Some Refinements of Existence Results for SPDEs Driven by Wiener Processes and Poisson Random Measures

We provide existence and uniqueness of global (and local) mild solutions for a general class of semilinear stochastic partial differential equations driven by Wiener processes and Poisson random measures under local Lipschitz and linear growth (or local boundedness, resp.) conditions. The so-called “method of the moving frame” allows us to reduce the SPDE problems to SDE problems. 1. Introduction Semilinear stochastic partial differential equations (SPDEs) on Hilbert spaces, being of the type have widely been studied in the literature, see, for example, [1–4]. In (1.1), denotes the generator of a strongly continuous semigroup, and is a trace class Wiener process. In view of applications, this framework has been extended by adding jumps to the SPDE (1.1). More precisely, consider an SPDE of the type where denotes a Poisson random measure on some mark space with being its compensator. SPDEs of this type have been investigated in [5, 6], see also [7–12], where SPDEs with jump noises have been studied. The goal of the present paper is to extend results and methods for SPDEs of the type (1.2) in the following directions.(i) We consider more general SPDEs of the form where is a set with . Then, the integral represents the small jumps, and represents the large jumps of the solution process. Similar SDEs have been considered in finite dimension in [13, Section II.2.c] and in infinite dimension in [14].(ii) We will prove the following results (see Theorem 4.5) concerning existence and uniqueness of local and global mild solutions to (1.3):(1)if are locally Lipschitz and of linear growth, then existence and uniqueness of global mild solutions to (1.3) hold;(2)if are locally Lipschitz and locally bounded, then existence and uniqueness of local mild solutions to (1.3) hold;(3)if are locally Lipschitz, then uniqueness of mild solutions to (1.3) holds.In particular, the result that local Lipschitz and linear growth conditions ensure existence and uniqueness of global mild solutions does not seem to be well known for SPDEs, as most of the mentioned references impose global Lipschitz conditions. An exception is [3], where the author treats Wiener process-driven SPDEs of the type (1.1), even on 2-smooth Banach spaces, and provides existence and uniqueness under local Lipschitz and linear growth conditions. In [3], the crucial assumption on the operator is that it generates an analytic semigroup, while our results hold true for every pseudocontractive semigroup.(iii) We reduce the proofs of these SPDE results to the analysis of SDE problems. This is due to the “method of