratherthan to the interactionwith a heatbath. Eachmagnethasits own field ~, and :thesefields areconstantin time. The force exertedby a Ileat bath.on the otherhand,fluctuatesin time. If thesefluctuationsaresufficiently rapid. i.e., if the correlation time of the external

force is small, they do causea lossof memoryand the averagedoesobeya differential equationwith time-independentcoefficientssimilar to (5.2), although only approximately.This is theunderlying ideaof many physicalapplicationsof stochasticdifferential equations,and is demon-

stratedin the following modification of our example.

6. Secondexaniple: A dipole in a fluctuating field

Again considerthe equation(5.1) for a magneticdipole in a field along thez axis,but supposethat the field is a randomfunction of time:

a = —i~(t;~)u, u(0) = 1. (6.1)

The sameequationhasoften beenusedasa model for a randomharmonicoscillator,seeSec. 13,andto describeparamagneticresonance(AndersonandWeiss. 1953).

Equation(6.1) can againbe solvedexplicitly for eachsamplefunction ~ on averagingthe resultis

(u(t)) = (exp(_if ~(t’) dt’} ) a. (6.2)

Unfortunately it is rarely possibleto evaluateexplicitly the averageof the exponential,so that it i~necessaryto resortto an expansion;or to the methodof S.c. 23.

The integral over ~ is itself a stochasticquantity !. On expandingthe exponentialoneobtainsa

seriesin successivemoments(~°>,which may bewritten asmultiple integralsovermomentsof ~:

Ku(t)) = ~ 1 -- if(~(t1))dt1 — ~ff(~(t1)~(t2)) dt1 dt2 + ... } a. (6.3)

Although thesemomentsareamenableto evaluation,it will be arguedpresentlythat this expansiocannotbe usedto providesuccessiveapproximations.The reasonis that any finite numberoftermsconstitutesabad representationof the function definedby the whole series— just as the behavior ofet for larget is badly representedby anyfinite numberof termsof its expansion.

This difficulty is overcomeof the cumulantsor “semi-invariants”. They arecertain combinatioi

of the momentsandwill be indicatedby doublebrackets.Forthe singlerandomvariable theyare definedby meansof a generatingfunction

= (—it)tm

(e~1t~)= exp — ((Zm)) . (6.4)m1 m!

On substituting for the integral over ~ one obtains

182 N.C. Van Kampen,Stochasticdifferential equations

(exp~_itf~(t’)dt’}) = exp~~L~1f ~ dti...dtm } . (6.5)mnl in!

0 0 0

The connectionwith tile nlomentsis givell by the following hierarcily of equatiolls(we write1. 2, .. for ~(t1), ~(t2), ...)

(1) = ((1))

(12) = ((1))((2)) + ((12))

(123) = (K] ))(( 2)X(3)) + ((1 ))(K 23)) + ((2))((13)) + ((3))((12)) + ((123))

(6.6)

The generalrule is that first the digits in tile momenthaveto be partitioned in subsetsin allpossibleways(not coullting the empty subset);for eachpartition oi~ewrites the product of thecumulantsfor the severalsubsets;finally one addsall such productsobtainedfor the different

partitionings.*Wllat is the reasonfor preferringthe expansionin curnulantsrathertilan in fllOnlelitS? Suppose

that ~(t) hasa short correlation timeT~. Moreprecisely we supposethat ~(t1) and~(t2) are statistic-ally independentquantitieswhen It1 — t21 ~ r~.Thenthe momentK~(t1)~(t2)) factorizesinto(~(t1)) (~(t2)). and it is seenthat the cumulant ((~(t1)~(t2))) vanishes,~* More generally,the m-thcumulant vanishesassoonas the sequenceof times t1, t2 tm containsa gaplargecomparedtoT~. This is a formal renderingof what is meantintuitively by “a rapidly fluctuating randomprocess”.Henceforth,wheneverwe say that a function hasa correlation time ~ we imply thisproperty of

the cumulants.The consequenceis that eachintegrandin (6.5) virtually vanishesunlesst1, t2,..., t,~are close

together.The only contribution to the integralconiesfrom a tube of diameterof orderr~alongthe diagonal in the rn-dimensionalintegrationspace.Hencefor large t thecontribution of eachterm is proportional to t. sothat

Ku(t)) = C1ct,~(0). (6.7)

Here C is sonic complex nuniber,which can be found from the cumulantsof the stochasticprocess~(t; w).

This leadsto the following cOnclusion.When the coefficient in (6.1) is not a random constantasin (5.1), but a randomfunction of time with a correlation time r~, the exponentialdecay(6.7),rathertllan being an artifact of a specialdistribution as in Sec.5, is generallytrue, albeit in anapproximatesense.The decayof (u(t)) is again dueto the lossof phasecoherence,but the phaseitself variesin a random fashion with time (“phasediffusion” rather thanphasemixing). Thefactthat for short r~the average(u(t)) again obeysa simpleequation is essentialfor many physical

*l.’or the generalformulasconnectingmomentswith cumulantsof a single variablesee Lukacs(1960);Prohorovand Rozanov

(1969) p. 165 (with misprints). l’or the multivariatecasesecMeeron(1957)and thearticleof J. AaseNielsenin: Muus andAtkins (1972).

**One saysthat themomentshavethe “productproperty”andthe cumnulantshavethe “clusterproperty”. The hierarchy (6.6) iscalled the“cluster expansion”.

N.G. Van Kampen,Stochasticdifferential equations t83

applications;it is implicit in mostapproximatetreatmentsof stochasticdifferentialequations(Sees.8, 10, 1], 12) and is also the basicideaof the presentarticle (until Ch. IV).

The approximationconsistsin the fact that a transienttime of orderr~musthaveelapsedafter

the initial time, at which the initial valueof u wasfixed. In practicethis transientshould be soshort that (6.7) becomesapplicablebefore(u(t)) hasbecometoo small. Another limitation appearsfrom the fact that the transienttime for the m-th cumulant is actuallyof order mr~,so that one

niust require the cumulantsto be negligible after a certainm. This fact also betraysthat the ex-pansionin cumulantsis likely to constitutean asymptoticratherthan a convergentseries,but the

precisemathematicalnatureis not knnwn.

II. LINEAR EQUATIONS WITH SHORT CORRELATION TIME

7. Preliminaries about linear equations

Much of our work will be concernedwith linear differential equations.The universalform of an

n-th order equationis

A(t; ~) u, (7.1)

whereu is a vectorwith n componentsandA(t; ~) is an n X n matrix whoseelementsarestochasticfunctionsof time. Formally thereis no difficulty in allowing n to be infinite, that is, in

allowingA to be an operatorin a linear vectorspace.However,becauseof possibleconvergenceproblemsit is convenientto think of finite vectorsandmatrices.

The problemis to find the stochasticfunctionu(t; ~) thatobeys(7.1) and the initial conditionu(0; ~) a. Ratherthana fixed initial vectoronemight alsoconsideran ensembleof initial vectorsa, with a certainprobability distributionQ(a). However, if this distributionis statistically indepen-dentof thatof ~, this generalizationwould be trivial, becauseonecaneasilyaverageafterwardsthe resultobtainedfor a fixed but arbitrarya, comp. Table I. If, on the otherhand,the distributionof a is correlatedwith ~, additionalcomplicationsarise,which will not be investigatedhere.

Insteadof trying to find all stochasticpropertiesof the solutionu(t; ~) we shall concentrateonits averagevalue(u(t)), as in the exampleof Sec.6. It will appearthat, in contrastwith nonlinearequations,this averageapproximatelyobeysa differential equationof its own, without the highermomentsenteringinto the picture. Of course, it mayhappenthatoneis interestedin knowingthe highermomentsof u(t; ~) as well. It will be shownin Sec. 16 that theycan be foundby the

samemethod,so that thereis no real restriction in studyingonly the average.In Sec. 1 8 a moresophisticatedmethodis developedby which the wholeprobability distributionof u at time t can

be found.Anotherremarkis that in manyapplicationsA appearsas the sumof a surepart anda fluctuating

part

A(t; l~)= A0(t) + etA 1(t; ~), (7.2)

whereais a parameterdeterminingthe size of the fluctuations.Sinceusuallya is small, a perturba-tion expansionin a is indicated.It is alsooften true that

154 N.C. Ian KanmtpenStochasticdifferential equations

(A~(t))= 0. (7.3)

Of coursethis can alwav~be achievedby’ definingA 0 as the averageof’ A. hut in that way tile

pil~’sicalsignificanceof’ tlte dcconlposition(7.2) may be lost. Also, it is necessaryf’or perturbationtlleory tllat tile equation with /1~aloneshould be soluble. Hence,although we shall often use ( 7.3)I’or simplicity, we shall not adopt it asa ulliversal truth.

Agaill, ill ntany applicationsA is a collstallt matrIx not depetldingon t. In that casethe ott-

perturbedequation can be solved formally

li°~(t) = exp(A0t)a. (7.4)

It 15 Ottell convenIent to tise tile interactIon representatIonby se

u(l) = exp(A0t)v(t). A (t) = cxp(A0t) Vtt) exp( A0t).

if, however,A0 dependson t one llas to replaceexp(A0t) with a more gelleral “evolution matrix”Y(i[t’) definedby

aY(t[t’)=A0(t) Y(tlt’). Y(t’[t’) = 1. (7.5)

at

‘fite interaction representatIonis then def’itled by

11(t) = Y(tIO)v(t). A1(t) = }‘(tIO) V(t) Y(OIt). (7.~)

Tile I’ollowing sontewliatmore explicit expressionfor Y(t 10) will be used in Sec. 1 2 and may

servehereto introducetile conceptof a time-orderedproduct of operators.It is readilyverifiedthat (7.5) is solved by

Y(tIO) = I + fdt1A0(11)+ /dti f dt2A0(t1)A0(t2)+

= t ft I

=n~oJdt1~fdt2 ...f dt0 A0(t1)A0(t2) ...Ao(t,~). (7,7)

Whentile factorsA 0 commuteall integrationscan be extendedfrom t~= 0 to t~= t (i = 1, 2

provided that a factor I/n! is suppliedto compensatef’or the larger integration domain.The resultwould be tile one in (6.3). However, evenwhen theA0(t1) do not commuteolle may still write

Y(tIO) = ~ fdt1 fdt2 ... fdtn [AotiA0t2 ... AO(t~1)~.

wheretile “time-ordering symbols” [...~indicatethat the operatorshaveto be shuffled soas toappearin the order of decreasingvaluesof their time arguments(Dyson. 1 949).The result mayalso he written in a more condensedforni analogousto (7.4)

N.G. VanKampen,Stochasticdifferential equations 185

Y(tlO) = [exp~JAo(t’) dt’}1~ (7.8)

where the time orderingsymbolsmeanthat oneshouldfirst expandthe exponentialand in eachterm order theoperatorschronologically. Alternatively onemay interpret(7.8) by the followingprescription:subdividethe interval (0, t) into N small intervalsof lengtht/N,

= 0. t1 = t/N, t2 = 2t/N tN = t

and read (7.8) as

t t t

lim exP~A(txi)} exP~A(tN2)) ... exp~A(to)j

=lirn(l +~A(tNl)} (1 ~ 2)) ... (i +~A(to)}.

Finally we remarkthat in manyapplicationsA hasthe evenmore specialform

A(t; ~) = A0 + a~(t; ~) B, (7.9)

with fixed matricesA0,B anda scalarrandomfunction~.Of courseall one-variableequations,e.g.,those in Sees.5 and6, can be regardedas to be of this type. The specialcasesthat ~ is aGaussianrandomprocess(Sec. 22) or evendelta-correlated (ItO equation)hasreceivedsomuch attentionthat it is necessaryto emphasizethat in the presentwork theseassumptionsarenot made.

8. Bourret’s integral equation

Considerthe equation(7.1) with the decomposition(7.2). In the interactionrepresentationde-fined by (7.6) the equationreads

va V(t)v, v(0)a. (8.1)

An obviousway of solvingit would be the expansionin a; this gives a resultanalogousto (7.7),andon averaging

(v(t)> = fI + dt1 (V(t1)> + a2 /d~

1f dt~K V(t1) V(t2)) + ... )a. (8.2)

However, this is not a suitableexpansion,becausethe successivetermsarenot merelyof increasing

order in a but also in t. That is, it is actually an expansionin powersof at, and is thereforeonlyvalid for limited time.* As indicatedin Sec.6 this difficulty is overcomeby usingan expansionincumulantsrather than in moments,as will be donein Sec. 12.

*It is true that in scattering theory — where (8.2) is known as the Schwinger—Dyson formula our objection does not apply, The

reasonis that in a scatteringprocesstheinteractionHamiltonian only actsduringtile collision andvirtually vanishesat all othervaluesof t. Accordinglyin that case(8.2) is anexpansionin powersof

155d. where Td is thedurationof the collision,

186 N.C. VanKannpen,Stochasticdifferential equations

First, however,we demonstrateasimpler. moreIleuristic approach.wllicil hasbeenusedby

severalauthors(e.g. Redfield, 1957. 1965;Howe, 1971) hut wasclearly formulatedby Bourret(1 962a, 1 962b). Forthis purposeit is necessaryto adopt(7.3), that is ( V(t)) = 0. Equation(8.1),includingthe initial condition, is strictly equivalentwith the integralequation

v(t) = a + af V(t’) v(t’) dt’. (8.3)

On iterating thisequationonceand averagingai’terwardsoneobtains

(v(t)) = a + a2 fdt’f dt”( V(t’) V(t”) v(t”)). (8.4)

This equationis still exact, but of’ no llelp in finding Kv(t)), becauseit relatesKv(t)) to amore

complicatedaverage.Tile essentialpointof the methodis that onesupposesthat the latteraveragemay be brokenup:

(V(t’) V(t”)v(t”)) (V(t’) V(t”))(v(t”)). (8.5)

With thishypothesis(8.4) becomesan integral equationt’or (v(t)). This integralequationcanbesimplified by differentiating once,

(v(t)) = a~fdt”( V(t) V(t”)) (v( t’)). (8.6)cit

0

For future usewe rewrite this result in the original representation,assumingfor convenience

tilat A0 doesnot dependon time

(11(t)) = A0(u(t)) + a2 f(A

1(t) exp~A0(t-- t’) }A 1(t’)) (u(t’)) dt’. (8.7)

This we silall call Bourret’s integral equation(without vouchingfor historical accuracy).It is a striking result, becauseit assertsthat the time dependenceof the averageof ii obeysan

equationall by itself. Thus onecall find (11(t)) without going throughthe processof first solving(7. 1) for eachindividual ~ andaveragingafterwards.This cannotbe true, asdemonstratedby theexamplesin Sec. 5. Actually it is all approximation for the casethat the fluctuationsare small andrapid. The preciseformulation of theseconditions is the main task of Sec. 10.

