Yamada-Watanabe Theorem for Stochastic Evolution Equation Driven by Poisson Random Measure

The purpose of this paper is to give a detailed proof of Yamada-Watanabe theorem for stochastic evolution equation driven by pure Poisson random measure. 1. Introduction The main purpose of this paper is to establish the Yamada-Watanabe theory of uniqueness and existence of solutions of stochastic evolution equation driven by pure Poisson random measure in the variational approach. The classical paper [1] has initiated a comprehensive study of relations between different types of uniqueness and existence (e.g., strong solutions, weak solutions, pathwise uniqueness, uniqueness, and joint uniqueness in law) arising in the study of SDEs (see, e.g., [2–4]) and the study is still alive today. New papers are published (see, e.g., [2, 3, 5–7]). In this paper we are concerned with the similar question for stochastic evolution equation driven by Poisson random measure by using the method of Yamada and Watanabe. Yamada and Watanabe's initial work [1] proved that weak existence and pathwise uniqueness imply strong existence and weak uniqueness. For -dimensional case, see [8, 9]. For infinite dimensional stochastic differential equation, Ondreját [6] proved similar result for stochastic evolution equation in Banach space driven by cylindrical Wiener process, where the solutions are understood in the mild sense. Lately, R？ckner et al. [7] proved similar result for stochastic evolution equation in Banach space driven by cylindrical Wiener process under the variational framework. On the other hand, Kurtz [2, 3] obtained a pleasant version of Yamada-Watanabe and Engelbert theorem in an abstract form, which covered most of the work mentioned above. However, we will consider the following concrete stochastic evolution equation by using a different method. In this paper, we will consider the following stochastic evolution equation driven by pure Poisson random measure under the variational framework: This type of equations can be applied to many SPDEs, for example, stochastic Burgers equation, stochastic porous media equation, and stochastic Navier-Stokes equation (see, e.g., [9–13]). We will introduce the above equation precisely in Section 2. Our aim is to obtain this jump-case Yamada-Watanabe theorem; that is, weak existence and strong uniqueness (which will be stated in Section 2) imply strong existence and weak uniqueness and vice versa. We note that there are some differences between the jump-case case and the Brownian motion case. It is well known that a Brownian motion can be treated as a canonical map on or (for some Hilbert space ), while for jump-case we have