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

TL;DR

This paper establishes strong law of large numbers and central limit theorem for vector-valued stochastic processes driven by random measures, focusing on cases where the associated nonlinear semigroup extinguishes in finite time, with applications to p-Laplacian evolution.

Contribution

It proves SLLN and CLT for solutions of stochastic evolution inclusions with finite extinction, extending results to vector-valued functionals and specific PDEs.

Findings

Proved SLLN and CLT for vector-valued processes

Applied results to p-Laplacian evolution equations

Demonstrated finite extinction property in stochastic setting

Abstract

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.