On the stochastic Allen-Cahn equation on networks with multiplicative noise

TL;DR

This paper studies stochastic Allen-Cahn equations on finite networks with multiplicative noise, establishing existence, uniqueness, and regularity of solutions using semigroup methods.

Contribution

It introduces a framework for analyzing stochastic Allen-Cahn equations on networks with multiplicative noise, proving well-posedness and regularity results.

Findings

Existence and uniqueness of solutions on finite networks.

Sample paths are continuous functions on the graph.

Enhanced space-time regularity of solutions.

Abstract

We consider a system of stochastic Allen-Cahn equations on a finite network represented by a finite graph. On each edge in the graph a multiplicative Gaussian noise driven stochastic Allen-Cahn equation is given with possibly different potential barrier heights supplemented by a continuity condition and a Kirchhoff-type law in the vertices. Using the semigroup approach for stochastic evolution equations in Banach spaces we obtain existence and uniqueness of solutions with sample paths in the space of continuous functions on the graph. We also prove more precise space-time regularity of the solution.