Research Article | Open Access
SPDEs with -Stable Lévy Noise: A Random Field Approach
This paper is dedicated to the study of a nonlinear SPDE on a bounded domain in , with zero initial conditions and Dirichlet boundary, driven by an -stable Lévy noise with , , and possibly nonsymmetric tails. To give a meaning to the concept of solution, we develop a theory of stochastic integration with respect to this noise. The idea is to first solve the equation with “truncated” noise (obtained by removing from the jumps which exceed a fixed value ), yielding a solution , and then show that the solutions coincide on the event , for some stopping times converging to infinity. A similar idea was used in the setting of Hilbert-space valued processes. A major step is to show that the stochastic integral with respect to satisfies a th moment inequality. This inequality plays the same role as the Burkholder-Davis-Gundy inequality in the theory of integration with respect to continuous martingales.
Modeling phenomena which evolve in time or space-time and are subject to random perturbations are a fundamental problem in stochastic analysis. When these perturbations are known to exhibit an extreme behavior, as seen frequently in finance or environmental studies, a model relying on the Gaussian distribution is not appropriate. A suitable alternative could be a model based on a heavy-tailed distribution, like the stable distribution. In such a model, these perturbations are allowed to have extreme values with a probability which is significantly higher than in a Gaussian-based model.
In the present paper, we introduce precisely such a model, given rigorously by a stochastic partial differential equation (SPDE) driven by a noise term which has a stable distribution over any space-time region and has independent values over disjoint space-time regions (i.e., it is a Lévy noise). More precisely, we consider the SPDE: with zero initial conditions and Dirichlet boundary conditions, where is a Lipschitz function, is a second-order pseudo-differential operator on a bounded domain , and is the formal derivative of an -stable Lévy noise with , . The goal is to find sufficient conditions on the fundamental solution of the equation on , which will ensure the existence of a mild solution of (1). We say that a predictable process is a mild solution of (1) if for any , , We assume that is a function in , which excludes from our analysis the case of the wave equation with .
To explain the connections with other works, we describe briefly the construction of the noise (the details are given in Section 2). This construction is similar to that of a classical -stable Lévy process and is based on a Poisson random measure (PRM) on of intensity , where for some , with . More precisely, for any set , where is the compensated process and is a constant (specified by Lemma 3). Here, is the class of bounded Borel sets in and is the Lebesgue measure of .
As the term on the right-hand side of (2) is a stochastic integral with respect to , such an integral should be constructed first. Our construction of the integral is an extension to random fields of the construction provided by Giné and Marcus in  in the case of an -stable Lévy process . Unlike these authors, we do not assume that the measure is symmetric.
Since any Lévy noise is related to a PRM, in a broad sense, one could say that this problem originates in Itô’s papers [2, 3] regarding the stochastic integral with respect to a Poisson noise. SPDEs driven by a compensated PRM were considered for the first time in , using the approach based on Hilbert-space-valued solutions. This study was motivated by an application to neurophysiology leading to the cable equation. In the case of the heat equation, a similar problem was considered in [5–7] using the approach based on random-field solutions. One of the results of  shows that the heat equation: has a unique solution in the space of predictable processes satisfying , for any . In this equation, is the compensated process corresponding to a PRM on of intensity , for an arbitrary -finite measure space with measure satisfying . Because of this later condition, this result cannot be used in our case with and . For similar reasons, the results of  also do not cover the case of an -stable noise. However, in the case , we will be able to exploit successfully some ideas of  for treating the equation with “truncated” noise , obtained by removing from the jumps exceeding a value (see Section 5.2).
The heat equation with the same type of noise as in the present paper was examined in [8, 9] in the cases and , respectively, assuming that the noise has only positive jumps (i.e., ). The methods used by these authors are different from those presented here, since they investigate the more difficult case of a non-Lipschitz function with . In , Mueller removes the atoms of of mass smaller than and solves the equation driven by the noise obtained in this way; here we remove the atoms of of mass larger than and solve the resulting equation. In , Mytnik uses a martingale problem approach and gives the existence of a pair which satisfies the equation (the so-called “weak solution”), whereas in the present paper we obtain the existence of a solution for a given noise (the so-called “strong solution”). In particular, when and , the existence of a “weak solution” of the heat equation with -stable Lévy noise is obtained in  under the condition which we encounter here as well. It is interesting to note that (6) is the necessary and sufficient condition for the existence of the density of the super-Brownian motion with “”-stable branching (see ). Reference  examines the heat equation with multiplicative noise (i.e., ), driven by an -stable Lévy noise which does not depend on time.