As the initial valuea hasdisappearedon differentiating (8.5) to obtain (8.6), the result (8.6)or (8.7) appliesto all solutionsthat are obtainedfor different choicesof a.The initial time t = 0,however,still enters.Hencethe equation(8.7) only appliesto thosesolutionsof (7.1) that arestatisticallyuncorrelatedwith A at that particular initial time (seeSec.4). Thisrestriction will beeliminatedin Sec. 10, but first we give two examplesto illustrate Bourret’s idea.

N.G. VanKampen,Stochasticdifferentialequations 187

9. Wave propagationwith randomrefractiveindex

Considera planewavepropagatingin the x-directioll in amediumwith refractiveindex n(x).As n is supposednot to dependon time onemaysplit off a factore”1°2 andwrite an equation

for the amplitude i,!i as a function of x

(9.1)

This first-order equationis obtainedfrom the actualwaveequationby neglectingscatteringintothe(—x)-direction (“parabolic approximation”or “quasi-optics”). That is a good approximationif n(x) variessmoothlycomparedwith the wave length.

Now supposethatn(x) equalsa constantplus a randompart,

(9.2)

with constantn0, smalla. andrandomlybut smoothlyvarying i~(x).We absorbthe factor wn0/c

in the variablex, so that the distanceis measuredin unperturbedwave lengths.Furthermore,weset ~!(x) = e~u(x)andobtain for u the equation

u’(x) jet i~(x)u(x).

This is the sameequationas (6.1) with x in lieu oft. Its solution is given in (6.5):

(~(x))= exP[i(l + a(~))x— ~a2ff ((~(x1)~(x2)))dx1~2 + }~(o). (9.3)

It hasbeenassumedthat r~(x)is astationaryprocesssothat (~?~is constantand simply addsto the

refractive index. The secondcumulantcausesa decreasein amplitude.Note that if theprobability

distributionof i~is symmetricalaboutzero,all oddcumulantsvanish,so that the phasevelocity isnot affected,but only the amplitude.The reasonwhy (9.3) could be obtainedwithout the assump-tion (8.5) is that (9.1) is a first order equationfor a singleunknown ~4i,so that no non-commutablematricesoccur.

Next we improveon the approximationinvolved in (9.1) by starting froni an actualwaveequa-

tion. Consider an electromagneticplane wavepropagating in the x-direction in a medium withdielectric constante(x) and permeability 1. It is sufficient to takea linearly polarizedwave withelectric field F alongthe y-axis andmagneticfield H along the z-axis.Maxwell’s equationsthenreduceto

dEiw dlliw(9.4)

It follows that E by itself obeysthe secondorderequation(2.2), but it is more convenienttomaintain thetwo-componentversion,which is theuniversalform (7.1) with x in lieu oft.

Again supposethat e(x) is a constantplus a random part

(9.5)

188 N.C. VanKampen,Stochasticdifferential equations

Tile relation with (9. 1). (9,2) is given by �~= n~and

(~)= 2(n) + a(~2). (~(x1)~(.v2))= 4(~(x1)~(x2)). (9.6)

wilen irrelevallt ordersof’ a are omitted. We cllOoseC1 such tllat (~)= 0. Again we :tbsorh tilefactor (w/c)~~0into ,v atid we set E = iii. I! = l12~~. Then tile specialform (7.9) applieswith

,0 l~ 00(9.7)

1 0’ 1 0

In order to appiy Bourret’s ititegral equationwe compute:

exp(A~v)= cos.v+ A1) 5i11,V. (d)5)

Tllis is simply acondensedway of writing the solution of’ tile unperturbedproblem. I ilsertlllg(9.7) and(9.8) into (8.7) we get after somematrix multiplications.

d ,0 0= A0(o(x)) + a2( (~(x)~(.v’)) sin(x - x’) (1d,v’))

dx 1 0 o

This nlay againbe written as a singleequationfor (li’) = (L

- ~ (u1(x)) +(l11(x)) a2f(~(.v)~(x))siti(.v ,v’) (ii1(v’)) dv’.dx

This equationcanbe solved by meansof Fourier or Laplacetransformation;one t’inds that thesolution behavesapproximately as a dampedplanewave,whosephasevelocity is also slightlycilangedby the fluctuations.We do not give tile explicit torlll becausea more convenientequationfor (u ~(t)) ~vill be found in the next section.and is solved ill Sec. 14.

Filially we apply Bourret’smethod in the propagationof electromagneticwavesin threedimen-sionsin a mediumwith dielectric constante(r) andunit permeability. Maxwell’s equationslead to

- - eE V(E’ Vioge). (9.9)

The right-handside is small coniparedto tile termson the left if

IVlogeI ~ (w/c)~ or IVeI ~ w/c,

that is. if e varies little over a wave length. Whenc is randomthiscondition mustbe imposedonthe largemajority of functions e(r) occurringin the ensemble.This is usually assumedto he true.

and as a consequeiice(9.9) reducesto threeseparateequations(2.2) for the threecompollentsofE. Whenmoreover� hastile f’orm (9.5) andc/we0 is chosenasunit of length the equation to bestudiedtakesthe forni

V2~+ ~ = ~a ~(r) ~. (9.10)

This is a partial differential equationandcannotreadily he written into our universalform (7.1).

N.G. Van Kampen,Stochasticdifferential eavations 189

Neverthelesswe sketchhow it canbe reducedto a sureequation for (11), becauseof the importanceof the problem.andbecauseit illustratesBourret’s idea.

The “unperturbed” equation

(9.11)

hasmany solutions.We selecta uniqueoneby giving a fixed incident wave. Let ~o be that solutionthat hastIle prescribedingoing part. Thenthe differential equation(9.11) can be rewritten as anintegral equation

~(r) = ~~(r) — afG(r — r’) ~(r’) ~(r’) dr’. (9.12)

Here G is the outgoingGreen’sfunction of (9. 11)

erG(r) =

47rr

This equationis still exact.A first approachto solving it would consistin replacing i/i(r’) underthe integralby 11i0(r’), whicil amountsto first order perturbationtheory in a:

~(r) = ~~(r) — afG(r r’) ~(r’)~~(r’) dr’.

Of coursethis simply yields for the averageamplitude(11.1(r)) = i1..’0(r). but the averageintensityis’~’

(I ~(r)I2) = I ~

0(r)I2 + a2ffG(r — r’) G*(r — r”) (~(r’)~(r”)) ~

0(r’) ~i~(r”) dr’ dr”.

This formula is the basisof most treatmentsof scatterilig by inhomogeneities,in particularthemonographsof Tatarski(1961)andChernov(1961), and aU the work on scatteringby fluids and

nearthe critical point**. It is valid when the total scatteringin R is so small that the attenuationof the incident wavemay be neglected.In otherwords therescatteringof the scatteredwavesisnot included;that is, we are dealingwith the Born approximation.

To obtain higherapproximationsonehasto iterate the equation(9. 1 2)

~(r) = ~0(r) — afG(r — r’) ~(r’) ~0(r’) dr’ + a2fG(r — r’) ~(r’) dr’JG(r’ — r”) ~(r”) ~(r”) dr”.

If one againreplaces11i(r”) in the last integralby 11i0(r”) the processesconsistingof two successivescatteringeventsaretakeninto account.Yet this still limits the validity to casesin which theincidentbeamis not much attenuatedby its passagethroughR. The ideanow is thatonecanim-proveon this and takethe attenuationduring the passageinto account,by replacing ili(r”) with(i1.i(r”)) ratherthan with i110(r)Thatleadsto an equationfor (11’>,

(~(r))= ~0(r) + a2fG(r — r’) dr’fG(r’ — r”) dr” (~(r’)~(r”)) K

For elegance,andto eliminate i1i~,apply the differential operatorV2 + I:

*As the result is of order 02 one should also include the cross-term between~ and the secondorder of ~. They have the effect

of attenuatingtheamplitudein theforward direction(“optical theorem”)but do not appear in the scattered waves.**Although in that caseoneis alsointerestedin time dependentfluctuationst(r, t), but they canbetreatedin a sintilar way (Van

Hove, 1954;Komarov and Fisher, 1963;Van Kampen, 1969).

190 N.C. Van Kampen,Stochasticdifterential equations

(V2 + 1) (~(r)) = a2f( ~(r) G(r - r’) ~(r’)) K ~(r’)) dr’.

which iS tile analogof (8.7). This is an integral equation. but the essentialpoint is tilat it is anequationfor tile averagealone.

10. The differential equation for the average

The ilypothlesis(8.5) is unsatisfactory.It hasbeencriticized by Keller (1962).who proposedtile name“disilonest” for all methodsin which the averageof’ a product is replacedby a productof averages,i.e., which are basedon an uncontrolledneglectof certain correlations.Yet it shouldbe recognizedthat tile statisticalmechanicsof transportprocesseswould be in a very sorry state

indeedwithout suchl hypothesesin tIle form of a ‘‘Stosszahlansatz’’,‘‘l’nolecular chlaosassumptioll’’,

or “random phaseapproximation . Ratherthancondemningtile hlypothleses(8.5) we shall try tounderstandwily it works.

Equation (8.1) determinesthreetime scales.The first oiie is the scaleon wit jell V varies;this 15

measuredby a~’(taking V to be of’ order unity). The secondone is tile scaleon which V(t) varies,andmay be calledr~.but it is not relevantfor the following. The tilird one is given by the correla-tion tinie r~of V; that is the time scaleon which the randomnatureof the function V(t) becomesappreciableThe expansion(8.2) wasunsatisfactorybecauseit involves successivepowersof’ at;we aim at an expansionin ar

5 . It will be shown that Bourret’sintegral equationis the first stepinsuchan expansion.

If etr~is small it is possibleto subdividethe time axis in intervals ~t, suchthat ~t ~ , andyeta i.~t~ 1. That is, V doesnot nary much during a time ~t in which V hasforgotten its past. Thenit is possibleto use the expansion(8.2) during ~t, witll (V(t)) insteadof’ V(0) on the right. Thisexpresses(V(t + it)) in termsof(V(t)). For tile next interval one may usethe sameexpansionagain to express(V(t + 2~t))in termsof(V(t + st)). The crucial point is that the valuesof V(t’)during tile secondinterval are practically uncorrelatedwith thoseduring the previous~t. Thatniakesit possibleto usein the secondinterval the sameunbiasedaveragesof’ V(t’), ratherthan theaveragesconditionedby the ktlowledgeilow v behavedin the previousinterval.

To put it differently. on tile coarse-grainedlevel determinedby ~t the processis (approximately)Markovian. In generalthat meansthat the probability distribution of V obeysa differential equation(the “masterequation”). In our caseit evenleadsto an equationfor the averagealone, owing tothelinearity of (8.1).

Bourret’sequation(8.6) is obtainedby using(8.2) to secondorder for expressingv(t + ~t) in

termsof V(t). That meantomitting termsof order (a~t)3andhigher.As the lower bound of ~t is

determinedby r~,it amountsto neglectingin (V(t)) termsof order(ar~)3.i.e., of orderetTc relativeto tile termsthat arekept. Thus Bourret’shold assumptioii(8.6) siniply amountsto neglectingtermsof order(ar~)3.

The readermay be surprisedto find that our argumentled usto expect a differential equation

for (V(t)) rather thanBourret’s integral equation.We shallactually derivesucha differentialequa-tion, but first anotiler stephas to be taken.

At the endof Sec.8 it wasremarkedthat (8.6), (8.7) still containthe initial time and thereforeare restrictedto solutionsthat are uncorrelatedwith A at that time. Sincewe now know thatA(t)

musthavea finite correlation time i~, it maybe expectedthat the influence of this initial time dis-

N.G. Van Kampen,Stochasticdifferential ec~jtionr -191

appearsafter a transientperiodof order r~,aswas actually found in (6.7). This will 110W be shownin general.

Changethe integrationvariable in (8.6) by settingt” t —

~(v(t)) = a2fKV(t) V(t — r))KV(t — T)) dT. (10.1)dt 0

Accordingto (7.6) onehas(V(t) V(t — r)) 0 fort> T~. aswe haveset (V(t)) 0. Henceassoonas t> no error is madeby extendingtheintegral to +00:

= a2f (V(t) V(t — T)%V(t — r)) dr. (10.2)

The initial time hasdisappeared,so that this equationappliesto all solutionsof(8.6), regardlessof the initial value andof the time at which the initial value is imposed,provided that a transientperiodof order ‘r~haselapsed.

