TRANSPORTATION COST-INFORMATION INEQUALITY FOR A STOCHASTIC HEAT EQUATION DRIVEN BY FRACTIONAL-COLORED NOISE∗

(李瑞囡)

School of Statistics and Information, Shanghai University of International Business and Economics,Shanghai 201620, China

E-mail: ruinanli@amss.ac.cn

Xinyu WANG (王新宇)†

Wenlan School of Business, Zhongnan University of Economics and Law, Wuhan 430073, China

E-mail: wangxin yu2000@hotmail.com

Abstract In this paper,we prove Talagrand’s T2 transportation cost-information inequality for the law of stochastic heat equation driven by Gaussian noise,which is fractional for a time variable with the Hurst index H ∈(,1),and is correlated for the spatial variable.The Girsanov theorem for fractional-colored Gaussian noise plays an important role in the proof.

Key words stochastic heat equation;transportation cost-information inequality;fractionalcolored noise

1 Introduction

The study of stochastic partial differential equations(SPDEs)driven by fractional Brownian motion (fBm) or other fractional Gaussian noises have become increasingly popular in recent years,as there are many applications in biology,electrical engineering,finance,physics,etc.;see,e.g.,[7,8,11,16,18].The purpose of this paper is to study Talagrand’s T2transportation cost-information inequality for the stochastic heat equation

whereu0(x) is the initial state,Lis the generator of a symmetric Lévy process taking values in Rd,bis a Lipschitz function with a Lipschitz constantKb,andBis a fractional-colored Gaussian noise with the Hurst indexH ∈（,1）in the time variable and with the spatial covariance functionfas in Balan and Tudor [1].That is,

is a centered Gaussian field with the covariance

whereS(Rd) is the space of all smooth functions with compact supports in Rd.

Transportation cost-information inequalities have been studied a great deal recently,especially in terms of their connection with the concentration of measure phenomenon,the log-Sobolev inequality,Poincaré’s inequality,and Hamilton-Jacobi’s equation;see [2,4,5,14,19,22,24,27,29]etc.

Let us first recall the transportation inequality.Let(E,d)be a metric space equipped with theσ-fieldBsuch thatd(·,·) isB×Bmeasurable,and letP(E) be the class of all probability measures onE.Givenp ≥1 and two probability measuresνandν′inP(E),the Wasserstein distance is defined by

where the infimum is taken over all of the probability measuresπonE×Ewith marginal distributionsνandν′.The relative entropy ofν′with respect to (w.r.t.for short)νis defined by

Definition 1.1The probability measureνis said to satisfy the transportation costinformation inequality Tp(C) on (E,d) if there exists a constantC>0 such that,for any probability measureν′∈P(E),

The T2(C)inequality was first established by Talagrand[27]for the Gaussian measure with the sharp constantC=2.The approach of Talagrand was generalized by Feyel and Üstünel[13]on the abstract Wiener space w.r.t.the Cameron-Martin distance.With regard to the path of the stochastic differential equation,by means of Girsanov’s transformation and the martingale representation theorem,the T2(C) w.r.t.theL2-distance was established by H.Djelloutet al.in [12].

Recently,the problems of transportation inequalities regarding stochastic heat equation have been widely studied.Wu and Zhang [32]studied the T2(C) w.r.t.theL2-norm by Galerkin’s approximation.By Girsanov’s transformation,Boufoussi and Hajji [6]obtained the T2(C) w.r.t.theL2-metric for the stochastic heat equations driven by space-time white noise and driven by fractional-white noise.Khoshnevisan and Sarantsev[17]also established the transportation inequalities under theL2-distance for a stochastic heat equation driven by spacetime white noise.Shang and Zhang[26]established the T2(C)w.r.t.the uniform metric for the stochastic heat equation driven by multiplicative space-time white noises;this was extended to the time-white and space-colored noise case by Shang and Wang [25].Wang and Zhang [31]studied the T2(C) for SPDEs with random initial values.Li and Wang [20]established the T2(C) w.r.t.the weightedL2-norm for a stochastic wave equation on R3.Ma and Wang [21]studied the T1(C) for the stochastic reaction-diffusion equations with Lévy noises.

