Linear parabolic differential equations as limits of space-time jump Markov processes
type: Item
instance of: scholarly article
title: Linear parabolic differential equations as limits of space-time jump Markov processes
zbMATH Open document ID: 0605.60039
DOI: 10.1016/0022-247X(86)90045-4
published in: Journal of Mathematical Analysis and Applications
publication date: 1986-01-01T00:00:00Z
review text: The paper is motivated by chemical reaction-diffusion systems in a bounded domain \(D\subseteq {\mathbb{R}}^ n\). First the author constructs a local stochastic (mesoscopic in \textit{H. Haken}'s terminology [Advanced synergetics. Instability hierarchies of self organized systems and devices. (1983; Zbl 0521.93002)]) model which is a space-time Markov jump process \(X^ N\), by dividing the volume into N cells of size v (determined by physical considerations). He then proves a strong first order functional limit theorem for the sequence \(X^ N\) (N\(\to \infty)\) as well as a related functional central limit theorem. The (deterministic) limit is obtained as the solution of a parabolic evolution equation (2.2), represented under its mild-solution form. In order to apply this technique he uses scaled Hilbert spaces \(H_{\alpha}\subset H_ 0\subset H_{-\alpha}\) and Maurin's embedding.
reviewed by: Gunter Ritter
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60J75
zbMATH DE Number: 3978048
zbMATH Keywords: distribution valued generalized Ornstein-Uhlenbeck
zbMATH Keywords: mild solution of stochastic partial differential equation
zbMATH Keywords: thermodynamic limit
zbMATH Keywords: chemical reaction-diffusion systems
zbMATH Keywords: functional limit theorem
author: Peter M. Kotelenez
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3978048