The secondstepconsistsin showingthat (10.2) is equivalentwith the differential equationob-

tained by replacing(u(t — r)) in the integrandwith (v(t)):

~=a2~fVtvt_T)dr}V(t~. (10.3)

Sincethe integral virtually only extendsovera timer~,the relativeerror madeby this replace-ment is of order

d(V)r~

Accordingto the equationitself

d(v)a2r Ky).

dt C

Thus the orderof the relativeerror is(ar~)2.As alreadytermsof relativeorder etTc were neglectedin the derivation of(10.2), this error is of no consequence,andwe may replacethe integralequa-tion (8.6) by the differential (10.3). In the original representationthis amountsto replacing(8.7)with

~(u(t)) (Ao+a2f (Ai(t)exp(Aor)Ai(t_r))exp(_Aor)dr}(u(t)). (10.4)

This is thefundamentalresulton which mostofour application will be based. It has beenestablishedin severalways,two of which areoutlined in Sees.11 and 12. For otherapproachesseeKhas’minskii (1966), PapanicolauandKeller (1971),Deutch(1972),Cogburn andHersh(1973),BrissaudandFrisch(1974). It is possibleto extendit to higherorders,

192 v. C. Van Kampen, Stochasticdifferential equations

d(u(t)) = K(u(t)) = {K0 + aK1 + a2K

2 + a3K

3 + ... } (u(t)). (10.5)

dt

The K’s are matricesactingon tile componentsof(u); they niay dependon tinle but they do not

act asoperatorson the time dependenceof(u(t)). ‘file successivetermsare.oforder et(aT~Y

1

f’or n = 1 , 2 The presentreview.however,will be restricted throughoutto tile first nontrivialterm. n = 2, which in mostapphcationsis the only term of interest.

It should be cleartilat the differential equation (10.4) is no less accurate than the integral equa-tion (10.]), except during the transientperiod.Nor is its validity morerestricted,becausebotil re-quire ar~to be small. When that requirementis not met one cannotmakethe transition froni tileintegralequationto the differential equation,but at the sametime tile hypothesis(8.5) used inderiving the integral equationis invalidated.An extremeexampleof this eventis the problemtreatedin Sec.5, wherer~= ~ anotherexanlpleoccursin Sec. 15.

The existenceof a transientPeriodhas tile effect that tile average(11(t)). whlen extrapolatedbackwardsdown to t = 0 by meansof (10.4).will not in getieralcoincidewitil u(0) = a. To put itdifferently, the averageof thesolution of (7. 1) with iiiitial valueit(0) = a is a solution of (10.4)

with a slightly different initial valuea’. It is easilyverified that tilis mismatcil is of order(ar~)2.Inpracticetile transientbehavioris usually irrelevant,since in mostapplicationsthereis no well-de-fined initial condition. What is physically importantis that all solutions tendto obey thedifferen-tial equatio~i(10.4).As a coilsequencetilis equatioil hastile physical meaningof a renorniahizationof the unperturbedoperatorA

0. The reasonwhy we startedout by emphasizingthe initial condi-tion is that it is neededin order to define the solutions of’ the stochasticdifferential equation.

11. Keller’s expansionmethod

We haveseenin Sec. 8 that a straightforwardexpansionin powersof a is unsatisfactorybecause

it amountsto expandingin powersof at. We ilave seenin Sec. 10 that Bourret’sniethod providesa first stepin an expansionin powersof are. Keller’s method(Keller, 1964;Karal and Keller,

1964)may be regardedas a way to rearrangetile termsin the a expansionin sucha mannerthlat it

actuallybecomesan expansionin ar~ratherthan in at.The stochasticequation(7.1) may be temporarily supplementedwith an inilonlogeneous

“sourceterm” g(t), which is a surevectorfunction of t,

Lu~(~_A) ug. (11.1)

The stochasticoperatorL = — A(t; ~) doesnot only act on the componentsof the vector ii,

but also on its time dependence.Its reciprocalU’ is the Green’sfunction of the equation,andone

has

uL”g

(u)(L’)g

(L’Y’(u)g. (11.2)

,‘V.G. Van Kampen,Stochasticdifferential equations 193

Remember,however,that this averageis not well-definedunlessoneprescribes,by meansof aninitial condition or otherwise,which of the niany solutioiis of (11 .1) hasto be taken(Sec.4). Inthe formalism usedherethat meansthat onehasto specifywhich Green’sfunction should be used.In Keller’s application to the propagationof wavesin a randomniedium it wasnatural to takefor

g the incident wave, sothat the Green’sfunction at largedistanceconsistsof only outgoingwaves.Sincewe are hereinterestedin tile initial valueproblem.however,we set g(t) = a~(t)and take for

L”’ the Green’sfunction G(tlt’) definedby

~a ~i,—— A) G(tlt’) = ~(t — t’), G(tlt’) = 0 for t < t’.

Equation(11 .2) is merelyan exactniathematicalconsequenceof (11 . 1) andappliesevenwilen

r~is infinite. But the operator(L’Y’ involvesan integrationover all previoustimes down to theinitial time t = 0. The problem is to eliminatethis infinite memory,which of courserequiresafinite T~.

In order to find a moreexplicit form of (11.2) we introducean expansionparametera with theaid of(7.2) and set

L=L0+aL1, L0=a~—A0, L1= —A1(t;~).

A0 is sure;for conveniencewe takeA0 independentoft and (A1) = 0. One easilyverifies the ex-pansion

L’ = L~’— aL~’L1 L~’+ a2L~’L

1 L~’L1L~’+,,,.

Hence,to secondorder in a,

(L”1) = L

01 + a2L~’(L

1L01L

1)L~, (11.3)

= — a2(L

1L~’L1). (11.4)

Now the unperturbedGreen’sfunctionL~= G0(tlt’) is given by the solution of the unperturbedequation

G0(tlt’) 0(t — t’) exp{A0(t —

where0(t) is theHeavisidestepfunction.Combiningthis with (11.1), (11.2),and(11.4)onefinds

to secondorder in a

(-~- — A0)(u(t)) = a2f(A 1(t) exp{A0(t — t’) }A ,(t’)) (u(t’)) dt’.

This is againBourret’s integralequation(8.7).So it seemsthat from the exactequation(11 . I) Bourret’s result can be obtained by a formal

expansionin a,without evenmentioningthe correlation timer~.The following testcase,however,will show that this is deceptive.Takethe seconditem in Table I,

U = —i(c + ay)u, .P(y) = (2ir)”1”2 exp(—~v2).

194 N. C. VanKampen. .S’tochasticdifferential equations

Rememberthat T~ = 00 aIld that tile exactsolution is

(ii(t)) = exp{---ict --- a2t2 ‘11(0).

Now first observethat Bourret’s integral equationtakesthe form

d I -

= --icKu(t)) --a2 fexp’, -ic(l -- t’)~ (lift’)) dt’cit

and is readily solved to give

(11(1)) = ec cosa t’ Ii) 0).

Tllis doesagreewith the exactsolution to order a2. hut the over-all behaviorof’ the solutionis

grosslydifferellt, owing to tile f’act that tile Iligiler ordersin a are also higherordersill t. Thus cx-hibits the evil of cutting off an expansioncontainingsecularterms.

Wheredid tilese seculartermsenter?To seetllis we write the exactGreen’sfunction for tilis

1110del,

(L~) (G(t~t’)) 0(t — t’)exp~ ic(t-- 1’)- ~a2(t -- t)2}.

The expansion(11.4) chlangesthus illtO

(G(t~t’))= 0(t --- 1’) exp~ ic’(t t’)} [I -~ ~a2(t - t’)21.

and it is clear that the omitted terms aresecular.The conclusionis tilat. although Keller’s metilod leadsto a final result which happensto be

correctwllen ~ is finite, it doesnot avoid secularternis in the interniediatesteps.That is the rea-son why it leadsto a wrong result wilen r~is infinite, in spite of the f’act that no condition onis evermentioned.Since tile term “dishonest” hasbeenpreempted,I proposethe term “illegal”for derivationswhichl concealthe limited validity of’ tile result that they purport to derive.

1 2. The cumulant expansion

In this sectiona method is given which avoidsboth Bourret’sassumption(8.5) and Keller’ssecularternls (Kubo, l962a; Freed. 1968, 1972;Van Kampen, 1973a, l973b;Terwiel, 1973).The

idea is to expandin cumulantsasdescribedin Sec. 6. Unlike the examplein Sec. 6, however,wenow haveto dealwith non-conlmutingi’natrices ratherthan with the scalarfunction ~.

Tile equationto be studiedhlas the universalform (7.1). There is n~needto separateA explicitlyinto a surepart andstochiasticterm. but it will be supposedthat the magnitudeof the fluctuationsis still measuredby a parametera. For eachiiidividual valueof ~ tile solution of (7.1) may bewritten in termsof a time orderedproduct,as usedfor the unperturbedequationin Sec. 7.

u(t) = ~exp~fA(t’) dt’}l u(0). (12.1)

N.C. VanKampen,Stochasticdifferentialequations 195

That gives for the average

(u(t)) = r(exp(f A(t’) dt’ } ~1 ii(0), (12.2)

since averagingandtime orderingobviously commute.Furthermore,inside the time orderingsymbolsonemay freely commuteoperators,becausethey haveto be put ultimately ill chronolog-ical order anyway.Hencethe samecumulantexpansiontheoremas ill (6.5) may be employed,

‘(11(t)) = [exp(fK(A(ti))) dt1 + ~ff t1)A(t2))) dt, dt2 + , )1~~(0) (12.3)

This formula is the startingpoint for the coming developments.Notice that, apartfrom the firstterm, the successivetermsin tile exponentareof order a

2, a3 Moreover,they all grow linearlywith t when t ~ r~.asarguedin Sec.6. It is understoodthatA(t) hasa finite correlationtime r~in tile sensethat all cumulantsof matrix elementsof A vanish,

((A~(ti)A~K(ts)... Ap0(t,~))) = 0,

assoonas the time argumentsin them havea gaplargecomparedto T~.The successivetermsarethenof order amtr~

Actually oneoughtnot to talk about the successivetermsin (1 2.3) asseparatemathematicalobjects,becauseafter expandingthe exponentthe time-ordering’will breakthiem up andmix theoperatorsthat occur in them.Yet it is easilyseenthat the abovestatementsaboutordersof mag-

nitude remaincorrect.Our next taskis to reducethe highly symbolicalexpression(1 2.3) to ausableequation.

As a first stepwe omit all termsof ordera2.

(u(t)) = [exp~J(A(ti)) dti)lu(0). (12.4)

By definition of the time orderingsymbol this is the solution of tile differential equation

d= (A(t)) (u(t)), (u(0)) = ri(0). (12.5)

Thus this first stepleadsto a first approximationto the operatorK in equation(10.5),

(A(t)) = K0(t) + aK1(t).

WhenA hasthe form (7.2) onehasof courseK0 = A0.To preparefor the nextstepswe utilize thisresult for definingan interactionrepresentation.

Let Y(tlt’) be theevolution operatorbelongingto (12.5);setu(t) = Y(tIO)v(0)as in (7.6).andaccordinglytransformA(t) into V(t). In this representation the equation (1 2.4) takesthe form(8.1),andits solutioncanbe expressedformally in analogywith (12.3)

196 S.C. Van Kainpen..S’toehastic differential equation.)

a2 -

(v(t)) = [exp ~- fJ (K i”(t) V(t2ft) dr dt2 + ... 0). (1 2.6)

‘fhie secondstepcotlsistsin omitting the terms of ordera3 and Ilighler ill tile eXpOllellt. vu., tile

termsindicatedby the dots. In principle oneshould now expand the exponentialalld in eachtermof the expallsionrearrangetile operatorsI’ chronologically. In partial f’ulf’ilnlcnt of tilis require—

nient we write

(v(t)) [exp~a2Jdttf dt2 ((i~7t1)V(t2)))~1 u(0). (12.7)

Define tile operator

L(t1) = dt2(( V(t~)Vt2)>).

and considerthe dlffercntial equation

d (V(t)) = a2 L(t) (VU)). (12.8)

Its solution is

(V(t)) = [exp~a2fL(tt)dtt}1 V(0). (12.9)

Tllis expressionis ahmostidentical with (1 2.7). but not quite: it will presentlyhe shownthat thedifferenceis of higher order in are.

First we investigatethe collsequcncesof substituting(12.9) for (12.7).Thlen (V(t)) obeys(12.8),which yields in tue original representation

-~ (u(t)) = ~K0(t) + a K,(t) + a

2 K2(t) } (11(t)). (12.10)

dt

HereK2(t) is the operatorLU) transformedback to the original representation,

K2(t)= Y(tIO)L(t)Y(OIt)

=Jdt’ Y(tIO)((V(t) VU’))) Y(OIt)

= dt’ A(t) Y(tlt’)A(t’))) Y(t’It).

NC. VanKampen, Stochasticdifferentialequations 197

Here Y is still the evolution matrix belongingto (1 2.5). Whien AU) = A0 + aA 1(t) with(A,(t)) = 0 onehas

Y(tlt’) = exp~A0(t—

In that caseour result (1 2. 10) may be written moreexplicitly

(u(t)) = A0 + a2fdr ((A 1(t) exp(A0r)A ,U — r))) exp( A0r) (Ut)). (12.1!)cit

0

Again whien the t ~ r~the upperlimit of the integral may he taken +00, and the result is identical

with (10.4).No hypothesisaboutabsenceof correlations wasneeded,nor hai’e secular termsbeencut off, to arrine at the equation(10.5) to secondorder in a.