To conclude the literature review, we should point out that there are many references related to stochastic differential equations with -stable Lévy noise, using the approach based on Hilbert-space valued solutions. We refer the reader to Section 12.5 of the monograph  and to [13–16] for a sample of relevant references. See also the survey article  for an approach based on the white noise theory for Lévy processes.
This paper is organized as follows.(i)In Section 2, we review the construction of the -stable Lévy noise , and we show that this can be viewed as an independently scattered random measure with jointly -stable distributions.(ii)In Section 3, we consider the linear equation (1) (with ) and we identify the necessary and sufficient condition for the existence of the solution. This condition is verified in the case of some examples.(iii)Section 4 contains the construction of the stochastic integral with respect to the -stable noise , for . The main effort is dedicated to proving a maximal inequality for the tail of the integral process, when the integrand is a simple process. This extends the construction of  to the case random fields and nonsymmetric measure .(iv)In Section 5, we introduce the process obtained by removing from the jumps exceeding a fixed value , and we develop a theory of integration with respect to this process. For this, we need to treat separately the cases and . In both cases, we obtain a th moment inequality for the integral process for if and if . This inequality plays the same role as the Burkholder-Davis-Gundy inequality in the theory of integration with respect to continuous martingales.(v)In Section 6 we prove the main result about the existence of the mild solution of (1). For this, we first solve the equation with “truncated” noise using a Picard iteration scheme, yielding a solution . We then introduce a sequence of stopping times with a.s. and we show that the solutions , coincide on the event . For the definition of the stopping times , we need again to consider separately the cases and .(vi)Appendix A contains some results about the tail of a nonsymmetric stable random variable and the tail of an infinite sum of random variables. Appendix B gives an estimate for the Green function associated with the fractional power of the Laplacian. Appendix C gives a local property of the stochastic integral with respect to (or ).
2. Definition of the Noise
In this section we review the construction of the -stable Lévy noise on and investigate some of its properties.
Let be a Poisson random measure on , defined on a probability space , with intensity measure , where is given by (3). Let be a sequence of positive real numbers such that as and . Let
For any set , we define
Remark 1. The variable is finite since the sum above contains finitely many terms. To see this, we note that , and hence .
For any , the variable has a compound Poisson distribution with jump intensity measure ; that is, It follows that and for any . Hence, for any and . If , then is finite. Define This sum converges a.s. by Kolmogorov’s criterion since are independent zero-mean random variables with .
Lemma 2. The family defined by (10) is an independently scattered random measure; that is,(a)for any disjoint sets in , are independent;(b)for any sequence of disjoint sets in such that is bounded, a.s.
Proof. (a) Note that for any function with compact support , we can define the random variable where . For any , we have
For any disjoint sets and for any , we have using (12) with for the second equality and (9) for the last equality. This proves that are independent.
(b) Let and , where . By Lévy’s equivalence theorem, converges a.s. if and only if it converges in distribution. By (13), with for all , we have This clearly converges to , and hence converges in distribution to .
Recall that a random variable has an -stable distribution with parameters , , , and if, for any , or (see Definition of ). We denote this distribution by .
Lemma 3. has a distribution with , and . If , then .
Proof. We first express the characteristic function (11) of in Feller’s canonical form (see Section XVII.2 of ): with and . Then the result follows from the calculations done in Example XVII.3.(g) of .
We say that is an -stable Lévy noise. Coming back to the original construction (10) of and noticing that it follows that can be represented as Here is the compensated Poisson measure associated with ; that is, for any relatively compact set in .
In the case , we will assume that so that is symmetric around , for all , and admits the same representation as in the case .
3. The Linear Equation
As a preliminary investigation, we consider first equation (1) with : with zero initial conditions and Dirichlet boundary conditions. In this section is a bounded domain in or .
We define now the stochastic integral of a deterministic function :
If , this can be defined by approximation with simple functions, as explained in Section 3.4 of . The process has jointly -stable finite dimensional distributions. In particular, each has a -distribution with scale parameter:
More generally, a measurable function is integrable with respect to if there exists a sequence of simple functions such that a.e., and, for any , the sequence converges in probability (see ).
The next results show that condition is also necessary for the integrability of with respect to . Due to Lemma 2, this follows immediately from the general theory of stochastic integration with respect to independently scattered random measures developed in .
Lemma 4. A deterministic function is integrable with respect to if and only if .
Proof. We write the characteristic function of in the form used in : with , if and if . By Theorem 2.7 of , is integrable with respect to if and only if where and . Direct calculations show that, in our case, if , if , and .
Proposition 5. Equation (23) has a mild solution if and only if for any , In this case, has jointly -stable finite-dimensional distributions. In particular, has a distribution.
