Large deviations and averaging for systems of...

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Large deviations and averaging for systemsof slow–fast stochastic reaction–diffusion

equations.

Wenqing Hu. 1

(Joint work with Michael Salins 2, Konstantinos Spiliopoulos 3.)

1. Department of Mathematics and Statistics, Missouri S&T.2. Department of Mathematics and Statistics, Boston University.3. Department of Mathematics and Statistics, Boston University.

Systems of slow–fast stochastic reaction–diffusionequations : Equation.

I

∂X ε,δ

∂t(t, x) = A1X ε,δ(t, x) + b1(x ,X ε,δ(t, x),Y ε,δ(t, x))

+√εσ1(x ,X ε,δ(t, x),Y ε,δ(t, x))

∂W Q1

∂t(t, x) ,

∂Y ε,δ

∂t(t, x) =

1

δ2

[A2Y ε,δ(t, x) + b2(x ,X ε,δ(t, x),Y ε,δ(t, x))

]+

1

δσ2(x ,X ε,δ(t, x),Y ε,δ(t, x))

∂W Q2

∂t(t, x) ,

X ε,δ(0, x) = X0(x) , Y ε,δ(0, x) = Y0(x) , x ∈ D ,N1X ε,δ(t, x) = N2Y ε,δ(t, x) = 0 , t ≥ 0 , x ∈ ∂D .

(1)

Systems of slow–fast stochastic reaction–diffusionequations : Equation.

I Here ε > 0 is a small parameter and δ = δ(ε) > 0 is such thatδ → 0 as ε ↓ 0.

I The operators A1 and A2 are two strictly elliptic operatorsand bi (x ,X ,Y ), i = 1, 2 are the nonlinear terms.

I The noise processes W Q1 and W Q2 are two cylindrical Wienerprocesses with covariance matrices Q1 and Q2, andσi (x ,X ,Y ), i = 1, 2 give the corresponding multiplicativenoises.

I The initial values X0 and Y0 are assumed to be in L2(D).

I The boundary conditions are given by operators Ni , i = 1, 2which may correspond to either Dirichlet or Neumannconditions.

Systems of slow–fast stochastic reaction–diffusionequations : Averaging and Large Deviations.

I Fast–slow system characterized by the small parameter δ > 0 ;

I The noise term√εσ1(x ,X ε,δ(t, x),Y ε,δ(t, x))

∂W Q1

∂t(t, x) in

the equation for the slow process X ε,δ(t, x) has a smallparameter

√ε, and both the deterministic as well as the noise

term in the equation for the fast process Y ε,δ(t, x) contain

large parameters1

δ2and

1

δ.

I In the limit, we expect an interplay between an averagingeffect in the fast process Y ε,δ(t, x) and a large deviationeffect in the slow process X ε,δ(t, x).

Systems of slow–fast stochastic reaction–diffusionequations : Previous work.

I Large deviations of stochastic partial differential equations ofreaction–diffusion type has been considered in previous works,but without the effect of multiple scales.

I Results in the case of slow–fast systems of stochasticreaction–diffusion equations have been considered indimension one, with additive noise in the fast motion and nonoise component in the slow motion.

I In finite dimensions the large deviations problem for multiscalediffusions has been well studied.

I To the best of our knowledge, the problem of large deviationsfor multiscale stochastic reaction diffusion equations inmultiple dimensions, with multiplicative noise is beingconsidered for the first time in the present work.

Systems of slow–fast stochastic reaction–diffusionequations : Application background.

I Fast–slow stochastic partial differential equations ofreaction–diffusion type have many applications in chemistryand biology.

I In the classical chemical kinetics, the evolution ofconcentrations of various components in a reaction isdescribed by ordinary differential equations. Such a descriptionturns out to be unsatisfactory in a number of applications,especially in biology.


I There are several ways to construct a more adequatemathematical model. If the reaction is fast enough, one shouldtake into account that the concentration is not constant inspace in the volume where the reaction takes place. Then thechange of concentration due to the spatial transport, as a rolethe diffusion, should be taken into consideration and thesystem of ordinary differential equations should be replaced bya system of partial differential equations of reaction–diffusiontype.


I In some cases, one should also take into account randomchange in time of the rates of reaction. Then the ordinarydifferential equation is replaced by a stochastic differentialequation. If the rates change randomly not just in time butalso in space, the evolution of concentrations can be describedby a system of stochastic partial differential equations.