In this paper,we shall study Talagrand’s T2-transportation inequality for the law of the solution of the stochastic heat equation (1.1) on the path space w.r.t.the weightedL2-norm.Our result generalizes the corresponding results of [6]and [17].The Girsanov theorem for the fractional-colored Gaussian noise plays an important role in the proof.

The rest of this paper is organized as follows: in Section 2,we give the properties of equation(1.1),and then state the main result of this paper.In Section 3,we shall prove the main result.We generalize the main result to stochastic heat diffusion equations with random initial values in Section 4.The existence and uniqueness of the solution to SPDE (1.1) is provided in the Appendix.

2 Background and Results

2.1 The Fractional-colored Noise and Stochastic Heat Equation

We first recall some facts about the integration of deterministic functions w.r.t.the fractional-colored noiseBfrom [1].

LetX={Xt}t≥0be a Lévy process taking values in Rd,withX0=0 and the characteristic exponent Ψ(ξ) given by

LetLbe the generator ofX.The domain ofLis given by

whereF-1denotes the inverse Fourier transform inL2（Rd）.

LetD（(0,T)×Rd）denote the space of all infinitely differentiable functions with compact support contained in (0,T)×Rd,and letHPbe the completion ofD（(0,T)×Rd）w.r.t.the inner product

whereqH=H(2H-1),F0,T ϕis the restricted Fourier transform ofϕin the variablet ∈(0,T) defined by

It follows from (2.1) and (2.2) that

LetB={B(ϕ):ϕ ∈D（(0,T)×Rd）} be a centered Gaussian process with covariance

For anyt>0 andA ∈B(Rd),one can defineBt(A)=B（1[0,t]×A）as theL2(Ω)-limit of the Cauchy sequence{B(ϕn)},whereconverges to 1[0,t]×Apointwisely.By a routine limiting argument,one can show that(2.3)remains valid whenϕandψare functions of the form 1[0,t]×Awitht>0 andA ∈B(Rd).LetEbe the space of all linear combinations of indicator functions 1[0,t]×A,wheret ∈[0,T],A ∈Bb(Rd),which is the class of all bounded Borel sets in Rd.One can extend the definition of E[B(ϕ)B(ψ)]toEby linearity.Then we have that

i.e.,ϕ →B(ϕ) is an isometry between (E,〈·,·〉HP) andHB,whereHBis the Gaussian space generated by｛B(ϕ),ϕ ∈D（(0,T)×Rd）｝.

Since the spaceHPis the completion ofEwith respect to〈·,·〉HP,the isometry (2.4) can be extended toHP,giving us the stochastic integral ofϕ ∈HPw.r.t.B.We denote this stochastic integral by

We assume that the Lévy processX={Xt}t≥0has a transition density which is given by

Denote thatgt,x(s,y)=pt-s(x-y)1{s

For any given positive bounded function Φ from Rdto R+satisfying that

is the weightedL2-metric.For example,for anyδ>0 satisfies (2.5).

For the existence and uniqueness of the solution to SPDE (1.1),we have the following result,which is a generalization of the linear case considered in [15](the proof of this result is inspired by [9]and is provided in the Appendix):

Then(1.1)has a unique solution{u(t,x),(t,x)∈[0,T]×Rd}in([0,T]×Rd)satisfying that,for all (t,x)∈[0,T]×Rd,

Remark 2.2Some concrete examples of symmetric Lévy processes satisfying condition(2.7) are the isotropicα-stable process,the independent sum of an isotropicα-sable process,and an isotropicγ-stable process withγ<α;see [15,Example 2.5].

Remark 2.3According to [15,Theorem 2.2],(2.7) implies that,for anyT>0,

2.2 Main Results

For anyµ ∈P(L2(Rd)),letPµdenote the distribution of the solution of SPDE (1.1) on the space([0,T]×Rd) such that the law ofu0isµ.In particular,ifµ=δu0for someu0∈L2(Rd),we writePu0:=Pδu0for short.

