Non-autonomous Mild Solutions for Spde with Levy Noise in Weighted L-spaces

In this paper we prove a comparison theorem for the stochastic di¤erential equation dX(t) = (A(t)X(t) + F (t;X(t))) dt+M (t;X(t)) dW (t) + R L2 M (t;X(t))x ~ N(dt; dx), t 2 [ 0; T ], driven by a Wiener noise W and a Poisson noise ~ N . The di¤usion coe¢ cients M and M are given by multiplication operators. The equation is considered in Lpspaces with a …nite weight measure over a (possibly unbounded) domain Rd. With the help of this comparison theorem we prove an existence result for the above equation in the case of non-Lipschitz drift F being a Nemytski-type operator of (at most) polynomial growth. 1. Introduction In recent years there has been large interest in SDEs with general, not necessarily continuous, semimartingales as driving noises. This is re‡ected in a growing number of papers going beyond the well-known framework of SDEs with Wiener noise, e.g. by considering compensated Poisson random measures or Lévy processes as noise. Stochastic evolution equations in in…nite dimensions are often used to describe complex models in natural sciences. Numerous examples of SDEs with Wiener noise in in…nite dimensions can be found e.g. in the introductory chapter of the monograph by DaPrato and Zabczyk [11]. SDEs with compensated Poisson random measures or Lévy processes as driving random forces are candidates to model situations, where the system does not develop in a time-continuous way. The theory of SDEs with jumps in in…nitedimensional spaces plays a role in modelling critical phenomena. Among areas of application let us mention neurophysiology, environmental pollution and mathematical …nance. Consider R, d 2 N, with Euclidean norm j j, Borel -algebra B(R) and Lebesgue measure d . Let us …x a (possibly unbounded) 2 B(R) and = 0 (in the case of bounded ) resp. > d (for unbounded ). Denote by a …nite measure on (R;B(R)) given by (1.1) (d ) := (1 + j j) 2 d : Given some 1, let L := L ( ) be the Banach space of Borel-measurable, 2 -integrable functions w.r.t. the measure on . Such weighted spaces are of 2000 Mathematics Subject Classi…cation. Primary 60H15; Secondary 60G51, 60G57. Key words and phrases. Stochastic PDE, Poisson random measures, comparison theorems. This work was …nancially supported by the DFG through SFB 701"Spektrale Strukturen und Topologische Methoden in der Mathematik". 1 2 SIMON MICHEL, TANJA PASUREK, AND MICHAEL RÖCKNER common use in the theory of (deterministic and stochastic) parabolic di¤erential equations, see e.g. [12], [22] resp. [26]. In this paper, given some …xed T > 0, we study the following SDE in L dX(t) =(A(t)X(t) + F (t;X(t)))dt+M (t;X(t))dW (t) (Eq.1) + Z L2 M (t; ;X(t))x ~ N(dt; dx); t 2 [ 0; T ]; X(0) = : We will look for solutions to Eq. (1) in the Banach space G (T ) of all predictable processes [ 0; T ] 3 t 7 ! X(t) 2 L such that jjXjjG (T ) := sup t2[ 0;T ] EjjX(t)jj L2 1 2 <1: The precise setting will be given in Section 2 below. Here, we only point out, what kind of coe¢ cients and noises are present in Eq.(1). For simplicity, we we shall supress explicit dependence on ! 2 of all random elements, if no confusion can arise. Everywhere below, we assume that: the family (A(t))t2[ 0;T ] generates a strong evolution operator U = (U(t; s))0 s t T in L ; F , and are time-dependent, random Nemitskii-type nonlinear operators de…ned through predictable functions f , , : [ 0; T ] R! R; M andM are the multiplication operators corresponding to and ; (W (t))t2[ 0;T ] is a Q-Wiener process in L with a trace class correlation operator Q 0; ~ N : [ 0; T ] L ! R is a compensated Poisson random measure with a Lévy intensity measure on L. All necessary technical assumptions on the growth of coe¢ cients and regularity properties of noises will be given in Section 2. Note that by the Lévy-Itô decomposition it will be possible to have similar results also for stochastic evolution equations with Lèvy noise, see Remark 2.11 below. The solutions to Eq.(1) will be understood in the mild sense (see De…nition 2.6). In the particular case = 0 such type of in…nite dimensional equation was considered in [26]. Compared to [26], Eq.(1) has an additional multiplicative (i.e. solution-dependent) jump noise, which needs a careful analysis. In this paper, we will extend the comparison method of [26] to the case of Poisson noise. Such extension is non-trivial, since now we are dealing with di¤usion processes with jumps. To control the e¤ect caused by a Poisson noise, we need additional assumptions (as compared to the Wiener case) on the jump di¤usion terms (e.g., their monotonicity in ( ) below). Thus, in Section 2 (see De…nition 2.6 there) we formulate a solution term, which di¤ers from the one applied in [26] (cf. De…nition 2.7, p. 55 there) in a more restrictive integrability condition and a weaker pathwise property (càdlàg paths vs. pathwise continuity). SPDE WITH LEVY NOISE IN L-SPACES 3 Note that, given a strong evolution operator (U(t; s))0 s t T generated by (A(t))t2[ 0;T ], we have to ensure the well-de…nedness of the so-called Poisson stochastic convolution (1.2) I ~ N (t) := t Z