I On the other hand, the rates of chemical reactions in thesystem and the diffusion coefficients may have, and as a rulehave, different orders. Some of them are much smaller thanothers and this leads to mathematical models based onslow–fast stochastic reaction–diffusion equations.

Systems of slow–fast stochastic reaction–diffusionequations : Problem Formulation.

I We want to have large deviation results for the slow motionX ε,δ in terms of Laplace principle.

I For every bounded and continuous functionh : C ([0,T ]; H)→ R we have

limε↓0−ε ln EX0,Y0

[exp

(−1

εh(X ε,δ)

)]= inf

φ∈C([0,T ];H)[SX0(φ) + h(φ)] .

(Laplace principle.)

I Calculate the action functional (rate function) SX0(φ).

Large deviations in terms of Laplace principle : Weakconvergence method.

I One of the most effective methods in analyzing large deviationeffects is the weak convergence method.

I Roughly speaking, by a variational representation ofexponential functionals of Wiener processes, one can representthe exponential functional of the slow process X ε,δ(t, x) thatappears in the Laplace principle (which is equivalent to largedeviations principle) as a variational infimum over a family ofcontrolled slow processes X ε,δ,u(t, x).


I In particular, we have, for any bounded continuous functionh : C ([0,T ]; L2(D))→ R, that

−ε ln E

[exp

(−1

εh(X ε,δ)

)]= inf

u∈L2([0,T ];U)E

[1

2

∫ T

0|u(s)|2Uds + h(X ε,δ,u)

].


I Here U is the control space, and the infimum is over allcontrols u ∈ U with finite L2([0,T ]; | • |)–norm.

I The controlled slow motion X ε,δ,u that appears on the righthand side of the Laplace principle comes from a controlledslow–fast system (X ε,δ,u,Y ε,δ,u) of reaction–diffusionequations corresponding to (1).

Large deviations in terms of Laplace principle : AssociatedControl System.

I

∂X ε,δ,u

∂t(t, x) = A1X ε,δ,u(t, x) + b1(x ,X ε,δ,u(t, x),Y ε,δ,u(t, x))

+σ1(x ,X ε,δ,u(t, x),Y ε,δ,u(t, x))(Q1u(t))(x)

+√εσ1(x ,X ε,δ,u(t, x),Y ε,δ,u(t, x))

∂W Q1

∂t(t, x) ,

∂Y ε,δ,u

∂t(t, x)

=1

δ2

[A2Y ε,δ,u(t, x) + b2(x ,X ε,δ,u(t, x),Y ε,δ,u(t, x))

]+

1

δ√εσ2(x ,X ε,δ,u(t, x),Y ε,δ,u(t, x))(Q2u(t))(x)

+1

δσ2(x ,X ε,δ,u(t, x),Y ε,δ,u(t, x))

∂W Q2

∂t(t, x) ,

X ε,δ,u(0, x) = X0(x) , Y ε,δ,u(0, x) = Y0(x) , x ∈ D ,N1X ε,δ,u(t, x) = N2Y ε,δ,u(t, x) = 0 , t ≥ 0 , x ∈ ∂D .

(2)

Large deviations in terms of Laplace principle : AssociatedControl System.

I Need to analyze the limit as ε→ 0 (and thus δ → 0) of thecontrolled fast–slow system (X ε,δ,u,Y ε,δ,u) in (2).

I Recall that for any bounded continuous functionh : C ([0,T ]; L2(D))→ R, that

−ε ln E

[exp

(−1

εh(X ε,δ)

)]= inf

u∈L2([0,T ];U)E

[1

2

∫ T


].

I u is in terms of a feedback control !

Set–up and assmptions.

I H = L2(D) ;

I W (t) =⊗∞

k=1 βk(t) ;

I Qi : R∞ → H where i = 1, 2 add color to the noise and alsodecide if the noise are independent ;

I W Qi (t, x) = QiW (t)(x).

I U = {x ∈ R∞,∞∑i=1

x2i <∞} with norm |x |2U =

∞∑i=1

x2i is the

space in which the control lives in.

Limit of the controlled fast–slow process.

I We only know that the control u ∈ L2([0,T ]; U) ;

I This makes the passage to the limit in the controlledfast–slow system (X ε,δ,u,Y ε,δ,u) as ε→ 0 (and thereforeδ → 0) difficult.

I Need more Hypotheses.

Hypothesis 1.

