Linear parabolic differential equations as limits of space-time jump Markov processes
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.
distribution valued generalized Ornstein-Uhlenbeck
mild solution of stochastic partial differential equation
scientific article; zbMATH DE number 3978048
scientific article