Theorem 2.4Assume thatu0∈L2(Rd) satisfiesand that (2.7)holds.Then the probability measurePu0satisfies T2(CT) on the space([0,T]×Rd).

As indicated in [3],many interesting consequences can be derived from Theorem 2.4 (see also Corollary 5.11 of [12]).For example,we give the following application of Theorem 2.4:

Corollary 2.5Under the condition of Theorem 2.4,the following statements hold for the constantCT:

(a) for any Lipschitz functionUon([0,T]×Rd) with

(b) (Hoeffding-type inequality) for any Lipschitz functionV:R→R with

we have that,for anyr ≥0,

ProofNote that T2(C)⇒T1(C),so part (a) holds automatically,by applying the equivalent condition of T1(C) given by [4,Theorem 3.1]to all Lipschitz functions.Moreover,the function

is Lipschitzian w.r.t.the weightedL2-metric‖·‖ΦandHence,part (b)follows from [12,Theorem 1.1].

3 The Proof of the Main Theorem

3.1 The Relationship Between the Fractional Time Noise and the White Time Noise

Recall that the kernel function is defined by

(see [23]).

LetPbe the completion ofD（(0,T)×Rd）w.r.t.the inner product

Define the transfer operator by

Due to (3.1),we know that

For anyφ ∈P,define that

that is,M={M(φ);φ ∈P} is a Gaussian noise,which is white in time and has a spatial covariance functionf.Denote that

HereM(φ) is Dalang’s stochastic integral w.r.t.the noiseM.Then

We will apply Girsanov’s theorem to prove Theorem 2.4.To do this,we need the next lemma,which describes all probability measures which are absolutely continuous w.r.t.Pu0.This is analogous to [12,Theorem 5.6]in the setting of finite-dimensional Brownian motion,[17,Lemma 3.1]in the setting of space-time white noise,and [25,Theorem 3.1]in the setting of time-white and space-colored noise.For completeness,we give the proof here.

Lemma 3.1For every probability measureQ ≪Pu0on the space,define a new probability measure Q on the probability space (Ω,F,P) by

Then there exists an adapted processh={h(t,x),(t,x)∈[0,T]×Rd} such that‖h‖HP<∞,Q-a.s.,and,for anyφ ∈HP,

is a centered Gaussian process with the covariance

whereεeQdenotes the expectation under the probability measure Q.This means that,under the probability Q,={(φ);φ ∈HP} is a fractional-colored Gaussian noise with a Hurst indexin the time variable and spatial covariance functionf.Furthermore,the relative entropy is given by

ProofFrom (2.8) and (3.3),we know that

Here,Mis a time-white and space-colored Gaussian noise with the spatial covariance functionf.Denote byHthe Hilbert space obtained by the completion ofS(Rd)w.r.t.the inner product

The norm induced by〈·,·〉His denoted by‖·‖H.

According to [25,Lemma 3.1],we know that,for every probability measureQ ≪Pu0on the space([0,T]×Rd),there exists an adaptedH-valued processk={k(t),t ∈[0,T]} such that

and,for anyφ ∈P,

is a centered Gaussian process with the covariance

Hence,by [25,Theorem 3.1],we know that under the probability Q,is a time-white and space-colored Gaussian noise with the spatial covariance functionf.Furthermore,

The proof is complete.

3.2 The Proof of Theorem 2.4

SincePu0is a probability measure on the metric space (([0,T]×Rd),‖·‖Φ),in order to prove our main result,we need to prove that

holds for any probability measureQon([0,T]×Rd) and some positive constantCindependent ofQ.Obviously,it is enough to prove the result for any probability measureQon([0,T]×Rd)such thatQ ≪Pu0and H(Q|Pu0)<∞.Let(Ω,F,)be a complete probability space on whichBis a fractional-colored Gaussian noise with a Hurst indexH ∈（,1）in the time variable and spatial covariance functionf.Let

Letu={u(t,x);(t,x)∈[0,T]×Rd} be the unique solution of (1.1) with the initial conditionu0.Then the law ofuisPu0.Consider that

LetM,,kbe the same as in the proof of Lemma 3.1.Then

Thus,to prove the main result,it is enough to prove it forQn.Without loss of generality,we

According to Lemma 3.1,we couple (Pu0,Q) as the law of a process (u,v) under Q as

By the definition of the Wasserstein distance,we have that

In view of (3.11) and (3.15),it remains to prove that

for some positive constantCindependent ofk.

