Phương trình khuếch tán phản ứng ngẫu nhiên với nhiễu kiểu nhảy và hệ số phi Lipschitz

Potential Anal DOI 10.1007/s11118-017-9651-9 Stochastic Reaction-diffusion Equations Driven by Jump Processes Zdzisław Brzeźniak1 · Erika Hausenblas2 · Paul André Razafimandimby3 Received: 6 January 2014 / Accepted: 4 September 2017 © The Author(s) 2017. This article is an open access publication Abstract We establish the existence of weak martingale solutions to a class of second order parabolic stochastic partial differential equations. The equations are driven by multiplicative jump type noise, with a non-Lipschitz multiplicative functional. The drift in the equations contains a dissipative nonlinearity of polynomial growth.

Keywords Itô integral driven by a Poisson random measure · Stochastic partial differential equations · Lévy processes · Reaction diffusion equations Mathematics Subject Classification (2010) 60H15 · 60J75 · 60G51 This work was supported by the FWF-Project P17273-N12  Zdzisław Brzeźniak zdzislaw.uk Erika Hausenblas erika.at Paul André Razafimandimby paul.za 1 Department of Mathematics, University of York, Heslington, York YO10 5DD, UK 2 Department of Mathematics and Information Technology, Montanuniversität Leoben, Franz Josef Strasse 18, 8700 Leoben, Austria 3 Department of Mathematics and Applied Mathematics, University of Pretoria, Lynwood Road, Pretoria 0083, South Africa Z. Brzeźniak et al. 1 Introduction The aim of this paper is to study a class of stochastic reaction-diffusion equations driven by a Lévy noise. A motivating example is the following stochastic partial differential equation with a Dirichlet boundary condition ⎧   ⎪ ⎪ du(t, ξ ) = u(t, ξ ) dt + u(t, ξ ) − u(t, ξ )3 dt ⎪ ⎪ ⎪ ⎪ √ ⎨ + 1+√|u(t,ξ )| dL(t), t > 0, |u(t,ξ )| ⎪ ⎪ u(t, ξ ) = 0, ξ ∈ ∂ O, ⎪ ⎪ ⎪ ⎪ ⎩ u(0, ξ ) = u0 (ξ ), ξ ∈ O , where L = {L(t) : t ≥ 0} is a real valued Lévy process whose Lévy measure ν has finite p-moment for some p ∈ (1, 2] and O ⊂ Rd is a bounded domain with smooth boundary.

One consequence of our main results is that for every u0 ∈ C0 (O) there exists a C0 (O)- valued process u = {u(t) : t ≥ 0}, which is a martingale √ solution √to problem (1. The diffusion coefficient in Eq., the function g(u) = |u|/(1 + |u|), is just a simple example of a bounded and continuous function, the drift term f (u) = u − u3 is an example of a dissipative function f : R → R of polynomial growth, and the Laplace operator  is a special case of a second order operator. Our results allow us to treat equations with more general coefficients than those in the model problem (1. In fact we can study stochas- tic partial differential equations with second order uniformly elliptic dissipative operators with non-constant coefficients and an unbounded nonlinear map as drift term.

Moreover, our results are applicable to equations with infinite-dimensional Lévy processes as well as systems with more general initial data, for instance, elements of the Lebesgue or Sobolev γ ,q spaces Lq (O) or W0 (O). The details are presented in Sections 4, 5 and 6. In our paper we adopt the approach used in [10] by the first named author and Ga̧tarek, in which a similar problem but driven by a Wiener process was treated. The major differences of the current work with respect to [10] are as follows.

Firstly, the approach [10] heavily depends on the use of the theory of Itô integral in martingale type 2 Banach spaces with respect to a cylindrical Wiener process, while we use an approach which relies on stochastic integration in martingale type p Banach spaces with respect to a Poisson random measure, see [13]. Secondly, the compactness argument in [10] relies on the Hölder continuity of the trajectories of the corresponding stochastic convolution process. However, the trajecto- ries of the stochastic convolution process driven by a Lévy process are not continuous, then the use of the counterpart of the Hölder continuity, i. the càdlàg property of the trajecto- ries seems natural to be used.

Unfortunately, as many counterexamples have shown, see for instance a recent monograph [65] as well as the recent papers [19] and [11], the trajectories of the stochastic convolution processes driven by a Lévy process may not even be càdlàg in the space in which the Lévy process lives. Hence, this issue has to be handled with special care. Thirdly, we do not use martingale representation theorem, unlike the authors in [10] who used a result by Dettweiler [27]. Instead we use an ad hoc method based on a generali- sation of the Skorohod representation theorem, see Theorem C.

Finally, in order to control certain norms of the approximate solutions, instead of using stopping times as in [10], we apply interpolation methods. Summarizing, this paper contains general results on the exis- tence of martingale solutions to stochastic reaction-diffusion equations with dissipative type coefficients of polynomial growth and with multiplicative Lévy noise. We should also point out that, as a by-product of our results, we are able to fill a gap in the proof of the main result in the article [43]. The gap is related to the Step I, see page 8, in the current paper.

