Moment bounds of a class of stochastic heat equations driven by space–time colored noise in bounded domains
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 .
Consider the fractional stochastic heat equation on the open unit ball subset of with zero exterior Dirichlet boundary conditions: where , is the generator of a symmetric stable process killed upon exiting the domain . The initial condition is a bounded and nonnegative function. The coefficient denotes the level of the noise, is a globally Lipschitz function. The mean zero Gaussian process is a space–time colored noise, i.e. where and are general nonnegative and nonnegative definite(generalized) functions satisfying some integrability conditions. The Fourier transform of the latter, 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 . Following [25], a random field is called a mild solution of (1.1) in the Walsh–Dalang sense if is jointly measurable in and ; for all , the stochastic integral is well-defined in ; Moreover, ; The following integral equation holds in : where and 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 . Please refer to Section 2 for a short description of the latter.
When , following Dalang [8], it is well-known that if the spectral measure satisfies the following condition: 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
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Space–time white noise: in which case and (1.4) holds only when which implies and .
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Riez Kernel: . Here and (1.4) holds whenever .
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Bessel kernel: . and (1.4) implies .
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Fractional Kernel: . and (1.4) holds whenever .
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 , the identity map. An adapted random field such that for all is a mild solution to (1.1) in the Skorohod sense if for any , the process is Skorohod integrable and the following integral equation holds:
It is well-known that a unique mild solution (1.5) exists in the Skorohod sense provided that the time correlation is locally integrable and the space correlation 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 be the space of test functions on . Then on a complete probability space , we consider a family of centered Gaussian random variables indexed by the test function with covariance We write Eq. (1.6) formally as (1.2). Let be the completion of with respect to the inner product
The mapping is an isometry which can be extended to . We denote this map by Note that if ,
Furthermore, contains the space of measurable functions on such that For , denote by the th Wiener-chaos of . Recall that is just and for , is the closed linear subspace of generated by the random variables where is the th Hermite polynomial. For , we denote by the th tensor product (resp. the th symmetric tensor product) of . Then, the mapping can be extended to a linear isometry between (equipped with the modified norm ) and , see for example [10] and [21] and the references therein.
Consider now a random variable measurable with respect to the field generated by . This random variable can be expressed as where the series converges in and the elements are determined by . 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 is locally integrable.
Assumption 1.4 There exist constants and and such that for all ,
Assumption 1.5 There exist positive constants and such that for all ,
Assumption 1.6 There is such that where
Throughout the remainder of this paper, , 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. represents the open ball of radius centered at the origin. We are now ready to state our main results.
Theorem 1.7 Assume and satisfies Assumption 1.5 . Then for all and , there exist positive constants and such that for all and , where 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 , the noise level and the noise term via the quantity . This result provides an extension to [22] where the author used equation (1.1) with , 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 , it also implies that there exist such that for all and , and there exists such that for all and for all , , These results were proved in [15, for the case ] and [11, for ] but without showing the explicit dependence of the moments on .
The results provided in this Theorem also lead to yet another phenomenon known as intermittency. Define the th upper Lyapunov moment of the random field at as Following [12], the random field is said to be weakly intermittent if: It is said to be fully intermittent if:
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 , the solution is not weakly-intermittent. However, quite the opposite situation occurs for the same random field when .
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 ) i.e. 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 and is given by (1.7). Then for all and , there exist positive constants and such that for all and ,
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 and we define the modified upper Lyapunov exponent (of index ) by is weakly intermittent if It is said to be fully intermittent if: Theorem 1.8 shows in fact that is weakly-intermittent for all since . Similar result was obtained in [1] and [2] and [16] for the case , but the authors worked on the entire Euclidean space . Furthermore, in [1], the bounds were not obtained for all values of . In addition, the value of matches the value of found in [1] and found in [2] for .
Corollary 1.10 Under the assumptions in Theorem 1.8, we have where .
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”: where is the first exiting time.
Define then where is the transition density of the “unkilled process” . Note that is also written in some literature.
When , corresponds to a Brownian motion (Wiener process) 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 th power of the mild solution to get Where is as in Theorem 1.7, See for example [12]. Note that we have also used the following fact straight from Hölder’s inequality:
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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