Condition (29) can be easily verified in the case of several examples.
Example 6 (heat equation). Let . Assume first that . Then , where and condition (29) is equivalent to (6). In this case, . If is a bounded domain in , then (see page 74 of ) and condition (29) is implied by (6).
Example 7 (parabolic equation). Let where is the generator of a Markov process with values in , without jumps (a diffusion). Assume that is a bounded domain in or . By Aronson estimate (see, e.g., Theorem 2.6 of ), under some assumptions on the coefficients , , there exist some constants such that for all and , . In this case, condition (29) is implied by (6).
Example 8 (heat equation with fractional power of the Laplacian). Let for some . Assume that is a bounded domain in or . Then (see, e.g., Appendix of )
where is the fundamental solution of on and is the density of the measure , being a convolution semigroup of measures on whose Laplace transform is given by
Note that if , is the density of , where is a -stable subordinator with Lévy measure .
Assume first that . Then , where
If , then is the density of , with being a symmetric -stable Lévy process with values in defined by , with a Brownian motion in with variance 2. By Lemma B.1 (Appendix B), if , then (29) holds if and only if
If is a bounded domain in , then (by Lemma 2.1 of ). In this case, if , then (29) is implied by (36).
Example 9 (cable equation in ). Let and . Then , where and condition (29) holds for any .
Example 10 (wave equation in with ). Let and with or . Then , where Condition (29) holds for any . In this case, if and if .
4. Stochastic Integration
In this section we construct a stochastic integral with respect to by generalizing the ideas of  to the case of random fields. Unlike these authors, we do not assume that has a symmetric distribution, unless .
Let where is the -field of negligible sets in and is the -field generated by for all , and for all Borel sets bounded away from . Note that where is the -field generated by , , and .
A process is called elementary if it is of the form where , , and is -measurable and bounded. A simple process is a linear combination of elementary processes. Note that any simple process can be written as with and , where are -measurable and are disjoint sets in . Without loss of generality, we assume that .
We denote by the predictable -field on , that is, the -field generated by all simple processes. We say that a process is predictable if the map is -measurable.
Remark 11. One can show that the predictable -field is the -field generated by the class of processes such that is left continuous for any , and is -measurable for any .
Let be the class of all predictable processes such that for all and . Note that is a linear space.
Let be an increasing sequence of sets in such that . We define
We identify two processes and for which ; that is, a.e., where . In particular, we identify two processes and if is a modification of ; that is, a.s. for all .
The space becomes a metric space endowed with the metric : This follows using Minkowski’s inequality if and the inequality if .
The following result can be proved similarly to Proposition 2.3 of .
Proposition 12. For any there exists a sequence of bounded simple processes such that as .
By Proposition 5.7 of , the -stable Lévy process has a càdlàg modification, for any . We work with these modifications. If is a simple process given by (40), we define Note that, for any , is -measurable for any , and is càdlàg. We write
The following result will be used for the construction of the integral. This result generalizes Lemma 3.3 of  to the case of random fields and nonsymmetric measures .
Theorem 13. If is a bounded simple process then for any and , where is a constant depending only on .
Proof. Suppose that is of the form (40). Since is càdlàg, it is separable. Without loss of generality, we assume that its separating set can be written as where is an increasing sequence of finite sets containing the points . Hence,
Fix . Denote by the points of the set . Say for some . Then each interval can be written as the union of some intervals of the form : where . By (44), for any and ,
For any , let , and, for any , define , , and . With this notation, we have Consequently, for any
Using (47) and (51), it is enough to prove that for any ,
First, note that This follows from the definition (40) of and (48), since .
We now prove (52). Let . For the event on the left-hand side, we consider its intersection with the event and its complement. Hence, the probability of this event can be bounded by We treat separately the two terms.
For the first term, we note that is -measurable and is independent of . By Fubini’s theorem where and is the law of .
We examine the tail of for a fixed . By Lemma 3, has a distribution. Since the sets are disjoint, the variables are independent. Using elementary properties of the stable distribution (Properties 1.2.1 and 1.2.3 of ), it follows that has a distribution with parameters: By Lemma A.1 (Appendix A), there exists a constant such that for any . Hence,
We now treat . We consider three cases. For the first two cases we deviate from the original argument of  since we do not require that .
Case 1 . Note that where is a submartingale. By the submartingale maximal inequality (Theorem 35.3 of ),
Using the independence between and it follows that Let . Using (57) and Remark A.2 (Appendix A), we get Hence,
From (59), (60), and (63), it follows that
Case 2 . We have where and .
We first treat the term . Note that is a zero-mean square integrable martingale, and Let . Using (57) and Remark A.2 (Appendix A), we get