arXiv subjects

Alexander Nerlich

Publications and source records attributed to Alexander Nerlich.

5 recordsLinked to original sources

arXiv2018

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*} η(t,z) N_Θ(dt \otimes z)\in dX(t)+\mathcal{A} X(t)dt, \end{align*} where $\mathcal{A}$ is a multi-valued operator and $N_Θ$ is the counting measure induced by a point process $Θ$. 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.

math.PR↗

arXiv2018

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

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 $(T(t))_{t \geq 0}$ is a (nonlinear) time-continuous, contractive semigroup acting on a separable Banach space $(V,||\cdot||_{V})$, $(α_{m})_{m \in \mathbb{N}}$ 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 $p \in (2,\infty)$.

math.PR↗

arXiv2018

A randomized weighted $p$-Laplacian evolution equation with Neumann boundary conditions

The purpose of this paper is to show that the randomized weighted $p$-Laplacian evolution equation given by \begin{align} \label{eveqrand} \begin{cases} U^{\prime}(t)(ω) =\text{Div} \left( g(ω) |DU(t)(ω)|^{p-2}DU(t)(ω) \right) \text{ on } S, g(ω)|DU(t)(ω)|^{p-2}DU(t)(ω)\cdotη=0 \text{ on } \partial S, U(0)(ω)=u(ω),\end{cases} \end{align} for $\mathbb{P}$-a.e. $ω\in Ω$ and a.e. $t \in (0,\infty)$ admits a unique strong solution and to determine asymptotic properties of this solution.

math.FA↗

arXiv2017

Abstract Cauchy Problems in separable Banach Spaces driven by random Measures: Existence and Uniqueness

The purpose of this paper is to study stochastic evolution inclusions of the form \begin{align*} η(t,z) N_Θ(dt \otimes z)\in dX(t)+\mathcal{A} X(t)dt, \end{align*} where $\mathcal{A}$ is a multi-valued operator acting on a separable Banach space and $N_Θ$ is the counting measure induced by a point process $Θ$. Firstly, we will set up the concepts of strong and mild solutions; then we will derive existence as well as uniqueness criteria for these kinds of solutions and give a representation formula for the solutions. The results will be formulated by means of nonlinear semigroup theory and except for separability, no assumptions on the underlying Banach space are required.

math.PR↗

arXiv2017

Asymptotic Results for Solutions of a weighted p-Laplacian evolution Equation with Neumann Boundary Conditions

The purpose of this paper is to investigate the time behavior of the solution of a weighted $p$-Laplacian evolution equation, given by \begin{align} \label{eveq} \begin{cases} u_{t} = \text{div} \left(γ|\nabla u|^{p-2}\nabla u \right) & \text{on} (0,\infty)\times S, \\ γ|\nabla u|^{p-2}\nabla u\cdotη=0 & \text{on} (0,\infty)\times \partial S, \\ u(0,\cdot)=u_{0} & \text{on} S,\end{cases} \end{align} where $n \in \mathbb{N}\setminus \{1\}$, $p \in (1,\infty)\setminus \{2\}$, $S\subseteq \mathbb{R}^{n}$ is an open, bounded and connected set of class $C^{1}$, $η$ is the unit outer normal on $\partial S $, and $γ: S \rightarrow (0,\infty)$ is a bounded function which can be extended to an $A_{p}$-Muckenhoupt weight on $\mathbb{R}^{n}$. It will be proven that the solution converges in $L^{1}(S)$ to the average of the initial value $u_{0} \in L^{1}(S)$. Moreover, a conservation of mass principle, an extinction principle and a decay rate for the solution will be derived.

math.AP↗