Abstract Cauchy Problems in separable Banach Spaces driven by random Measures: Asymptotic Results in the finite extinction Case

The aim of this paper is to prove the strong law of large numbers (SLLN) as well as the central limit theorem (CLT) for a class of vector-valued stochastic processes which arise as solutions of the stochastic evolution inclusion \begin{align*} \eta(t,z) N_{\Theta}(dt \otimes z)\in dX(t)+\mathcal{A} X(t)dt, \end{align*} where $\mathcal{A}$ is a multi-valued operator and $N_{\Theta}$ is the counting measure induced by a point process $\Theta$. The SLLN and the CLT will be proven not only for real-valued, but also for vector-valued functionals and the applicability of these results to the (weighted) $p$-Laplacian evolution equation (for "small" $p$) will be demonstrated. The key assumption needed in this paper is that the nonlinear semigroup arising from the multi-valued operator $\mathcal{A}$ extincts in finite time.