But we still haveto justify tile useof (1 2.9) in lieu of (1 2.7). The diff’erencebetweenthieni is

that on expandingthe exponentialill (1 2.7) andapplying the time orderingit may happen tilatthe two operatorsV(t,) and VU2) in the cumulant areseparatedby oneor moreoperatorsfronithe other factors;whereasin (12.9) tue operatorsV ill a cumulant stay togetherto make L. and

only the operatorsL are time orderedamongthemselves.To estimatethusdift’erence. take atypical term of tile expansionof (I 2.7),

~ [(J~~1J dt2 ((V(t1) V(t2)))} 1, (12.12)11! 0 ~

There are n pairsof tinie arguments(t,, t2), andonehasto integrateovera 2ii-dirnensionaldomain.There is a subdomainwhere no two pairsoverlap, andwhich theref’oreis correctly repre-sentedin (12.9).The volume of that subdomainis of order t’

1r~.becauseeachpair rangesover aninterval of order t, while the pair t

1, t2 must be within r~from one another.Tue contribution ofthis subdomainis thereforeof’ order a

2~t’~r~’.In the remainingsubdomainthe orderingof the operatorsV is not the samein (12.7) as in

(1 2.9). However, if two pairsilave to overlap,all four timesinvolved mustbe within a distailceoforder r~from oneanother,so thlat the rangeof integrationfor thosetwo pairs is only of ordertr~.Hencethe differencebetween(12.7) alld (12.9),as far as the term (12.12) is concerned,is atmostof order a2~t° ~ ~.As this is proportionalto t° ~, it has to be comparedwith the prin-ciple contributionof the previousterm in the expansionof(l 2.7), which is a2’°2t°’ r~ Thisshowsthat the differencebetween(1 2.7) and(12.9) is of ordera2T~.

Thesehigherorder termscan be evaluatedanda generalexpressionof all ternis in the expansionof K hasbeenfound(Van Kampen, l973a, 1973b).An alternativederivationhasbeengiven byTerwjel (1973)andEmid (1 974). As announcedin Sec. 10, however,the higherordersareoutsidethe scopeof the presentreview.

1 98 ,\. C. VanKampen .5’ tocha~ticdifferential equations

1 3. The random harmonic oscillator — General remark

Numerousauthors” Ilave studied tile harmonicoscillator with fluctuating frequencyto illustratethe effect of ranciolll coefficientsin differential equations,in collnection with mechallical sys-

tems, turbulellce. wave propagatlon,line broadening.and lasers.Mainly two modelshavebeenused.Tile first model is describedby the equation

(13.1)

with randomw. or in our universalnotation.

d /X\ / 0 1~IXI = I I . I. H3.2

dt\xt w2 0!\.v

‘[lie secondmodel is describedby a single equationfor a complex variablez.

(13.3)

whicil is tile equationstudied ill Sees.5 and6. When w is constant(independentof time) tileseequationscanof coursetrivially be transformedinto oneanotiler by the substitution

w.v + ~= - a--. (13.4)

~2w

We herewant to stresstile f’act that (lw involvesrandomfluctuations(13.]) and (13.2) are nolonger equivalent, and in fact many niore ‘‘random harmonicoscillators’’ exist, wilicil are notequivahentwith eachother.

It is obviousthat tile substitution(13.4) no longertransforms(13.3) into (13.1) whenw is afunction of time. In fact, it is no longerpossibleto write (13.1) or (13.2) as a first order equatioll

for a single complexquantity.The decisionwhich equationshouldbe useddependson the situa-tion to be described.Actually it turns out that often neitheris correct,but thiat a randomtwo-by-two matrix is requiredof the generalform

/ 0 l\ /Ei ~2\A

0=I I, A=I I , (13.5)\ —h 0! \~3 ~41

with four randomfunctions ~.

As an exampletakea pendulumwith massI, length 1, subjectto a randomlyfluctuating gravita-tional force g. The equationfor its angle~ with the vertical is

gi~= ----1ip,

*The following list is not exhaustive: Kraichnan (1961);Bogdanoffand Kozin (1962);Caugheyand Dienes(1962);Kubo (1963);

Glauber(1965); Louisell andWalker (1965); Bourret (1965, 1971);Weidlich and Haake (1965); Lax (1966); Oppenheim et al,(1967); Klauder and Sudarshan (1968);Papanicolauand Keller (1971); Fox (1972);Soong(1973); Bourretet al. (1973);Van Kampen (1973, 1974a, 1974b); Klyatskin and Tatarskii (1974).

N.G. VanKampen,Stochasticdiff/rential equations 199

which hasthe form (13.1). However,suppose1 is a random function aswell, due for instancetoheatt’luctuations in the string (Fox, 1972). In that casethe Lagrangefunction

L(~,~, t) = ~(l2~2 + 12) — gl( I — cos~)

leadsto tue linearizedequationof motion, for ~

0dt ~ I - -g/l -21/1/ \~‘

This is not of the form (13.2), let alone(13.3), butof’ courseis coveredby the generalform (13.5).As a secondexampleconsidera two-level atom with level distance~ subject to a random

electric field E. The Hamilton operatoris a specialcaseof (2.4)

(0 ~)+E(t)(° ~), (13.6)

wherea is thematrix elementof the dipole momentP. Again this cannotbe cast into the form

(13.2);andiii termsof the variablez it takesthe form

z=—~i~z+{~i~~~aE(t)}z*,

which’is not (13.3). Henceequation(1 3.3) is not a properdescriptionof line broadeningby arandomelectric field. In fact, I kiiow of no randomharmonicoscillatorwhicil is properlyde-scribedby (13.3) with randomw, exceptapproximatelyas in (9.1).That was tile reasonwily inSec.6 dipoleswerechosento illustrate the equation(6.!).

Evenwhenw is a randomnumberconstantin time, as in Sec. 5. onecannotuse(13.1) and

(1 3.3) interchangeably.The reasonis that, althoughthe equationsareequivalent,the correspond-ing initial valueproblemsarenot. Prescribingfixed valuesfor x(0) and *(0) is hot equivaleiit toprescribingz(0), inasmuchas the connectionbetweenx andz involvesthe randomnumberw, see(13.4). The stochasticprocessdefinedby (13.1) or (13.2)wth fixed initial i-’aliies is not thesame

as thatdefinedby (13.3) andfixed initial value. Onecannotevenescapeby sayingthat one is onlyinterestedin the differential equation,becausewhenw is independentof time the initial valueinfluencesthe behaviorat all later times.Thesefactscan easilybe verified by an explicit calcula-tion similar to the work in Sec. 5. They are the reasonwhy in Sec. 5 dipoleswere chosento illu-

stratethe equation(5.1).The moral is that onecannotgive a stochasticdescriptionof a physicalsystemsimply by taking

oneof thefamiliar equationsfor the sure caseanddeclaringsomeof the coefficientsto be random.One hasto start from the fundamentalequations,taking into accountthat the coefficientsdependon time. This moral appliesto all applications,not just harmonicoscillators.For instance,in thewaveequation(2.2) the refractiveindex n maybe takento be a randomfunction of x in the casethat the fluctuationsaredueto � alonewhile the permeability/1 is constant(Sec. 9). If, however,ji also fluctuatesit is no longerpossibleto reduceMaxwell’s equationsto the form (2.2). Similarlyin threedimensionsthe exactequation(9.9) only reducesapproximatelyto the waveequationwith fluctuatingrefractive index(9.10).

200 .V. C. Van Kampen, Stochasticdifferentialequations

14. The randomharmonicoscillator —- Calculations

Bearingtheselimitations in mind, we silall now apply tile result 110.4) to equation (13.2) with

a stochasticfrequency

(14.1)

where~(r) is a stochasticIil’ocesswith zero meanandcorrelation time T~ . Note that this is thesameas the equation(9.4) for propagationof a planeelectromaglleticwave ill arahidonl medlidihil.

Tile fact that it is of secondordernlakesit essentiallymore ciiff’icult thaii the equations(1 3.3)or (9.1), whicll were solved in Sec. 6. We could haveincluded a dampingterm in (13.2).hut thatwould maketile algebramorecomplicatedwithout beingmore illuminating.5

For the presentproblem eqdlatiOtl (1 0.4) takestile f’orm

d(u(t)) = -

= ~A0 + a2f (~(t)~(t — r)) B exp(/1

0r)B exp(- A0T) di (11(t)). (14.2)

where/10 andB are tile sameas in (9.7). Witil tile aid of (9.8) one finds

/0 0 \13 exp(A0r)B exp( - A 0r) = I . ‘ 2 (14.3)

\ sill i cosT - sin T

1

lt remainsto computethe two coefficients

= f (~(t)~(t— r)) sin 2r di (14.4)U

= f (~(t)~(t --- r)) (I - cos 2r) dr. (14.5)

The latter one is readily expressedin the f’luctuation spectrum(4.3) of ~,

c2~{S(0) S(2)}. (14.6)

The otheronemay be written c2 = ~S~(2), whiere

2 1 = S(w’)S~(w)--f (~(t)~(t— i)) sin wr dT= — f —~- — dw’

‘iT ir w w

15 tile Hilbert transforniof S (Titchmarsii. 1948).Insertingthesequantitiesinto (14.2) one obtains

*The harmonic oscillator with damping hasbeen treatedby means of Bourret’s integral equation (Bourret, 1965. 1971)and by the

method of our Sec.23 (Bourret et al., 1973; Kubo, 1969).

N. C. Van Kampcn, Stochastic difti’rential equatIons 201

d (x) (1 0 l\ a2/O 0 ~(x)HI I+—( I .

dt (.v)~ ~\ —l 0! 2 \c1 —c2 \(.v)

In a niore familiar form thus final result is

+ ~a2c2~2+ (1 — ~a2c

1)(x) = 0. (14.7)dt

2 dit

Thus tile fluctuations in the frequencycausea dampingof’ tile averageamplitude.According to(14.6) this dampingmay be negativewilen the fluctuationsareparticularly strongat twice the un-

perturbedfrequency.There is also a shift in the frequencydeterminedby the Hilbert transformofS. Tilese two effectsare quiteanalogousto the iiatural line broadeningahid energysluft in the

theory of’ scattering.In contrastwe computetile averagedsolution of (13.3).with

w(t) I +a’i7(t), (14.8)

wh’iere 11(t) is relatedto ~(t) by (9.6). The solution is obtainedby substituting(14.8) into thegeneralresult (10.4) (modified to take into accountthat (ri) ~ 0. or elsedirectly from (9.3)):

(z(t)) = exp~—it— ia(11)t — a2f (11(t) 11(t — r)) di) z(0).

In termsof the process~ thusbecomes.with the aid of (9.6).

(z(t)> = exp i—it — ~a2(~2)t — a25(0) t) z(0), (14.9)

It appearstilat the dampingis dif’ferent than found in (14.7), viz..

ira2 ira25(0) versus -—~ ~S(0)— S(2)~.

Thus tile dampingcannotnow becomenegative,owing to the lack of the term that representsthe

resonancebetweentue fluctuations and the double f’requencyof’ the oscillator.Thus denionstratesthat (13. 1) and( i 3.3) are niaterially diff’erent wilen w has randomfluctuations.

The reasonfor this resonancewith the double f’requencvcan be bettervisualizedin the caseof a

planewave in a mediumwhoserefractive index variesrandomly with x. Thesevariationsgive riseto a scatteredwave(order a) in the oppositedirection, which in turn is rescatteredinto the for-warddirection(ordera2). Eachi of thesescatteringprocessesconsistsiii changingthe wavevectork into -—k or vice versa,which requiresFouriercomponents2k in tile refractive index.This effectdoesnot occur in the simplified equation(9.1), the analogof (13.3). as it is an approximationforthe casethat the variationsare sosmooththat suchcomponentsdo not occur. In fact, one easilyverifies that if (~(t)~(t — r)) variesslowly with r the equations(14.5) and (14.4) reduceto

c2~ f(~(t)~(t_r)dr. ci~

202 ‘s. C;. VanKampen,Stochasticdifferential equations

In tllis approximation(1 4.7) beconlesequivalentWitll (14.9).The readermay wonderhow spatialvariationsof tile ref’ractive index can result in a negative

damping.The answeris tilat our initial value problem is not a descriptionof what ilappenswhiena given wave impingeson a medium. All we haveshown is that if both incident and reflectedwavesare given at x = 0. the wave inside tile medium hasto grow expotientiallv with .v (whenC

2 < 0).Of’ coursefor a tinie-dependetitproblem theseare tile proper initial conditions,but in that case

tile negativedanipingdueto tluctdlationsis not paradoxical.

15. Diffusion iii a turbulent fluid

A fluid, for instancewater, is moving with a local velocity u (r, I). The motion is supposedto be

turbulent,so that is too irregulara furlction to he known. bdit its stochasticpropertiesare sup-

posedto be describedliv reasonablyknown functions. We supposethat (u(r, t)) = 0 and

(v~(r,t) v1(r’, t’)) = 111(r -- r’. t t’). (15.1)

In this flLud a small cloud1 of extraneousiiiolecules, for instanceink, is insertedat time t = 0. Their

densityn(r. t) obeystue stochasticdifferential equation( 2.3). This eqdlationhias beenstudiedbymany authlors*, btit will servehlere mainly to demonstratetue hinlitations of’ our expansionin T

5

Althloughl (2.3) is a partial diff’erential equation.it is of the universal type ( 7. 1). wilen r istreatedas the label v that distinguishiesthe conlponents.The hllatrix A consistsof a sureand arandompart.

A0DV2. aA

1Vu.

Note that the symbol ~ should be read asan operatoractingon every function of r tilat appears

to the rugilt of it.The generalresult (1 0.4) takesthe form

-- (n(r, t)) = (DV2 + fdr( V’u(r, t) exp(rDV2)V’u(r. t — r)) exp(_TDV2)) (n(r, t)).

