  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 $p \in (2,\infty)$.
