Moment bounds of a class of stochastic heat equations driven by space–time colored noise in bounded domains

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Abstract

We consider the fractional stochastic heat type equation tut(x)=(Δ)α2ut(x)+ξσ(ut(x))Ḟ(t,x),xD,t>0,with nonnegative bounded initial condition, where α(0,2], ξ>0 is the noise level, σ:RR is a globally Lipschitz function satisfying some growth conditions and the noise term Ḟ behaves in space like the Riez kernel and is possibly correlated in time and D is the unit open ball centered at the origin in Rd. When the noise term is not correlated in time, we establish a change in the growth of the solution of these equations depending on the noise level ξ. On the other hand when the noise term behaves in time like the fractional Brownian motion with index H(12,1), We also derive explicit bounds leading to a well-known intermittency property.

Introduction

Stochastic Partial Differential Equations (SPDEs) have been studied a lot recently due to many challenging open problems in the area but also due to their deep applications in disciplines that range from applied mathematics, statistical mechanics, and theoretical physics, to theoretical neuroscience, theory of complex chemical reactions [including polymer science], fluid dynamics, and mathematical finance, see for example [18] for an extensive list of literature devoted to the subject. On the other hand SPDEs driven by a random noise which is white in time but colored in space have increasingly received a lot of attention recently, following the foundational work of [8]. One difference with SPDEs driven by space–time white noise is that they can be used to model more complex physical phenomena which are subject to random perturbations. Two phenomena of interest are usually observed when studying these SPDEs, “intermittency” and “phase transition”. See for example [1], [2], [3], [4], [16] and [12] for the former and [11], [15], [19] and [26] for the latter.

In this article, we consider an SPDE driven by a space–time colored noise. This type of equation has received a lot of attention recently, see for example [1], [2], [4], [16] and the references therein. The novelty is that we assume the space to be a proper bounded open subset of Rd.

Consider the fractional stochastic heat equation on the open unit ball D subset of Rd,d1 with zero exterior Dirichlet boundary conditions: tut(x)=(Δ)α2ut(x)+ξσ(ut(x))Ḟ(t,x)xD,t>0,ut(x)=0xDcwhere α(0,2], (Δ)α2 is the L2generator of a symmetric αstable process killed upon exiting the domain D. The initial condition u0(x) is a bounded and nonnegative function. The coefficient ξ denotes the level of the noise, σ:RR is a globally Lipschitz function. The mean zero Gaussian process Ḟ is a space–time colored noise, i.e. E(Ḟ(t,x)Ḟ(s,y))=γ(ts)Λ(xy),where γ:RR+ and Λ:RdR+ are general nonnegative and nonnegative definite(generalized) functions satisfying some integrability conditions. The Fourier transform of the latter, Λˆ=μ is a tempered measure. We first focus our attention on the case where the noise term is uncorrelated in time.

The objective of this paper is to provide lower and upper bounds for the moments of the stochastic fractional heat equation (1.1). But first, let us define some terms and expressions we will use in this paper.

Definition 1.1

Assume γ=δ0. Following [25], a random field {ut(x)}t>0,xD is called a mild solution of (1.1) in the Walsh–Dalang sense if

  • 1.

    ut(x) is jointly measurable in t0 and xD;

  • 2.

    for all (t,x)R+×D, the stochastic integral 0tDpD(ts,x,y)σ(us(y))F(dy,ds) is well-defined in L2(Ω); Moreover, supt>0supxDE|ut(x)|p<,for all p2;

  • 3.

    The following integral equation holds in L2(Ω): ut(x)=(Gu0)t(x)+ξ0tDpD(ts,x,y)σ(us(y))F(dy,ds),where (Gu0)t(x)DpD(t,x,y)u0(y)dyand pD(t,x,y) denotes the Dirichlet heat kernel of the stable Lévy process. It is the transition density of the stable Lévy process killed in the exterior of D. Please refer to Section 2 for a short description of the latter.

When γ=δ0, following Dalang [8], it is well-known that if the spectral measure satisfies the following condition: Rdμ(dζ)1+|ζ|α<,then there exists a unique random field solution of (1.1). Please refer to Appendix for the proof of existence and uniqueness of a random field solution in this case.

Some examples of space correlation functions satisfying condition (1.4) include

  • Space–time white noise: Λ=δ0 in which case μ(dζ)=dζ and (1.4) holds only when α>d which implies d=1 and 1<α2.

  • Riez Kernel: Λ(x)=|x|β,0<β<d. Here μ(dζ)=c|ζ|(dβ)dζ and (1.4) holds whenever β<α.

  • Bessel kernel: Λ(x)=0yηd2eye|x|24ydy. μ(dζ)=c(1+|ζ|2)η2dζ and (1.4) implies η>dα.

  • Fractional Kernel: Λ(x)=i=1d|xi|2Hi2. μ(ζ)=ci=1d|xi|12Hidζ and (1.4) holds whenever i=1dHi>dα2.

We refer the interested reader to [13] for more examples of such functions. We now turn our attention on the case where the noise term is also correlated in time.