In order to work out the operator { ) we take Fourier transfornisin spaceand obtain in obviousnotatioIl

(n(k. t)) = -Dk2 - (2irY~~J’drff(v1(q. t)v1(q’. t — i)) X

(k1+q1+q)exp~---rD(k+q’)2l(k,+q)exp(rDk2)dq dq’) (n(k.t)).

*Ior literature seethe footnole in Sec.2.

V. C. VanKampcn.Stochasticdifferentialequations 203

Oneeasily f’ihlds froni (15. 1)

(v1(q. t) v1(q’. t — r)) = 6(q + q’) (2ir)3’2 F~

1(q.r).

Hencetile aboveequatioii reducesto

— (n(k. t)) = -Did2 -- (23/2~ Jdrfr

11(q. i) k1(k1 --- q1) exp( —TDq2+ 2rDk .~)d~}(n(k.t)).

0 (15.2)

in principle thus solvesthe problem.If oneknows F onecan perform tile integration and obtainadifferential equation for (n(k, t)). If F vanishessuf’ficiently rapidly for r one~iiay eXtelldhthe integratioll to t = ~. so that the differential equationis independentof the initial time. More-over, if tue turbulenceis isotropic. the integralcontanisa (‘actor k2 and tile result may be writtell

a—(n(k, t)) ~--D(k)k2(n(k, t)), (15.3)at

wllerC D(k) is a renormahized,k-dependentdiffusion constant.Unfortunately thereis a difficulty. Although tue actualF for a turbulent fluid is not known,

hydrodynamicsof incompressiblefluids showsthat for smallq and larger

2/ q1q1\F11(q. r) ‘~ ‘y(q )y

5i1 — exp(—vq~r),

q

wherev is tue kinematicviscosity of the fluid. Hencethe mtegrandof (15.2) behaveslike

exp—[(D+v)q2-.- 2Dk’qlr

and fails to convergefor T -~ when Iql < 2Dk(D + r~)i. The integral cannotbe extendedtot = and thereforeno equationof type(15.3) exists.The physical explanationis that tile low q.long wave length fluctuationsin u areinsufficiently dampedby the viscosity to ensurethe exis-

tenceof a finite ‘re. Hencethe correlationsof n(r, t) also extendover infinitely long times.and 110

equationfor (n(r, t)) existsin which the initial time is no longerrelevant.The samelong memoryis responsiblefor the long time tails in thevelocity auto-correlationfunction of the moleculesof’a fluid (for literatureseeDorfman, 1 975).

16. Higher moments

So far only the averageof the solution of our stochasticdifferential equation(7.1) hasbeen

studied.It will now beshown that the sametheory permits to find highermomentsas well. Firsttake the secondmoments.When then componentsu

0 of u obey (7.1). then then2 quantities

alsoobeya linear differentialequation

(u~u~)= ~ ~~t) ‘(içu~), (16.1)

204 \. IL Van Kantpen.5toehasttedifferential equations

where5

= /17,5 ö05 +

By determining tile averageof tilis equationwith initial condition 11.(0) 1e~(O)= a,a0 oneobtainsthe secondmoments(u,( I) it~(1)) (Adomian, 1 970, 1 97 1). In tile sameway higiler moments ofId t) can be toutid, althoughithe required! laborgrows rapidly. Fourtil momentsIlave beenusedl fordescribingthe mean squareof’ the intensity distribtition of a light beampassingtlirough’i a focussingsystemwitil randonl inlperfections( Papanicolaou et al.. I 973; McLaugllhin, 1975). A more power-liii approach,\vhichl directly leadisto the entire proh;tbilit~’distributioti of lIft) will he given inSec. IS.

It silould be remarkedthat although (1 ~. I ) is a set of /12 equations(‘or ,i2 quantities it is ilOt

really a diff’erential systemof order ,12. becauseowing to the symmetry relations~ = u~u~

thereare only ~n(n + 1) litlearlv independentquantities.Moreover, owing to tile quadraticrela-tions (11,117,)~ 1i~) = (tilt0 )2 actually only it are itidlependlent. ti agreementwithi tile (‘act tilat(1 6. 1) containslid) more solutions tllan tile original edluation (7. 1). For the averages,however,thesequadraticrelationsdIO not holdi. and therefore~n(n + I ) linearly nidepetidentSOlLitiOils fortile secondmomentsare possible.T’hus illdistratestile discussionin Sec.4. wilicil showedtllat a

stochasticdif’f’erential equationhasa largerclassof solditionsthan a stire equation of the sameorder.

When it is conipld’x~,as for examplea quantdinlmecllanical wave function, one is usuallyuiter—

estedin tIle qdiantities (i1~l10) ratherthan (l1,1i~).They also obey a set ol’ linear equationssimilarto (I 6. 1). In (‘act. tiley are simply the elenletltsof the density nlatrix and the equationis identical

with the f’anuliar Neumailnequation.Tile secoild order approximation(1 0.4) for the (u~’u0) iscalled in this cotlnectlon the Redlf’ield eqdiation “‘

As ail examplewe computethe averagectle’rgy ol the ilarmonic oscillators in Sees. 1 3 and 14.For the model (I 3. 1) it is ohviodis tilat

w2x2 +

= — — = constant,2w

which statesthat the averageenergy-~(x -I w2x2) doesnot vary in time. Only the phlasebecomes

progressivelymore random,wilichi cadisestile averageamplitudeto decayaccordingto (14.9).The model (1 3. 1) is less trivial. The equation for tile threesecondmomeilts is found from

(h3.2

0 0 2 ‘s~2

=~ 0 0 2w2 ~2 =~ ~2dlt~ ~.

xx / w~ 1 0 / x.~/ “,

This is equivalentto (16.1), hut we haveeconomizedby writing on1y the ~n(n + I) hinearhymdc-

“~wouid be calleda ‘‘supcroperais)r” in lirussels.

**Redfieid (1957, 1965). Sec also: Slichtcr (1963); Argyres and Kelley (1964); Aleksandrov(1966);Cukier and Deutch (1969);

Aibers and Deuich (1971):and various contributions in Muus and Atkins (1972).

N.C. Van Kainpcn,StochasticdiffLrcntial equations 205

peiideiit momeilts. Inserting(14.1) leadsto a decompositionof ,a~in a sureaild a i’andoni part

0 0 2:, / 0 0 0\

0 0 -2 . ~i ~(t) 0 0 ---2 ~(t)~.

- 1 1 0) --- 1 0 0 1

~ has two eigenvalties±2i. correspondingto the eugeilvaluesof’ A0. These .~ ~ aild yl arenowsubstitutedin (10.4) with the result

(.v2) ~‘ 0 0 2 ‘, (.v2)

dl (x2) = a2c3 —-a

2c’2 ---2 (.v

2)

dit

\ (xx) / \ - I + a2c1 I ---a

2c2 / \ (xx)/

where C’~= ir~S(0) + S(2) }.

To investigatethe consequenceswe determinethe eigenvaluesto secondorder in a:

= ±21(1 ~a2ci) ~a2(3c2+ c5)16.2)

= ~a~(c5 -‘ c2).

Oneseesthat tile eigenfreqtiencies±2i of’ the unperturbedoscillator ilave undergonea shift andh,in addition, a dampingappears.The zero frequencyof the unperturbedcase.\vilichl correspondedto conservationof the energy ~(~2 + ~2) llas now becomeX0 = ~ira2S2), Thus the eiiergvgrows.owing to thosefluctuationsin the force that Ilave twice the proper frequencyof the oscillator.Hencealthough theaverageamplitudeis dampedthe oscillator is unstableenergy-wise.It’ we hladhincluded iii our unperturbedequation(13.1) a (sure) dampingterm. the result would be that theoscillator is stablewheii a

2S(2)is below a certaincritical valueand unstableaboveit (Bourret.1971). A harmonicoscillator with fluctuatingdaniping term has also beenworked out ( Van

Kampen. I 973b).

1 7. Randomly located scatterers*

The wave equationin onedimensionin a homogeneousmedium may be written, in suitable

variables,

duA

0u.dx

with constantniatrix A0. Supposeat thepointsy~,y2 v5 scatteringcentersare located;eachscattererconnectsthe componentsof u on its right with thoseon its left by a transmissionmatrixC

u(y0 + 0) = Cu(y0 — 0). (a = 1, 2,...,s). (17. 1)

*l~oldy(1945); Lax (1951, 1952); Twersky (1964);hibbink (1971).

206 N. C. Van Kampen, ,S’tochasticdifferentialequations

This is not a differential equationfor u, so tilat tile method of’ Sec. 12 cannotbe applied straight-away. Yet the problenl can be ilandled in a very similar manner(Ubbink. 1971 ).

Tue valueof it(x) is related to 11(0) by the explicit f’ormula

u(x) = exp{Ao(x - ‘s~) }CexpfA(i5 -- ~ f/C... Cexp~Ai’1}u(0),

taking 0 < i’ < y~< .. < ‘i’~ < .v. Define tile interactionrepresentation

11(x) = exp(A0x)v(x). exp( -A55i’0 )Cexp(A0r0) =

In tllis representatioilthe explicit fornlula f’or it(x) takestild’ forlll

v(x) = C(y5) C(y5 ~) C(i’1)v(0) = r01 C(,v0)1. (h7.2

wheretile orderingsyniholsindicate that tile operatorsin tile producthaveto be taken in orderof descendingvaluesof their argumentsi’. If the effect of eachscattereris weak it is naturalto setC I + aB. sothat

v(.v) =r0’~,~+ aB(vo)/1.

Now supposethat the scatterersare locatedat random,sothat tue j’~ are random(luantities, aswell as tIle numbers.Suchia set of randomdotson a line can be describedby a huerarchyof dis-tribdition functions t;~(n = I , 2, ...) definedby (Stratonovicil, 1 962)

f0(x1, v2,.., x~)~ dx2 ... dtv,7 = probability that eachof then intervalsx,...v~+ dx1 containsadot, regardlessof the location of the remainingdots.

The probability for more tllan one dot in x7.,x + dx1 is of iiigher order in dx1 andthereforenegligible.The f0 are densities.hut not probability densities.becausethey do not integrateoutto unity. f’i(x) is simply the densityof clots,f2Cx’ .x2) tells somethingabout their correlation.Onedefinestile correlationfunctionsg07 by a clusterexpansionsimilar to (6.6)

J’i(xi) =g1(.vi)

f2(xi, x2) = gi(xi)gi(x2) + g2(xi, x2)

etc.

Again we assumethat thereis a correhationlength x~.such that: if {x1, x2 x0 } containsa gaplarge comparedto x~.theng~0(xj,x2 x,5) = 0 (“cluster property”) andf~factorizes(“productproperty”).

Knowing how to describerandomlylocateddotswe cannow work out the averageof (17.2).

N.C. Van Kampen,Stochasticdifferentialequations 207

(v(x)) = r({l + aB(y5)}~h+ aB(y51)} ... ~h + aB(Yi)})1

= [I + a~(B(y5))+ (B(y51)) + ... + (B(y~))}+ a2 {(B(y

5) B(y51)) + ... } + a~{...} +

= [1 + afB(x)f1(x)~+~ ffB(xi)B(x2)g2(xi~x2)~i dx2 + ~

The essentialidentity analogousto (6.5) statesthat this may againbe written asan expansionin

the exponent:

(v(.v)) rexP(a[B(xi)~i(xi)~i+~ jfB(xi)B(x2)~2(xi~x2)dxidx2+...)~.

At this point we arebackon the trackof thederivation in Sec.12, comp. (12.3),and maythereforejump to the final formula(12.11)(to ordera

2x~):

(u(x)) = (A0 + aBgi(x) + a2/ B exp(A0r)Bexp(--A0r)g2(x,x — T) di) (u(x)). (17.3)

For a stationary distribution gi(x) = constant= f~andg2(x, x — r) =

The abovetheory is dueto Ubbink (1971), who appliedit to the stochasticdescriptionof aphotoconductor.Let ii0(t) be the probability that therearen electronsin the conductionbandattime t. In the absenceof photonsthereis a certainprobability per unit time, r(n), thatone of then electronsdropsbackinto the valenceband.Hence

/10=—r(n)u0+r(n+ l)u0÷1. (17.4)

This equationhasthe form /~=A0u (with infinitely manycomponents).Supposea photonentersat timey; then

u0(y + 0) = u~(y— 0) — a u0(y — 0) + a u~~(y— 0),

wherea is the quantumefficiency.Thisequationhasthe form (17.1)with C I + aB.In order to apply (17.3)onehasto solve the unperturbedequation(1 7.4). As no explicit solu-

tion for generalr(n) exists,we taker(n) = n. This correspondsto thesomewhatacademiccasethatn is small comparedto the numberof unoccupiedstatesin the valenceband.The exphicitsolution is then

Im~(e~

40t)nm= (, )e~t(h—

With the aid of this resultandsomealgebra(17.3) becomes

d(u,~)= —n(u

0)+ (n + l)(u0+i)—afi(u~)+ afi(u0.1) + a2c{(u~)— 2(u~~>+ (u~

2)},

208 V. I . I ‘cit Kern pen Stochastic differential ~‘qiialto? is

wi crc

= 1’ ~ r~2~ dli.

From this result oneobtainsfor instancethe eqduhihridunprohahihit~’d!istrlhlition of’ the num-

her of chargecarriers. Ratller than wri tutig that expressionexplicitly we use it to f’indi tileequilibrium f’Idictciations:

(/12) - (~)2 = (n) + a2c.

For c = 0 thi~reducesto tile Poissonformula, tiit’ clc’i’iatioii fioiii ii i~i/ldicatllC 0! i/ic correlation

betts’een 1/ic air al ti/lies o/ the p/totoils.

