A Markov Process Approach to the asymptotic Theory of abstract Cauchy Problems driven by Poisson Processes

TL;DR

This paper uses Markov process theory to analyze the long-term behavior of solutions to stochastic evolution problems driven by Poisson processes, establishing laws of large numbers and central limit theorems under certain conditions.

Contribution

It introduces a novel approach employing Markov process theory to derive asymptotic results for stochastic evolution inclusions driven by Poisson processes.

Findings

Proves that the process is a Markov process under certain assumptions.

Establishes strong law of large numbers for the process.

Demonstrates a central limit theorem under polynomial decay conditions.

Abstract

In this paper, we employ Markov process theory to prove asymptotic results for a class of stochastic processes which arise as solutions of a stochastic evolution inclusion and are given by the representation formula \begin{align*} \mathbb{X}_{x}(t)=\sum \limits_{m=0}\limits^{\infty}T((t-\alpha_{m})_{+})(x_{x,m})1\hspace{-0,9ex}1_{[\alpha_{m},\alpha_{m+1})}(t), \end{align*} where $(T(t))_{t \geq 0}$ is a (nonlinear) time-continuous, contractive semigroup acting on a separable Banach space $(V,||\cdot||_{V})$, $(\alpha_{m})_{m \in \mathbb{N}}$ is the sequence of arrival times of a homogeneous Poisson process, $x$ is a $V$-valued random variable and $(x_{x,m})_{m \in \mathbb{N}}$ is a recursively defined sequence of $V$-valued random variables, fulfilling $x_{x,0}=x$. It will be demonstrated that $\mathbb{X}_{x}$ is, under some distributional assumptions on the involved random variables, a time-continuous Markov process and that it obeys, under polynomial decay conditions on $T$, the strong law of large numbers (SLLN) and, if the decay rate is sufficiently fast, also the central limit theorem (CLT). Finally, we consider two examples: A nonlinear ordinary differential equation and the (weighted) $p$-Laplacian evolution equation for $p \in (2,\infty)$.