From (3.13) and (3.14),we can representu(t,x)-v(t,x) as

For everyt ∈[0,T],define that

By (2.9) and the boundedness of,we know thatη(t)<∞for anyt ∈[0,T].

By the Cauchy-Schwarz inequality w.r.t.dtand the probability measuregt,x(s,y)dyon Rd,respectively,and by the Lipschitz continuity ofb,we obtain that,for anyt ≤T,

By the Cauchy-Schwarz inequality,we have,for anyt ≤T,that

whereCg,Tis defined by (2.10).Putting (3.17)–(3.19) together,we have,for any 0≤t ≤T,that

Using Grönwall’s inequality,we obtain that

Thus,we have that

This implies that (3.16) holds with

The proof is complete.

4 Talagrand’s Inequality for SPDEs with Random Initial Values

In this section,we establish Talagrand’s inequality for the stochastic heat equation (1.1)with random initial values independent ofB.

LetCb(Rd) be the space of all bounded continuous functions on Rd,endowed with the uniform metric

Inspired by [31,Theorem 3.1],we get the following result for SPDE (1.1) with a random initial value inCb(Rd):

Theorem 4.1Assume that (2.7) holds.Letµ∈P(Cb(Rd)).Then

holds for some constantC>0 if

holds for some constantc>0.

ProofBy Theorem 2.1 in [31]and Theorem 2.4,to prove Theorem 4.1,it suffices to prove that

Letu(t,x) andv(t,x) be the unique solutions of (1.1) with initial valuesu0andv0,respectively.We know thatu(t,x) satisfies (2.8) andv(t,x) satisfies (2.8) withu0replaced byv0.Hence

For the first term,it is easy to get,for some constant>0,that

Again,defineηby

According to(2.9),we know thatη(t)<∞for anyt ∈[0,T].The same calculation of eq.(3.18)implies that,for anyt ≤T,x ∈Rd,

Applying the same procedure as that in the proof of Theorem 2.4,we get that

Since the law of (u(t,x),v(t,x))(t,x)∈[0,T]×Rdis a coupling ofPu0andPv0,we have that

The proof is complete.

Conflict of InterestThe authors declare no conflict of interest.

Appendix

Proof of Theorem 2.1For any givenT>0,we will prove that (1.1) has a unique solution in([0,T]×Rd),which satisfies that

Sincebis a Lipschitz function,there exists some constantK>0 such that

We will follow a Picard iteration scheme.Denote thatAssuming thatunhas been defined forn ≥0,set

First,we show that,for anyT>0 andn ≥0,

Assume by induction that,for anyT>0,

By (A.3) we have that

the sequence{un,n ≥0} is well defined.

Next,we prove that,forT>0,

Using (A.3),(A.2),the definition ofgt,x(s,y) and (2.10),there exists some positive constantsuch that

By the extension of Grönwall’s lemma ([9,Lemma 15]),we know that (A.7) holds.

We first show that the sequence{un(t,x),n ≥0} converges inL2(Ω).Let

Using (A.2) and (A.3),a similar calculation as to that above implies that

By the assumption onu0(x) and (A.4),we get that,which together with the extension of Grönwall’s lemma ([9,Lemma 15]),yields that,asm,n →∞,

This implies that the sequenceun(t,x) converges inL2(Ω),uniformly in time and space,to a stochastic processu(t,x) asn →∞,and thatu(t,x) is the solution of (1.1).Moreover,by(A.7),we have that

In addition,by the definition of the‖·‖Φnorm,Minkowski’s inequality and (A.12),we obtain that

Hence the random processuis inThe uniqueness is proven by the same argument.

The proof is complete.