Ill. NONLINEAR EQUATIONS

IS. Nonlinear equations reduced to the linear case

11w general stochasticnonhitieardii f’f’ereii tial equalion can be written in the utiiversal form

= I’Jll. ~ 11~1/ ~). (v = 1. 2. .... n. (lS.l

l’ogethier with a given itutial value11(0) = a, the equationdietcriliitles a stochasticprocessi,,.(i. ~) provided! of’ course that for each mdhividtial ~ E ~2the equationhasa titiique sOldition. Itis clear, however. tllat the methodsdevelopedso far no longerapply. Onecatinot expect to finda dli) fei’entiai eqdiation (‘or (iift)) by itself’, becausethe notil inearity necessarilybrings in tile

highermoments. Nevet’theicsswe shall now describea simpledlevice (‘or redticing tue nonlinear

problem ( I S. I) to the hi tear case.Representu by a point in an n-dimensional “phasespace”. Equation (I S. I ) dieterminesa

velocity in eachpoint of’ this phasespace:f’rom each initial point a a trajectory startsout whichdlescrihesthe correspotmtiiiigsolution of’ ( 1 8. 1). (For the moment ~ is treatedasa fixed parameter.)Now considlera clodidi of’ itiitial points. Jescrihedlby a diensify in pilasespacep( 11, 0). \Vhcti allpoints t[iOVC accordhitigto ( 1 8. 1 ) the density vai’ies in time accoi’ding to tile contindiity eqdiatiotl,‘sVilicil expressesthe conservationof’ tIle points.

apoe. 1) a- = - -- lEe. C ~) pEe. t). (18.2)

is i

Of course this equatiotl resemblesclosely Liodiville’s equation in statisticalillecilanics, but thereis an importatit difference: Liotivihle’s tiworein - statitig the consel’vatioil of phasespacevoluitte

doesnot appl~’: there is no genei’al reasonwily our (‘low shouldh he incompressible.I lence tuea/all, has to renlain in trout of the 1’,. As a consequencethe soldition (if (1 8.2) is not obtainedby

just takingp constatlt alongeach trajectory, hut a Jacobiandeterminantwill appearin ( 1 8.6).

5Sec :iiso: Siraionoviuh (1963) cli. 4: Kuho ) i 963):Sturrock (1966): Hail ansi Sturrock (1967): So ansi Yeit (1968); (‘rut~’ aiui So

1973): Van Kanipcn (I 975i.

S.C. Van Kampen Stochastic di! ferential equatlolts 209

Now considerall valuesof ~ e ~2with their probability distribution (4.1). Then(18.2) is astochiasticdifferential equationfor p. wilicil is linear, and cart tlleretore lie treatediby the llletilodisdevelohiedpreviously.Tile unknownp is analogousto a, in (7. 1). wblile the ii in (1 8.2) hias thesamerole as the subscriptv in (7. 1).

Supposeoiie hassolved (1 8.2). that is. f’or any prescribedinitial valuep( 11. 0) one knows how tofind (p(le, t) ) . The way in which tilis leadsto the solution of tile original equation(1 8. 1) is givenby the t’ollowing

Lemma(Van Kampen. 1975). Solve(1 8.2) with the initial condlition

pEe. 0) = 8~~(l1-- a). (18.3)

On tile other llandl. let pEe. 1) he tile probability density of’ ie( t) as dleterminedlby (1 8. 1) togetherwith the initial value11(0) = a. Then

(pEe, 1)) = pEt. 1). (18.4)

To prove this lemma we write tile solution of (1 8. 1) in the f’ornl

11 = U(a, t; ~).

For t’ixedl t and ~ thusconstitutesa mappitigof the initial vectora into 11. As it wasassumedthat

F obeystile disual solubilitv condhitionsof a dit’ferential equationthis mapping hasan inverse

a U~(It, t;~). (18.5)

The solutioii of (1 8.2) may then be written explicitly

r - 1 d(U’(u))pot. t; ~) = P1 U 1(11 t; ~), 0;~ I - - (18.6)

L J dOe)

wilere the last factor is tIle Jacobiandeterminantof tIle lllapping.* Using (1 8.3)

(pEe, t)) =fp(u, t; t~)P(&~)d&~

d(LC’)=J P( ~) d~O[ U’ (a, t; ~) -- aJit d(ii)

Witll tIle aid of the wehlknowntransformationproperty of the delta t’unction onef’inds f’or tllis

(p(u. t)) f P(~)d~8[11 —- U(a. t; ~)j.

But this is just tile probability that the solution U(a, t; ~) of’ (1 8. 1) takesthe value11. This coni—pletesthe proof’ of’ (1 8.4).

Thus we Ilave establishedan exactrelationshipbetweentile nonlinearstochasticdit’f’ereiitialequation(1 8.1) andthe linear one(1 8.2). It may be addedthatother solutionsof (18.2). i.e..

*In prineipie we ought to have taken the absolutevalue ot’ the Jacobian,1)ut it is positive,since it is equal to i ii = I) and is not

allowed in vanish,

210 N. C. Van Kampen Stochasticdifferential equations

with otherinitial values than just the delta(‘unction asin (1 8.3), havea nieaningas well. Theycorrespondto solutionsof’ (1 8.1) in which the initial valuea is also random;with probability dis-tribution p(a. 0). In that case(p(u, 1)) is identicalwith the probability densitypEt, t) of 11 arisingfrom the randomnessof the equationand of’ the initial value. It is indispensable,however,thatthe distribution of initial valuesis statistically independentot tilat of ~.

Finally it may be remarkedthat the samedevicemay he usedeven if F is linear in a. tilat is. ifF hasthe forni ( 7.1). It then leadsto tile entire probability distribution of u at time t. ratherthanjust its average.Thus the sametreatmentof linear equations,which at first only yielded(a). nowpermits to determinep(u, t) by the simple deviceof app/wag it to the partial d(ffèrential eqllatiO/1

(18.2) rather than to (7. 1) itself:

19. Nonlinearequationswith rapid fluctuations

For linear equationstile main result was(1 2.3). valid in the casethat the fluctuations are smalland havea short correlation time i~. This result carriesover the nonlinearcaseand will then lead

to an equationfor (pta, t)) = pta, t) of the form

öp(u, t)(19.1)

where K is an operatoracting~n the u-dependenceot’p, but tiot on its 1-dependence.Thus tile

probability densityof a obeysa niasterequation.whuch meansthat 11 is a Markov process(Sec. 22).To put it differently the expansionin ar~leadsto an approximatedescriptionof a as aMarkovprocess.This is a special exampleof a fundamentalideain statisticalniechanicsof irreversibleprocesses:the rapid motionsof nioleculesgive rise on a coarsetime scaleto a masterequationforaprobability distribution (which on an evencoarsertime scajegives rise to the deterministicequa-tionsof macroscopicphysics).This hasbeendiscussedin detail for a niore restrictedcasein

chapter4 of the nionographby Stratonovich(1 963).In order to utilize the resultsobtainedfor linear equationsit is convenientto take, in analogy

with (7.2), (7.3)

F(u, t; ~) = F0(u) + aF1(u, t; ~), (F~(n,t)) = 0. (19.2)

The averageis to be takenover ~ with fixed a and t. The basicassumptionis

(F~(u,t) F~(u’,t’)) 0 for It — t’l > r~.

andsimilarly for highereumulants.Equation (18.2) may now be written

p(u, 1) = (A0+ aAi) p~’t),

A0=V’F0,

The symbolV~is usedfor the operatorthatdifferentiateseverythingthat comesafter it withrespectto u~the index a hasbeensuppressedfor shortness.On substitutingall this in (1 2.3) oneobtainsto secondorder

N.G. Van Kampen.Stochasticdifferentialequations 211

ap(u,t) V —F0+ a2f (F~(t)exp(—rV’F0) V’ F~(t— r))exp(rV’ Fo)dr) p(u, t). (19.3)

The meaningof theoperatorexp(-—rv’ F0) can be gleanedfrom the fact that it providesthesolution of the equation

aJ(u, t)(19.4)

Similarly to (1 8.6) this solution can be found explicitly in termsof the characteristiccurves.Con-siderthe equation

a= F0(u). (19.5)

This equatioii determinesfor fixed t a mappingfrom the initial u(0) into ii(t). As we havesup-posedfor simplicity that F0(u) doesnot dependoii t, this mappingcan be indicatedsimply by

a —~ at, with inverse(ut)t = a.Then the solution of(1 9.4) is

d(ut)

flu, t) J~ut, 0)——-— = exp(—tV ‘ F0)flu, 0).

d(u)

Thusour operatoris specifiedby its effect on an arbitrary functioii f’:

d(u- t)exp(—tV’ F0)flu) =f(u-

t)d(u)

wherethelast factor is againthe Jacobiandeterminant.With the aid of this specificatioii equation(19.3) takesthe form

ap(u, 1) d(uT) d(u)

at V’ —F0(u)+ a

2/ d(u) (F~(u,t)V_7’ F1(u

5, t — r)) di) p(u, t), (19.6)

whereV~denotesdifferentiation with respectto u,~..This is the masterequation(19.1) to secondorder in a, that is, to ordera(ar~).Note that it involves two differentiationsof p(u, t) with re-spectto the componentsof a, andhasthereforethe form of a Fokker—Planckequation.

An alternativeform of (19.6) is sometimesconvenient

ap(u, t)

_____ = V’ —Fo(u)+ a2f (F1(u, t)V_~‘ F1(u

T, t — r)) di

+ a2 / (F1(u, t) F~(u

T,t — i)> A(uT)) p(u, t). (19.7)

HereA is an n-componentvector function (not an operator),

2 i 2 5’, C;. l’a,i Kampen. Stochastic differential equations

r dl(111’)

AOl) = VI hogL d(u)

In the specialcasethat a hasa singlecomponentonehas

a1~ a = aJ-’1(11, t) , die

- -/‘o)iI) af (——~_—---— l’~fieT.t——i))---— dli pat ale ate dill ~

a2 d11+ a2 --~ f (1’~(1i, 1) f (11 T -- i)) ~-- di p. (19,8)

Anotherspecializationobtainswhien Ru, t; ~) haszero divergence.V ‘F = 0. Then theJacobianof the unperturbedequationis uiiitv so that A = 0, and moreoverthe operatorV in(19,7) commuteswith F

1(i~ 5), Hence(19.7) takesthe form (Cruty and So. 1973)

apEs, t) a a = au5 a1~-- f(Fi~(ut)Fi~(l1 Ct—i)) - di --. (19.9)

at all ‘ ~ atl ‘ . dII~ anis p 5

20. Two applications

As an applicationconsiderthe heatingof a plasmain a randomelectric f’ield (Sturrock, 1 966;1 lalI amid Stdirroek. 1 967; Silevitch andGolden, 1973).The motion of’ a singlechargedparticle inone dlinlellsion is

.vaE(x, t;~). (20.1)

The stochasticfunction F is supposedto havezeromean,to be stationaryin time andspace.and

to Ilave a correlation time r~suchthat ai~is small.We takex anda = x as the two componentsui, 112. Then

J”0,1v.

/‘0,2 = 0, J’i,2 = L’(x, t)

The mapping ti —~ u~is found by solving tile unperturbedequation.

.vtxO+vOt, ut=v0.

Its Jacobiandeterminantequalsunity, ~o that A = 0. Substitution in (1 9.7) yields

ap(x,v.t) a a a- = v+- a2f (L(x, t) ‘ L(x

5,t—i))di pat ax av av—~

0

ap a a=—v--+a2- fE(x.t)L’(xT.t_T). --dip.

ax an 0 av--T

N.C. Van Kampen. Stochasticdifferential equations 213

TIle differentiation with respectto v1 can be written in ternls of .v andn

a ax a an a a a-- ----- = — --- - + - ---— —- = i -— + —

an--i av--~ax an—T an ax an

Hencetile final equationis

ap ap a ap a ap(20.2)

at a.v an ax av an

where the two eoet’ficientsare

c1(n) = / r(E(x, t) E(x -- ni, 1 — i)) di,

c0(n) = / (E(x, t)E(x -- ni, t — i)> di.

Note tllat tlley contain the field correlationsas felt by the particle movingwith velocity n.lntergrationoverx leavesuswith an equation for the velocity distribution I’.

aj(n, 1) ., a a~— = a~----- c0(n) —‘ (20.3)

at an an

From this one readily t’inds

d(n) dc0(n)- ~), (20.4)

dt dn

d(n2) d

= 2a2(~~~~nc0(n)). (20.5)

dt dn

Sturrock (1 966) obtainedtheseequationsfirst by perturbatioii theory and thlen usesthem to con-struct the Fokker--Planckequation(20.3). Surprisinglyhe arrivesat the sanie equationalthougil

his expressionfor d(n2)/dt is slightly different from (20.5),andnot compatiblewith (20.3).

As a secondapplicationconsiderthe motion of a particle in onedimension,subject to a forceK(.v) dependingon tile position x. a friction force —ax. anda random force a~(t):

mx + fIx = K(x) + a~(t). (20.6)

This probleni wasstudiedby Kramers(1940)as a model of certain chemicalreactions,andbyZwanziget a!. as a model for fluctuating nonlinearsystems(Bixon andZwanzig, 1971;NordhiolniandZwanzig. 1974).ForK 0 it is the Langevinequation(2.1).

We set in = 1 andtakex andn = x as the two componentsa1,a2.Then

F0,1 = a. F11 = 0,

F0,2 = - fIn + K(x), F12 = ~(t).

214 5, C;. 1-an Kainpen Stochasticdifferential equations

Supposingthat ~ hasa shiori auto—correlationtime T~ ( wifhlout necessarilybeing dielta—correlatedwe may apply the result ( I 97), The mapping ie -~ ii~ follows f’i’oni solving tile iinpertiirhesl equa-tiC)n

= a. a = + A’).v).