Stochastic Reaction-diffusion Equations Driven by Jump Processes Our paper confirms an observation that has already been made in earlier papers [13, 42] that the theory of stochastic integration with respect to a Poisson random measure in martingale type p Banach spaces is, to a large extend, analogous to the theory of stochastic integration with respect to a cylindrical Wiener process in martingale type 2 Banach spaces provided the space of γ -radonifying operators is replaced by the space Lp (ν), where ν is the intensity measure of the Poisson random measure in question. We would also like to point out that the theory of Banach space valued stochastic integrals with respect to a Poisson random measure is richer than the corresponding Gaussian theory, see the recent paper [30] by S. It would be of great interest to develop a theory of inequalities for stochastic convolutions and generalise the paper [13] in the framework of [30]. It is worth mentioning that although SPDEs with unbounded nonlinearity driven by a Lévy process have not been as extensively studied as their Gaussian counterparts, there exists a number of interesting recent publications on the subject.

For instance, Truman and Wu [74, 75], Jacob et al. [61], and Giri and Hausenblas [41] studied equations with Burgers type nonlinearities driven by a Lévy noise. In addition, Dong and Xu in [31] con- sidered the Burgers equation with the compound Poisson noise, thus in fact dealing with a deterministic Burgers’ equation on random intervals and random jumps. Some discussion of stochastic Burgers’ equations with additive Lévy noise is contained in [19], where it was shown how integrability properties of trajectories of the corresponding Ornstein-Uhlenbeck process play an important role in the existence and uniqueness of solutions.

In the recent paper [32] Dong and Xie studied the stochastic Navier-Stokes equations (NSEs) driven by a Poisson random measure with finite intensity measure. Fernando and Sritharan [37] and the first two named authours together with Zhu [17] studied the 2-D stochastic NSEs by means of a local-monotonicity method of Barbu [4]. This method seems to be restricted to the 2-D NSEs and does not require the use of compactness results. In their beautiful mono- graph [65] Peszat and Zabczyk studied classes of reaction-diffusion equations driven by an additive Lévy process.

In this case, the stochastic evolution equations can be transformed into an evolution equation with random coefficients, a method which usually does not work with multiplicative noise. In another recent paper [55] Marinelli and Röckner investigated a certain class of generalized solutions to problems similar to ours. Röckner and Zhang in [67] established the existence and uniqueness of solution to and a large deviation principles for a class of stochastic evolution equations driven by jump processes. Finally, Debussche et al.

in [25, 26] considered a stochastic Chafee-Infante equation driven by an additive Lévy noise and investigated the dynamics of the equation, for instance, the first exit times from domains of attraction of the stationary solutions of the deterministic equation. The stochastic PDEs driven by Lévy processes in Banach spaces have not been inten- sively studied, apart from a few papers by the second named author, like [42, 43], a very recent paper [13] by the first two named authors and [54] by Mandrekar and Rüdiger (who actually studied ordinary stochastic differential equations in martingale type 2 Banach spaces). Martingale solution to SPDEs driven by Lévy processes in Hilbert spaces are not often treated in the literature. Mytnik [60] constructed a weak solution to SPDEs with non-Lipschitz coefficients driven by space-time stable Lévy noise.

Mueller, Mytnik and Stan [59] investigated the heat equation with one-sided time independent stable Lévy noise. One should add that the noise in the paper [58] does not satisfy the hypothesis of the current work. The current paper is organized as follows. In Section 2 we introduce the notations used later on in the paper and we present the standing hypotheses and essential preliminary facts.

Our two main results, i.4 are stated in Section 3. In that section we also Z. Brzeźniak et al.5 which could be seen as the reformulation of some of our main results in terms of a Lévy process and not a Poisson random measure. Three examples illustrating the applicability of our results are presented in Sections 4, 5 and 6.

To be more precise, SPDEs with dissipative polynomial drift driven by a real-valued α-stable tempered Lévy process are treated in Section 4, SPDEs with a bounded drift driven by a space-time Lévy white noise are treated in Section 5 and finally, stochastic reaction-diffusion equations with dissipative polynomial growth drift, driven by a space-time Lévy white noise, are treated in Section 6. In Section 7 we state and prove several preliminary results about the stochastic convolution processes which we believe are interesting in themselves. Sections 8 and 9 are devoted to the proofs of our results. Unfortunately, these proofs are very technical and long and hence their brief outline is presented just after the statement of Theorem 3.

In the appendices we recall some definitions and well-known results in analysis and probability theory. We also prove new results, amongst them a modified version of the Skorohod representation theorem, see Theorem C.1, which are interesting in themselves. We finish the Introduction by pointing out that the approach presented in this paper (or rather it’s earlier arXiv version) has already been taken up and used for the proof of the existence of solutions to Stochastic Navier-Stokes equations and second grade fluids driven by Lévy noise, see [57] and [44], respectively. Notation 1 By N we denote the set of natural numbers, i., N = {0, 1, 2, · · · } and by N, respectively N∗ , we denote the set N ∪ {+∞}, respectively N \ {0}.

Whenever we speak about N, respectively N-valued measurable functions, we implicitly assume that the set N, respectively N, is equipped with the full σ -field 2N , resp. By R+ we denote the set [0, ∞) of nonnegative real numbers and by R∗ the set R \ {0}. If X is a topological space, then by B (X) we denote the Borel σ -field on X. By Leb we denote the Lebesgue measure on (Rd , B (Rd )) or (R, B (R)).

The space of bounded linear operators from a Banach space Y1 to a Banach space Y2 is denoted by L(Y1 , Y2 ). The norm of A ∈ L(Y1 , Y2 ) is denoted by A L(Y1 ,Y2 ). If O ⊂ Rd is a bounded domain with smooth boundary ∂ O, by C0 (O) we denote the space of real continuous functions on O which vanish on the boundary ∂ O. Suppose that (Z, Z ) is a measurable space.