I Hypothesis 1. For i = 1, 2, there exist complete orthonormalsystems {ei ,k}k∈N in H, and sequences of non–negative realnumbers {αi ,k}k∈N , such that

Aiei ,k = −αi ,kei ,k , k ≥ 1 .

The covariance operators Qi : U → H, i = 1, 2 arediagonalized by the same orthonormal basis {ei ,k}k∈N in thefollowing sense. For i = 1, 2, there exists an orthonormal set{fi ,k}k∈N. The set of {fi ,k}k∈N is not necessarily complete.There exist sequences of non–negative real numbers{λi ,k}i=1,2

k∈Nsatisfying

Qi fi ,k = λi ,kei ,k .

Notice that if span{f1,k}k∈N ⊥ span{f2,k}k∈N, then the drivingnoises of the fast and the slow motion are independent.

Hypothesis 1.

I Hypothesis 1 (cont’d). If d = 1, then we have

κi := supk∈N

λi ,k |ei ,k |0 <∞ , ζi :=∞∑k=1

α−βii ,k |ei ,k |20 <∞

for some constant βi ∈ (0, 1), and if d ≥ 2, we have

κi :=∞∑k=1

λρii ,k |ei ,k |20 <∞ , ζi :=

∞∑k=1

α−βii ,k |ei ,k |20 <∞

for some constants βi ∈ (0,+∞) and ρi ∈ (2,+∞) such that

βi (ρi − 2)

ρi< 1 .

Moreoverinfk∈Ni=1,2

αi ,k =: λ > 0 .

Hypothesis 2.

I Hypothesis 2. 1. The mappings bi : D × R2 → R andσi : D ×R2 → R are measurable, both for i = 1 and for i = 2,and

∑i=1,2

(LXbi

+ LXσi

+ LYbi

+ LYσi

) ≤ M for some M > 0.

Moreover,

supx∈D|b2(x , 0, 0)| <∞ , sup

x∈D|σ2(x , 0, 0)| <∞ .

2. Recalling λ, the constant introduced in (18), we have that

LYb2< λ.

Hypothesis 2.

I Hypothesis 2 (cont’d). 3. σ2 grows linearly in X , but isbounded in Y . There exists c > 0,

supx∈D

supY∈R|σ2(x ,X ,Y )| ≤ c(1 + |X |).

4. The Lipschitz constants LYb2

and LYσ2

are so chosen that

LYb2

λ+

√K2(LY

σ2)2

∫ ∞0

s−β2

ρ2−2ρ2 e

−λ ρ2+2ρ2

sds =: LY

b2,σ2< 1

where

K2 =

(β2

e

)β2ρ2−2ρ2

ζρ2−2ρ2

2 κ2ρ22

and λ, β2, ρ2, ζ2, κ2 are all from Hypothesis 1.

Hypothesis 3.

I Hypothesis 3. b1 and σ1 grow at most linearly in X andsublinearly in Y . To be precise, there exists0 ≤ ζ < 1− β1(ρ1−2)

ρ1and a constant C > 0 such that

supx∈D

(|b1(x ,X ,Y )|+ |σ1(x ,X ,Y )|) ≤ C (1 + |X |+ |Y |ζ) .

Limit of the controlled fast–slow process.

I Controlled pair of fast–slow process (X ε,δ,u,Y ε,δ,u).

I The right way to describe the limit is a pair (ψ,P), which wecall a viable pair.

I ψ ∈ C ([0,T ]; H) is the limit dynamics, andP ∈P(U × Y × [0,T ]) is a certain invariant measure.

I This is an averaging procedure.

I Constructions of the same type have been carried outthoroughly in the case of finite dimensions, but for the firsttime in infinite dimensions in our work.

Limit of the controlled fast–slow process : Tightness.

I We introduce the family of random occupation measures

Pε,∆(dudYdt) =1

∆

∫ t+∆

t1du(u(s))1dY

(Y ε,δ,u(s)

)dsdt .

I After very technical proof we can show that the pair(X ε,δ,u,Pε,∆) is tight in the spaceC ([0,T ]; H)×P(U × Y × [0,T ]).

I We take weak topology on U and norm topology on Y, andwe make use of a–priori bounds and tightness functions.

Limit of the controlled fast–slow process : Limiting slowdynamics.

I Tightness guarantees that (X ε,δ,u,Pε,∆)→ (X̄ ,P) as ε→ 0(and therefore δ → 0) for some limiting (X̄ ,P).