Withiout resortingto an explicit solution one seesthat the Jacobianobeysthe edhuation

d d(x’, a1) a if-----log - -- v~ ----~v+K(x)}= ~.

sit ci) x, a) iiv ôd)

thencethe Jacobianeqnals e ~ and! consequenthv A = 0. Equation ) 1 9,7) thereforeredlUcesto

ap(x, a, t) a1 a ap a a

-- = a- -+fI -up - A(x) -+n~- J (~(t)~(t i)) --~— dli /1. (20,7)ax c)tJ an ~au cia

[he term on tIle seeotldhline is actually a rathler complicatedoperatoractingon the (.v. a )-dle-

pendenceof’ p. wllichi involvesthe solution of’ the unperturbededluation.It simplifies, however,ifT5 is soshort that the velocity doesnot vary appreciablyduring i5, SC) that a -T may be i’eplaeedwith a. Formally thusamountsto considering~ as delta-correlated,

(~(t)~(t’)) = 2c0~(t1 12),

With tllis simplification the secondline of’) 20.7) becomes

a2p

a2c0 -‘ (20.8)

an

svhichi reduces(1 9, 1 0) to the Kraniersequation.As a next approximationone may take the variation of a diuring i~ into accountto first order

in i5:

= .v —- in. a -T = a + fIiv -- iK(x).

From tilis onededuces

a a a= (I fIi) --- + i---- + 0(i

2).an—T an ax

The secondline of (I 9, 1 0) therefore becomes

a2p a2pa2(c

0 — fIc1) ----— + a2c

1 -- -

an an ax

where ‘1 = f~(~(t) ~(t --- r)) i di. Note thiat the magnitudeof thesecorrection termsdependsonhow much the unperturbed x andn vary ddiringr~; it is thereforenot inconsistentto include themwhile neglectingiligiler ordersof ai~

NC. Van Kampen,Stochasticdifferentialequations 215

21. Useof the interactionrepresentation

In practiceit is often moreconvenientto introducean analogof the interactionrepresentation

asusedin the linear theory. Considerthe original equation(18.1) in the form*

= F0(u, t) + aFi(it, t; ~), (F1(u, t)) = 0. (21.1)

Let the generalsolution of tile unperturbedequationi~= F0(u, t) be

11 = ip(n, t).

wheren standsfor n independentintegration constants.Thesemay, but neednot be the initialvaluesof the coniponentsof ii. This relation betweena andn is now usedto transform(2 1. 1) to

new variablesn (“variation of constants”)

-~ + ~ -~ = Fo,~(n,t), t) + aF1 j~(n,1). t; ~).

at ,,~

By constructionthe first term on the left cancelsthe first term on the right andone is left with

= a F1,5(~(v,t), t; ~) a G0(n, t; ~). (21.2)an

This is the equation(1 8.1) transformedto tile newvariablesa.One may now apply to it thesamelinearizationdevice.The continuityequationf’or the distribution a(n. t) is

aa(n,t) a= —a — G~(n.t)u(n, t). (21.3)

at an~

The connectionbetweena(n, t) andp(u, 1) is of course

d(p(v, t))a(v. t) = p(ip(n, t), 1) —-——‘-—-‘- . (21.4)

d(n)

In particular, the initial value of a is

d(sp(n, 0))a(v, 0) = ö[ip(n, 0)—al —--—-——- = 6(v — b), (21.5)

d(a)

whereb is the initial valueof n correspondingto the initial valuea of u, that isa = p(b, 0).The averagedsolution of (21.3) is the probability densityofv

(a(v, t)) = q(v, t),

andq(v, t) is relatedtop(u, t) by the sameequation(21.4).Henceone is justified in solving (21.3)ratherthan (18.2) anddeterminingafterwardsthe distribution of u. To secondorder the equation -for q is

*We now allow F0 to depend on time becauseit doesnot give rise to additional complications.

216 v. C I ‘all Kampen ,S’tocitastic dt( fei’ential equations

aq(v. t)

- = ~2 V J (G(v. t) V fda. I - i)) di q(v. 1). (2 I.~)

This equationis equivalentto (1 9,3). hut obviously more sitiiple. It is easyto write it in the lorniof’ a Fokker Planekequation:

aq(v. 1) a a2~ -[(‘,q1+~ - {D

1, cii. (21,7)at av ~ on, all0

The coeft’teients m tIme cii ff’usion audi convectiotl terills are

D,0(v. 1) = a~f(05(v. 1) (~(v.t -- i)) cii. (21.8)

a05(v. 1)(‘~(a.1) = ~ f ( - G (a. 1-- T)) cIT. (21,9)

aa0

Of coursetius can he translatedlhack mto the Fokkei’ - Planck eqdiation (1 9,7) l’or p(11, 1) itself’.We shall now apply theseresults to a problem in populatiotl growth. ‘file way in which tile size

11 of’ an isolated hioptilation varieswith time is of’ten described!by tile nonlinear Nlaltllus---- Verllulst

equation( Lotka. 1 92 1 Montroll. 1 968; Batschleiet.1 97 1 Montroll and! Badger. 1 974)

ugi, sl1~. f21.lO)

‘[lie t’irst term represetitsthe getierationdue to excessof births overdeaths,and the secondtermservesas a rough descriptionof the adlditional death ratedue to tIle struggle f’or food. Both coef’t’i-cientsg atidi 5’ depetidon the etivironnlent amid may thereforepossessrandomfluctuationsin time,*1 lie sameeqdlatlon ( 2 1 .1 0) is the rate equationfor the autocatalyticchemical reaction(Glatisdorff and! Prigogine. 197 I; .Schlogl, 1972;Kei,,er andFox. 1974;Glansdorf’f’ et al. . 1 974)

A + X ~ 2X.

For fixed g and.c tile solutioti is

r.s / 1 .s 1’i-e(t) 1 - -+1 ~e gt -

Lg ~i,(0) g!

As a first examplewe allow s’ to be random andset

g = I. s I + a ~)fe ~). (~)= 0.

The abovesolution of’ the unperturbedequationmay be writteti witil a conveniently chosenillte-gration constant~

(21.11)

i+vet

*lt should be clear that this has nothing to do with the fluctuations due to the fact that the population consistsof a finite number

of individuals (Van Kampen. i 976).

N.C. Van Kampen,Stochasticdifferential equations 217

Froni (21 .2) onethen finds

G(v,t)= --et~(t),

so that ( 2 1 .3) takesthe form

aa(n. t) aa(v. t)—- ---——- = a et~(t)

at an

This equationis still exact.Tile secondorderapproximation(21.6) f’or (u(n. 1)) is

aq(a, t) = a2q(n. 1)

- at - = a e2tf (~(t)~ -- r)) e1 di~_~~n2~_When ~(t; ~) is a stationary processthe integral is a constantC’. The explicit solution, with the

initial condition ( 21 .5), is easily obtained,

r (n—b)2

q(v. t) = [2~a2c(e2t --- h)]~’2exp~-- 2tL 2ac(e -- I)

Transformingback to a by meansof’ ( 20.4)one f’inally has

r I ~ 1/u — I - (I a --- I )e t 2 1pta, t) [?~a2c(l ~C2t)I~l/2 expt ~2 -- ,-2t --~~J~(21.12)

a .ac I c

wherea is the initial valueof a. Observethat the probability duly vanishesat a = 0 and that the

edluilibrium distributionp0. ~) hasits maximumat I --- a2c.As a secondexamplewe set

g = I + am’~(t;~), s = I, O~(t))= 0.

The sametransformation(21.11) can be appliedwith the result

G(v, t) = —(et + a)77(t).

It is convenientto make useof(21 .7). The coefficientsare

D(n) = a2{e2tci + etn(ct + c0) + n

2c0 ~

C~(n)= a2(etci + nc

0),

where

c0/(77(t)77(t_i))di, cif(77(t)77(t_r))e~di.

The secondorder approximationis therefore

aq a a2

__a2_(etci+vco)q+a2~ {e2tci+vet(ci+co)+vsco}q. (21.13)at an an

218 SC. Van Karnpen ,.S’tochastiedifferential equations

The easiestway to solve thus Fokker - Platlek equationwith time dlependentcoefficientsis bytransformingto a newvariable.

a = etw. q(v. t) = e tr(tt, C).

The result is a time-independentFokker- Planck equation.which can be solved with the aid of’

hypergeonietricfunctions. Alternatively onemay draw someconclusionsdirectly from (21 . 13).e.c.,

—(a) = a2(c

1et+ c

0(v)).

From this one finds (a) a2c e~for large t amid hence

him (11(t)) = I --- a2c1.

‘rhus the equilibrium averageis lowered b the fluctuations.

IV. A METHOI) FOR ARBITRARY CORRELATION TIMES

22. Markov processes

Bef’ore proceedingto the next method we haveto recall somefactsaboutMarkov processes,seee.g. Bharucha-Reid(1960). Let ‘17(t; ~) be a stochasticprocess,i.e., an ensembleof functions

of t labelledby the parameter~ with given distribution D( ~). Let P1(y ~. 0) dy he the probabilitythat 77(t~; ~) lies between v1 and v~+ dl1.

P1(y1, t1) = f6[77(ti; ~) ---- .vi I D(w) d~. (22.1)

Sinularly let P2(,1 ~ t1 Y~12) dy dy2 be the joint probability that 77(t1) lies in Yi, y + dy i andalso77(t2) in Y2. Y2+ dy2. In this way one can continueto definea hlierarchy of probabilitiesP3. P4wilichl constitutesan alternativeway of describingstochasticprocesses.*

More important f’or our purpose,however,is the coilditional probability distribution of 77(t2)when77(t1) at t1 < t2 is known to be equal to .v1. According to Bayes’rule this conditional prob-

ability density is

P2(y1. t1y2, 12)T(y2. t2lyi, 0) — ‘ (22.2)

P1(y1. t1)

The process77(1; ~) is calledMarkovian if T obeysthe following “Chapman—Kolmogorov”

equation** for any t1 < t2 < 13

*This alternativedescriptionis more familiar in physics (WangandUhlenbeck,1945).The equivalenceof both descriptionswas

provedby Kolmogorov(1950).**Also called “Smoluchowskiequation”,but this namehasbeenapplied to slightly differentequationsaswell.

NC. Van Kampen, Stochasticdifferentialeq~i’~tions 219

T(y3, t31y1. t~) f T(v3.t3~2,t2) dy2 T(y2. t2Iy~.ti). (22.3)

The equation statesthat the probability t’or going f’rom ~‘ to ~ via 12 15 tile prodluct of theprobability for going froni y~to ~2’ timesthat f’or going subsequentlyfrom 12 to ~ that is,successivetransitionsare statistically independent.

A more manageablel’orm is obtainedby taking t3 = t2 + ~t witil small ~t. For a physicalprocessoneexpectsthat tile probability that 77 makesa junip during ~t will be proportionalto~t, apart from higherordersin ~t. Tllis is expressedby

T(y3, t,1t2. t2) = 6(~’~— .r2)1 I --- w ~t1 + ~t W(y31y2)+ 0 (~t2). (22.4)

W is the transition probability perunit time froni V

2 to ~. The term w~tis the total probabilitythat a transition took place.

wO’2) =fW(y31y2) dy3. (22.5)

Substitutionof(22.4) with (22.5) into (22.3)yields

~ t~v,t~)

— at’ f~W(,vIy’)T(y’, tIy1, Ci) -—- W(y’Iy) T(, tIy~,t1)}dj-’’. (22.6)

Slightly more general:selecta subensembleof’ the functions77(1; ~) by prescribinga distributionP(,y i. t1) at sonic initial time t ~. Then the distribution in this subeilsembleat t ~ t obeysthemasterequation*

aP(y, t)J~W(yiy’)P(y’, t) -- W(y’ly) P(v, t) } dy. (22.7)

The right-handside is a linear operatoractingon they-dependenceof P and may be abbreviatedWP. The formal solution is

P(y, t) = eS~~~tP(v,0).

In particular, for C > t’,

TO, tly’, t’) = eW~t’)6(y— y’).

The auto-correlationfunction (4.2) for astationaryMarkov processwith zeromeanis

(77(t)77(t’)) fyiP~(yi)dytfy2T(y2.tIy1, C’) dy2

fye~tt’)yP~(y)dy. (22.8)

The simplestMarkov process,dearto the heartof the model builders,is called the “two-leveljump process”,“two stateMarkov process”.“dichotomic Markov process”,or “random telegraph

*Unfortunately this namehas been used so indiscriminately in the literature that it has lost almost all denotativevalue. We use it

in the original sense(Nordsieck et al., 1940): thedifferentialform of theChapman--Kolmogorosequation characterizingtheMarkovproperty.

220 ‘s. C I’an Kasnpeii Stochasticdii es-esittaIequa f/os/s

process”.Here~ takesonly two possiblevalues±I . with equal prohahilites.atldl lumpswithprobability a cit per cit. that is.

/ VW=~ I. (229)

a a!

Its transition probability is f’or 12 -- It = T ~ 0

ii + e - 21’T - e~”m’~. fe v~.t~) = eW( ~2 (if =

I e2l’~ I + e 21T -

aiid its auto-correlatiotitcinctiotm is

= exp( 2P~1/ - [2~ }, (22,10)

A Gaussian processis defitieci as a stochasticprocesst’or Wilich all f’unctions t~./2. of thsabove-mentionedhierarchy are Gadissian.Dooh’s theorem assertsthat the otilv Gaussian.statiotlaryMarkov processis the Ornstein Ulmlenheckprocess.ehat’acteriteciby

/‘i(t’) = ( 2~u2~ 1 2 exp( ---~‘2,!2a2).