Definition 1.2

Assume σ=Id, the identity map. An adapted random field {ut(x)}t>0,xD such that E[ut(x)]2< for all (t,x) is a mild solution to (1.1) in the Skorohod sense if for any (t,x)R+×D, the process {pD(ts,x,y)us(y)1[0,t](s):s0,yD} is Skorohod integrable and the following integral equation holds: ut(x)=(Gu0)t(x)+ξ0tDpD(ts,x,y)us(y)F(δs,δy).

It is well-known that a unique mild solution (1.5) exists in the Skorohod sense provided that the time correlation γ is locally integrable and the space correlation Λ satisfies condition (1.4). When handling the mild solution in the Skorohod sense, we shall make use of the Wiener-chaos expansion.

Recall that the covariance given by (1.2) is a mere formal notation. Let C0(R+×Rd) be the space of test functions on R+×Rd. Then on a complete probability space (Ω,F,P), we consider a family of centered Gaussian random variables indexed by the test function {F(φ),φC0(R+×Rd)} with covariance E[Ḟ(φ)Ḟ(ψ)]=R+2×R2dφ(t,x)ψ(s,y)γ(ts)Λ(xy)dxdydtds.We write Eq. (1.6) formally as (1.2). Let H be the completion of C0(R+×Rd) with respect to the inner product φ,ψH=R+2×R2dφ(t,x)ψ(s,y)γ(ts)Λ(xy)dxdydtds.

The mapping φF(φ)L2(Ω) is an isometry which can be extended to H. We denote this map by F(φ)=R+×Rdφ(t,x)F(dt,dx),φH.Note that if φ,ψH, E[Ḟ(φ)Ḟ(ψ)]=φ,ψH.

Furthermore, H contains the space of measurable functions φ on R+×Rd such that R+2×R2d|φ(t,x)φ(s,y)|γ(ts)Λ(xy)dxdydtds<.For n0, denote by Hn the nth Wiener-chaos of F. Recall that H0 is just R and for n1, Hn is the closed linear subspace of L2(Ω) generated by the random variables {Hn(F(h)),hH,hH=1} where Hn is the nth Hermite polynomial. For n1, we denote by Hn(resp.Hn) the nth tensor product (resp. the nth symmetric tensor product) of H. Then, the mapping In(hn)=Hn(F(h)) can be extended to a linear isometry between Hn (equipped with the modified norm n!.Hn) and Hn, see for example [10] and  [21] and the references therein.

Consider now a random variable XL2(Ω) measurable with respect to the σfield FF generated by F. This random variable can be expressed as X=E[X]+n=1In(fn),where the series converges in L2(Ω) and the elements fnHn,n1 are determined by X. This identity is known as the Wiener-chaos expansion. Please refer to [10] and [21] for a complete description on the matter. We will need the following assumptions:

Assumption 1.3

γ:RR+ is locally integrable.

Assumption 1.4

There exist constants C1 and C2 and 0<β<αd such that for all xRd, C1|x|βΛ(x)C2|x|β.

Assumption 1.5

There exist positive constants lσ and Lσ such that for all xRd, lσ|x|σ(x)Lσ|x|.

Assumption 1.6

There is ϵ(0,12) such that infxDϵu0(x)>0,where DϵB1ϵ(0)

Throughout the remainder of this paper, α(0,2], the letter C or c with or without subscript(s) denotes a constant with no major importance to our study, Assumption 1.4, Assumption 1.6 hold unless stated otherwise. BR(0) represents the open ball of radius R centered at the origin. We are now ready to state our main results.

Theorem 1.7

Assume γ=δ0 and σ satisfies Assumption 1.5 . Then for all t>0 and p2, there exist positive constants c1,c2(α,β,d,lσ),C1 and C2(α,β,d,Lσ) such that for all ξ>0 and δ>0, c1pept(c2ξ2ααβμ1)infxDϵE|ut(x)|psupxDE|ut(x)|pC1pept(C2ξ2ααβzp2ααβ(1δ)μ1),where zp is the constant in the Burkhölder–Davis–Gundy’s inequality.

This theorem shows that the rate at which the moments of the solution to Eq. (1.1) exponentially grow or decay depends explicitly on the non-local operator (Δ)α2, the noise level ξ and the noise term via the quantity ξ2ααβ. This result provides an extension to [22] where the author used equation (1.1) with σ=Id, an essential assumption when using the Wiener-Chaos expansion in the proofs. However, the proof we provide for this theorem uses a different argument. This theorem also provides an extension to [14] where similar bounds were obtained but only for the second moments of the solution to Eq. (1.1). While showing explicitly the dependence of moments of solution of Eq. (1.1) with the noise level ξ, it also implies that there exist ξ0(p)>0 such that for all ξ<ξ0 and xD, <lim supt1tlogE|u(t,x)|p<0,and there exists ξ1(p) such that for all ξ>ξ1 and for all ϵ>0, xDϵ, 0<lim inft1tlogE|u(t,x)|p<.These results were proved in [15, for the case α=2] and [11, for 0<α<2] but without showing the explicit dependence of the moments on ξ.