I Recall that in mild form we can write

X ε,δ,u(t) = S1(t)X0 +

∫ t

0S1(t − s)B1(X ε,δ,u(s),Y ε,δ,u(s))ds

+

∫ t

0S1(t − s)Σ1(X ε,δ,u(s),Y ε,δ,u(s))Q1u(s)ds

+√ε

∫ t

0S1(t − s)Σ1(X ε,δ,u(s),Y ε,δ,u(s))dW Q1(s) .

I Limiting slow dynamics

ψ(t) = S1(t)X0+

∫U×Y×[0,t]

S1(t−s)ξ(ψ(s),Y , u)P(dudYds) ,

where

ξ(X ,Y , u) = Σ1(X ,Y )Q1u + B1(X ,Y ) .

Limit of the controlled fast–slow process : Limitingoccupation measure.

I In regards to the limit for the controlled fast process Y ε,δ,u,the limiting measure P characterizes simultaneously thestructure of the invariant measure of Y ε,δ,u and the controlfunction u.

I In general, these two objects are intertwined and coupledtogether into the measure P, so that the averaging withrespect to the measure P is different and hard to perform asin the classical averaging principle.


I P(dudYdt) = ηt(du|Y )µψt (dY |u)dt, and we cannotguarantee that µ is invariant measure to some process, as thisprocess is a controlled process and we have no regularityproperties of the optimal controls known other than beingsquare integrable.

I In finite dimensional case with periodic coefficients, theproblem can be resolved using the characterization of optimalcontrols through solutions to Hamilton–Jacobi–Bellmanequations.

I Such a characterization is not rigorously known in infinitedimensions and even if that becomes the case, one would needto establish sufficient properties of such equations that wouldthen imply that the resulting controlled process has a welldefined invariant measure that is regular enough.


I Is it possible to have P(dudYdt) = ηt(du|Y )µψt (dY )dt ?

I Recall that the fast motion can be written formally as

dY ε,δ,u(t) =1

δ2

[A2Y ε,δ,u(t) + B2(X ε,δ,u(t),Y ε,δ,u(t))

]dt+

+1

δ2

δ√ε

Σ2(X ε,δ,u(t),Y ε,δ,u(t))Q2u(t)dt+

+1

δΣ2(X ε,δ,u(t),Y ε,δ,u(t))dW Q2 ,

Y ε,δ,u(0) = Y0 ∈ H .

I Sendδ√ε→ 0 as ε→ 0 (and therefore δ → 0).

Hypothesis 5.

I Hypothesis 5. We assume that ε ↓ 0, δ = δ(ε) ↓ 0 and∆ = ∆(δ, ε) ↓ 0, such that

limε↓0

δ√ε

= 0, and limε↓0

δ

∆√ε

= 0 .


I From Hypothesis 5 we know that

limε↓0

δ√ε

= 0, and limε↓0

δ

∆√ε

= 0 .

I Fast motion without control but driven by the controlled slowprocess

dYε,δ(t) =1

δ2

[A2Y

ε,δ(t) + B2(X ε,δ,u(t),Yε,δ(t))]

dt+

+1

δΣ2(X ε,δ,u(t),Yε,δ(t))dW Q2 ,

Yε,δ(0) = Y0 ∈ H .

I One can show that

E1

∆

∫ T

0|Y ε,δ,u(t)−Yε,δ(t)|2Hdt → 0 ,

as ε→ 0 (and therefore δ → 0).


I Slogan : Ergodic properties are stable with respect to mildlyregular small perturbations.

I Knowing closeness of two processes in mean square sense isalready enough to draw conclusions about the equivalency oftheir invariant measures.

I

P(dudYdt) = ηt(du|Y )µψt (dY )dt .

Averaging of the controlled fast–slow process.

I DefinitionA pair (ψ,P) ∈ C ([0,T ]; L2(D))×P(U × Y × [0,T ]) will becalled viable with respect to (ξ,L), or simply viable if there is noconfusion, if the following are satisfied. The function ψ(t) isabsolutely continuous as a function in the space C ([0,T ]; H), P issquare integrable in the sense that∫

U×Y×[0,T ]

(|u|2U + |Y |2H

)P(dudYds) <∞

and the following hold for all t ∈ [0,T ] :

ψ(t) = S1(t)X0 +

∫U×Y×[0,t]

S1(t − s)ξ(ψ(s),Y , u)P(dudYds) ,


I Definitionthe measure P is such that

P ∈ P ={P ∈P(U × Y × [0,T ]) : P(dudYdt) = η(du|Y , t)µ(dY |t)dt,

µ(dY |t) = µψ(t)(dY ) for t ∈ [0,T ]

}where µX is the invariant measure for the uncontrolled fast processwith frozen slow component X , and

P(U × Y × [0, t]) = t .