1’~ t’t e ~ )2T0’~,t

2[r1, t~)= (2Tu2)’2exp ‘ -~ - -—

202( I ~-25T

Here i = ~2 - -- t and (J and ~ are positive cotistants.Its auto-correlationfunction is

(77(1)77(12))=ue i~dI

and time operatorW of time masterequation is

a2~= y~’- - ~ + ~i2 -—

am’ - am’2

Thus processwasoriginally designedto describetile random behaviorof the m’eloeitm’ of’ a heavyBrownian particle itunierseciin a gasof’ light moleeules**. hut on accountot’ its s’nple propertiesis now Otteil usedin other physical applicationsto simulatestochasticfunctions whose precise

propertiesare not ktiowim.The “white noise process” can he obtained from the Ornsteiti -Uhhenheckproces~’’Jsthe

liiiuting ease f’or

-~ ~, (52 ~ ~, ~ = ~‘ = constant.

It ilas zero meanand tile auto-correlation function is

(77(t) 77(1’)) = 2F8(t --- 1).

5-Doob(1943).‘I’his theorem is sometimesmisquoted by omitting one nt the threepremises.Our formulation is not strict either.

becausethere is one other Gaussian.stationary Markov process,vi,., the purely random processin which 7?(ti) and ri(t2 I arc

statistically independentas soon ast i ±t2.

**A selectionfrom the extensiveliterature on the connectionof’ Brosvnian motion with the Ornstein -Uhienbeckprocessis:

Rayieigh (1 891); Uhienbeckand Ornstein (1930); (‘handrasekhar(1943); I)ooh (1943): Wangand Uhienbeck(i 945);Feiier 11966),

SC. Van Kampen. Stochasticdifferential eqt,attons 221

These two propertiesare the sanie as thoseassumedfor the Langevin f’orce iii (2.1). One often

asstuiiesin additioii that the lligher momentsof the Langevin t’orces are also the sameas f’or thewhite noise process.although that is hard to justify on time basisof physical inttntion. If oneassumesthus, the solution of) 2. I ) is the Ornstein---Uhilenhc’ckprocess.

Tile R’ieiier pro c’es.s or Wiener—Lém’’m’processis anotherlinuting case.specified by

0, ~2 ~, a2y = constant= D.

In tilis hiniit

V (i’~ -— mi)2TO’2. t2hi1, 0) = (4i~DtY

172CXI) --- -

L 4Dt

As, however.P~(r’)doesnot havea hitiut onenlust selecta suhensemh!eby f’ixing an initial distribu-

tioti. e.g. Pi(m’, 0) = ~(j’). Consediuentlythe Wienerprocessis non-stationary.thoughGadissiali amidMarkoviami. It describesthe position of the Brownian particle on a coarsetinie scale.on whlichltimesof order I /y are ignored ( Kramers. 1940). It is also tile integral of’ the white noise process.amid as such is extensivelyused! in tIle niathiematical literature (Gillnman and Skoroimod.1972;Bunke. 1972;Arnold, 1973).

23. A rnethlod for arbitrary correlation times

So f’ar our treatnientof’ stochasticdifferential equationwasbasedon the assumptionthat thereis a short auto—correlationtime i~. If, on the otherhaiid, i~ is int’inite (or at least long comparedto the durationof the processone is interestedin) no simplifying approximationis available.Onesimply is t’aced witll an ensembleof solutionsrunning concurrently,and the equationIlas to besolved f’or eachone individually. Thus canactuallybe doneonly in a few cases,mainly whenthe

coefficientsare time-independent,assllown by tile examplesiii TableII. Time difficulty is evengreatert’or in termediatevaluesof’ i~,neitherlong nor silort comparedto the duration of the pro-cess.Yet the following methodmakesit possibleto treatall valuesof T~.at the expenseof a severerestriction of the type of stochasticheilavior. It hasbeenappliedby severalautilors to linearmodels(Pinsky, 1968; Kubo, 1969;SulemandFrischi. 1972;Hersll andPapanicohaou.1972;

Bourretet al.. Van Kanipen, 1973;BrissaudandFrisch, 1974) but may be used in nonlinearcasesas well (Cruty andSo, 1973;HeilmanandVail Kampeii, 1975).

Let a be againan n—dimensionalvector, f”O, t; ~‘) avector f’unction of 11, 1, and oneadditionalvariable m’’ Considerthe equation

a = Ru, t;77(t)). (23.1)

where 77(t) is a stochasticprocess.The restriction to be madenow is that 17 is taken to he a Markom’process,shell that its probability densityH(y, t) obeysa masterequation

fl(y, t) WH(y, t). (23.2)

Moreover,for simplicity we drop the explicit C-dependenceof F.The ideais that for equationsof’ this type the (n + 1)-componentaariable (a ~,a

2,...,11n’ )‘) is

222 .V C. Van Kainpen Stochastic differential equations’

again a Markom’ process. Its probability density~(mi, m’, t) varies imi tinie owing to the flow iii 11-spaceamid tIle jumpsof 1’. The corm’espondimigmastereqdiation is found by combiningthe comitiiiuitv

equationI’or the probability densityof 11 with time masterequation( 23.2) f’or c.

a~(ii,v, t) a= - {14a; a) ~1 +W~. (23.3)

at all

As imutial comiditioml we take

i’, 0) = ~(a — a) J1eq(1,). ( 23.4)

where fl~ is the equiiibriumii distribution of j-’, deternmineciby the equatiom’mWfF~h,i’)= 0. Otilerinitial conditionsmay occur andcan be treated iii tile sameway.1ii the specialcasethat F( a; i’) is linear andIlence ot’ the fornl /1(y)a omie cango onestepfurther. Define the ‘‘marginal averages’’

m,,(y, t) =f u,,P(a, ~‘, t) (j~i~(.

Multiplying (23.3) with a, amid integratingone obtainsaim equation f’or thesemarginalaverages:

am 0’, t)= ~ A

1,~,(v~n1~(i’,t) +Wm,(j’. 1). (23.5)at

The initial condition is accordingto (23.4)

m,,(y, t) = ~fleq (~ (23.6)

The quantity one wantsto find is

(u5(t)) fm5a, t) dy. (23.7)

The applicability of this approachis contingenton the onehandon the validity of the specialtypeof the stochasticcharacterof F, andon the otherhandon the questionwhether(23.3) or(23.5) cami he solved.An approximationmethod of the Chlapman---Enskogtype is possiblewhemiW is large(Heilman andVan Kampen, 1 974), but that brings ushackto the precedingexpansionin aTe.becauselarge W meansrapid fluctuationsin y andhenceshort r~.More cami be hearnedfrom specialniodehsfor which the equationscami be solvedexactly,althoughtheseniodelsarenecessarilysomewhatartificial.

24. Line broadening

Kubo (1 969) consideredthe following problem as a model for line broadeningdueto randomperturbations.*Take the equation(6.1) for an oscillator with random freqrmemicy~(t) = w0 +

U = --ifwo+ cE77(t)~a. (24.1)

*For more detailed treatments see Abragam (1961),Schhichter (1963),Cooper (1969),Muus and Atkins (1972).

N.G. VanKampen.Stochasticdifferential equations 223

Supposethat 77 is a Markov processdeterminedby (23.2). As our equation(24.1) is linear it ispossibleto use the result(23.5) for the marginalaverage

m(. t) = ---i(w0 + av)m +Wm. (24.2)

Tilis has to be solvedwith initial condition (23.6).First suppose17 ~5 the dichotomic Markov processgiven by (22.8).Then (24.2) reducesto two

coupledordinary dif’ferential equations

rn(+i) = -i(w0+ a) rn(+l) — vm(+l) + am(--l)

m(—1) = —i(’w0 — a) m(—h) — vm(—l) + vm(+l).

The initial condition (23.6) is.m(±I) = ~a. Tile solution of theseequationsis elementaryand

yields for (u(t)) = m÷(t)+ m_(t)

(u(t)) = a exp~—(a+ iw0)t} (cos(t~~2) + sin(Wa~— p2)). (24.3)

The caseof very long auto-correlation time obtainsin the limit a-÷0,

(u(t)) = a exp{—iw0t}cos at = ~aexp~—i(w0+ a)t} + ~a exp{—i(w0 — a)t}. (24.4)

This showsthatu(t) oscillateswith eitheronefrequency,eachwith probability ~-, which ispatently correct. The next powerof a gives rise to a damping~ andtherefore to a Loreiitzianfrequencydistribution abouteither frequency.There is also a phaseshift and the secondorder ofa introducesa frequencyshift as well, but the over-all behaviorremainsthe sameas long as a ~ a,

that is ar~~ I. The caseof aeryshortauto-correlation time obtainsby expandingin h/a,

/ a2\

(u(t)) a (1 + —i- ) exp~—iw0t— (a

2/2v)t}. (24.5)\ 41) /

This is a dampedwavewith a singlefrequencyandthereforeasingle Lorentz peak.It obeysthedifferentialequation

d(u(t)> a2

dt = (_~~

0 — —) (u(t)), (24.6)

which is the equation(10.4).The factor (1 + a2/4v2) in (24.5) is the mismatchbetweentheinitial

valuesdueto the transientmentionedin the lastparagraphof Sec. 10.The exactequation(24.3) of coursealso tellswhat happensbetweentheselimiting cases.

Clearly (u(t)) doesnot generallyobeyan equationof type (24.6). (It happensto obeya secondorder differentialequation,but that is only an artifact due to the specialnatureof the dichotomicMarkov process.)One alsoseesthat (u(t)> is dampedfor all valuesof a anda, evenfor a = a.

(u(t)) = a expf--(a+ iw0)t}( I + Vt).

The line shape,as a functionof the frequencyw, is given by theFouriertransformof the auto-correlationfunction(Anderson, i949;Cooper,1969).The “line shapefunction” is

224 .\ (I. Van Kanm~o’n. Stochastic differential equations

1(cc --- cc0) = J (a(0)* 11(1)) e ~‘ dht.-- 2~(ii(0)’~iiO)).

~[o find the auto-correlatiotit’cinction use I 22~). ( 21.6) and (23.7):

/ as(l1(0)* 1~(t))= exp (a+ mcc0)t}

2d’~2 .s’)~(9~~I) (1> 0).ts’

1 hius oneobtaimis. in termsof’ ~cc = cc cc0.

2a a’2

/(~cc)= ~-- - - ‘ (24.7)iT ‘~(~cc0 a2)2 + 4v2(~cc)’

For a2 < 2v2, that is for small aTe. this is a Loretittian with mliaxinidinl at cc0 and width determumed

by tue mlumbera of ram’mclom’n jumpsper utlit tinic’ as wasderivedby Loremitz (1909), For large

two peaksappearroughly at the frequemiciescc0 ±a witll a mmnimlium in betweenat cc0 thismeansthat the oscillationsoccur at either of’ the two possibletreqdic’ncies.with rare transitiomms

betweenthieni.In sunimarv omie seesthat thlere are two Lorenti lines, whucil spread!out with increasinga, then

merge into one broadlitie, f’romn wllmcbl agaimi a sharpercentral peak enierges.In a certain ramigeof

a thic’rel’ore increasinglyrapid thuctuatiotisgive rise to a narrowerline: “motional narrowing”.Drawingsof’ theseline prot’iles are given by Ktibo ( I 969).

Incidentally. on cornparimigthe soltmtion (24,3) with (6. I) one seesthat the resmmlt (24.3) amountsto the statementthat f’or tile dichotomic M arkov processTI

exp( ---ia177(t’) dlt2 = e et{cos(I~a’2-- a2) + ~ ~2)} . (245)

This result wasusedimphcitly by Hu and hlartniann(1 974) to (‘inch the influence on a centralspinof the magneticdipole fields of’ the surroundingspimis.The latter are supposedto flip at ramidoni.so that eachof theni createsa dichiotomimic Markov field at the center.The strength a depemiclsonthe locatiomi of time flipping spm. All thesef’ields haveto be added,whuch amotmntsto taking the

product of inamiy factors(24.8). becauseth~~spinsareassumedto flip independiently.Films mainproblem is the evaluationof tlus product.

Next we take the sameequatiomi(24.1), amid ilence also(24.2). but choosef’or 77 theOrnstein Uhlenbeckprocess,which meansthat W is the Fokker- -Planck operator,

aU(y, t) / a a2u -= --i(w

0+av)U+ yl—--- m-’U+ ~-

at \av’ ay2

This equation.with initial condition U(y, 0) = iTi/2 e ~2 a can be solvedby settingU(y, t) = exp(Ay2 + By + C) amid solvimig the resultingequatiomisfor A, B, C as fmmnctions oft. Tileresult is

,V, C. Van Kampest. Stochastic differential equatiosts 225

V ia a2 3a2 a2 a’2U(J’,t)=iT~2aeXpI---m’2——-(l--e t))’—iw

0t—~—t+-—~ -~e ~t+_~ e~L y _y 4y y

Integrationover a yields

(11(t)) = a exp~- -icc0t — (a2/2y)t} exp[~_

2-(I - e t)] . (24,9)

Tile first exponentialis identical \Vitll the result of’ otir equatiomi) 10.4).The reasonwhy tllatapproximation turnsout to he exact ill the presentcaseis firstly thiat no operatorsoccur. so that

equation(6.5) applies:andisecondly tilat tIme Ornstein Uhllenheck processis Gaussian.so thlat allcmmmulantsbeyond tile secondlvaimishm.Tue last factor in ( 24.9) is the transiemitet’f’ect.

It hasto be admittedthat .thie modiel iii this sectionviolates tile injunction of Sec. 13 by startingfrom the equation(24. 1) ratherthan f’ronm the actualSchirOdingerequatiomi( 2.4) of a two—level

atom in a randomfield. Such a more realistic treatmentinvolvesmore componentsit,,. aimd is

thereforeessentiallymorechf’fictmlt.

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