The results provided in this Theorem also lead to yet another phenomenon known as intermittency. Define the pth upper Lyapunov moment of the random field u{ut(x)}t>0,xD at x0D as γ¯(p)lim supt1tlogE|ut(x)|pfor all p(0,).Following [12], the random field u is said to be weakly intermittent if: for allxD,γ¯(2)>0andγ¯(p)<for all p(2,).It is said to be fully intermittent if: pγ¯(p)pis strictly increasing for all p2andxD.

It is also known that weak-intermittency can sometimes imply full intermittency, see for example [12] and the references therein. In Theorem 1.7, when ξ<(μ1C(p,δ))αβ2α, the solution u is not weakly-intermittent. However, quite the opposite situation occurs for the same random field u when ξ>(μ1C1(p))αβ2α.

The next result is concerned with the space–time colored noise case.

One of the time correlation functions that has received a lot of attention lately is the correlation function of the so-called fractional Brownian motion (of index H) i.e. γ(r)=CH|r|2H2,forH(12,1)andCH=H(2H1).Note that this function clearly satisfies Assumption 1.3. We refer the interested reader to [1] and the references therein for more information about this function

Theorem 1.8

Assume σ(x)=x and γ is given by (1.7). Then for all t>0 and p2, there exist positive constants c¯1,c¯2(α,β),C¯1 and C¯2(α,β) such that for all ξ>0 and δ>0, c¯1pep(c¯2t2Hαβαβξ2ααβμ1t)infxDϵE|ut(x)|psupxDE|ut(x)|pC¯1peC¯2p((p1)ααβt2Hαβαβξ2ααβ(μ1δ)t).

Though these bounds might not be very sharp, to the best of our knowledge, this is the first paper ever to examine the moments of the solution of SPDEs driven by such type of noise in bounded domains. Notice again the dependence of moments with the noise level.

Remark 1.9

Theorem 1.8 also holds for more general time correlation functions satisfying Assumption 1.3. Please refer to the proof of Theorem 1.8 for more details.

When Eq. (1.1) is driven by a noise correlated in time, we observe a different notion of weak intermittency. This is obtained by a modification of the Lyapunov exponent. For ρ>0 and xD we define the modified upper Lyapunov exponent (of index ρ) by γ¯ρ(p)lim supt1tρlogE|ut(x)|pfor all p(0,). u is weakly ρ intermittent if for allxD,γ¯ρ(2)>0andγ¯ρ(p)<for all p(2,).It is said to be fully ρintermittent if: pγ¯ρ(p)pis strictly increasing for all p2andxD. Theorem 1.8 shows in fact that u is ρweakly-intermittent for all ξ>0 since ρ2Hαβαβ>1. Similar result was obtained in [1] and [2] and [16] for the case α=2, but the authors worked on the entire Euclidean space Rd. Furthermore, in [1], the bounds were not obtained for all values of t>0. In addition, the value of ρ=2Hαβαβ matches the value of ρ found in [1] and ρh found in [2] for α=2.

Corollary 1.10

Under the assumptions in Theorem 1.8, we have 0<lim inft1tρlogE|ut(x)|plim supt1tρlogE|ut(x)|p<for allxDϵ,where ρ=2Hαβαβ>1.

The rest of the paper is organized as follows: in Section 2, we provide several estimates needed for the proofs of our results; Section 3 is devoted to the proofs of our main results and this paper ends with an Appendix where useful results from other authors are compiled.

Section snippets

Preliminaries

The Dirichlet heat kernel will play a major role in the proof of our results. Here we give a few details about it. We define the “killed process”: XtD=Xtt<τD0tτD,where τD=inf{t>0:XtD} is the first exiting time.

Define rD(t,x,y)Ex[p(tτD,XτD,y);τD<t],then pD(t,x,y)=p(t,x,y)rD(t,x,y),where p(t,.,.) is the transition density of the “unkilled process” Xt. Note that p(t,x,y) is also written p(t,xy) in some literature.

When α=2, Xt corresponds to a Brownian motion (Wiener process) (Bt)t0 with

Proofs of the main results

Proof of Theorem 1.7

For the upper bound, we combine the Burkhölder–Davis–Gundy’s, Minkowski’s and Jensen’s inequalities after taking the pth power of the mild solution to get E|ut(x)|p2p1{((Gu0)t(x))p+ξpzpp(0tD2pD(ts,x,y)pD(ts,x,z)Λ(yz)×E|σ(us(y))σ(us(z))|dydzds)p2}2p1{((Gu0)t(x))p+ξpzpp(0t(supyDE|σ(us(y))|p)2p×D2pD(ts,x,y)pD(ts,x,z)Λ(yz)dydzds)p2}

Where zp is as in Theorem 1.7, See for example [12]. Note that we have also used the following fact straight from Hölder’s inequality: E|σ(us(y))σ(us(z)

Acknowledgments

The authors would like to sincerely thank the editor and an unanimous referee for useful comments that improved the paper very much. The authors certify that they have NO affiliations with or involvement in any organization or entity with any financial interest (such as honoraria; educational grants; participation in speakers’ bureaus; membership, employment, consultancies, stock ownership, or other equity interest; and expert testimony or patent-licensing arrangements), or non-financial

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