I TheoremLet u ∈ PN

2 (U), (X ε,δ,u,Y ε,δ,u) be the mild solution to controlledfast process and T <∞. Let also Pε,∆(dudYdt) be the occupationmeasure. Assume Hypotheses 1, 2, 3 and 5, X0 ∈ Hθ

1 with θ > 0sufficiently small and Y0 ∈ H. Then, the family of processes X ε,δ,u

is tight in C ([0,T ]; H) and the family of measures Pε,∆ is tight inP(U × Y × [0,T ]), where U × Y × [0,T ] is endowed with theweak topology on U, the norm topology on Y and the standardtopology on [0,T ]. Hence, given any subsequence of{(X ε,δ,u,Pε,∆), ε, δ,∆ > 0}, there exists a subsubsequence thatconverges in distribution with limit (X̄ ,P). With probability 1, theaccumulation point (X̄ ,P) is a viable pair with respect to (ξ,L).

Large Deviations.

I Back to our large deviations problem : For every bounded andcontinuous function h : C ([0,T ]; H)→ R we want


[exp

(−1

εh(X ε,δ)

)]= inf

φ∈C([0,T ];H)[SX0(φ)+h(φ)] .

I Representation formula

−ε ln E

[exp

(−1

εh(X ε,δ)

)]= inf

u∈L2([0,T ];U)E

[1

2

∫ T


].

Large Deviations.

I TheoremLet (X ε,δ,Y ε,δ) be the mild solution to fast–slow SRDE and letT <∞. Assume Hypothesis 1, 2, 4, and 5 and let Y0 ∈ H andX0 ∈ Hθ

1 . Define

S(φ) = SX0(φ) = inf(φ,P)∈V(ξ,L)

[1

2

∫U×Y×[0,T ]

|u|2UP(dudYdt)

],

with the convention that the infimum over the empty set is ∞.

Large Deviations.

I TheoremThen, there exists θ̄ > 0 such that for all θ ∈ (0, θ̄] and X0 ∈ Hθ

1 ,and for every bounded and continuous functionh : C ([0,T ]; H)→ R we have


[exp

(−1

εh(X ε,δ)

)]= inf

φ∈C([0,T ];H)[SX0(φ)+h(φ)] .

In particular, {X ε,δ} satisfies the large deviations principle inC ([0,T ]; H) with action functional SX0(·), uniformly for X0 incompact subsets of Hθ

1 .

Large Deviations : Lower bound.

I By Fatou’s Lemma we have

lim infε↓0

(−ε ln EX0

[exp

(−h(X ε,δ)

ε

)])≥ lim inf

ε↓0

(EX0

[1

2

∫ T

0|uε(t)|2Udt + h(X ε,δ)

]− ε)

≥ lim infε↓0

(EX0

[1

2

∫ T

0

1

∆

∫ t+∆

t|uε(s)|2Udsdt + h(X ε,δ)

])= lim inf

ε↓0

(EX0

[1

2

∫U×Y×[0,T ]

|u|2UPε,∆(dudYdt) + h(X ε,δ)

])

≥ inf(φ,P)∈V(ξ,L)

[1

2

∫U×Y×[0,T ]

|u|2UP(dudYdt) + h(φ)

],

which concludes the proof of the Laplace principle lowerbound.

Large Deviations : Upper bound.

I In order to prove the Laplace principle upper bound, we haveto construct a nearly optimal control that achieves the upperbound.

I The nearly optimal control will be in feedback form withrespect to the Y variable,

I Due to infinite dimensionality we can only work it out in somespecial cases.

Hypothesis 4.

I Hypothesis 4. b1 is as in Hypothesis 3 and if d = 1, then thereare positive constants 0 < c0 ≤ c1 <∞ such that0 < c0 ≤ σ2

1(x ,X ,Y ) ≤ c1, or if d > 1 thenσ1(x ,X ,Y ) = σ1(x ,X ) is independent of Y and can grow atmost linearly in X uniformly in x ∈ D.

Thank you for your attention !

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