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

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-α_{m})_{+})(x_{x,m})1\hspace{-0,9ex}1_{[α_{m},α_{m+1})}(t), \end{align*} where $(T(t){)}_{t\ge 0}$ 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 $p\in (2,\infty )$.

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• Mathematical Models for Erosion and the Optimal Transportation of Sediment
• Abstract Cauchy Problems in separable Banach Spaces driven by random Measures: Existence and Uniqueness