Stochastic reaction-diffusion equations on networks

TL;DR

This paper studies stochastic reaction-diffusion equations on finite networks, establishing existence and uniqueness of solutions with Gaussian noise, and generalizing existing theoretical results to accommodate complex network structures.

Contribution

It extends the semigroup approach to stochastic reaction-diffusion equations on networks with Gaussian noise, allowing for polynomial reaction terms with stochastic coefficients.

Findings

Proves existence and uniqueness of solutions on finite networks.

Generalizes stochastic reaction-diffusion theory to network structures.

Handles polynomial reaction terms with stochastic coefficients.

Abstract

We consider stochastic reaction-diffusion equations on a finite network represented by a finite graph. On each edge in the graph a multiplicative cylindrical Gaussian noise driven reaction-diffusion equation is given supplemented by a dynamic Kirchhoff-type law perturbed by multiplicative scalar Gaussian noise in the vertices. The reaction term on each edge is assumed to be an odd degree polynomial, not necessarily of the same degree on each edge, with possibly stochastic coefficients and negative leading term. We utilize the semigroup approach for stochastic evolution equations in Banach spaces to obtain existence and uniqueness of solutions with sample paths in the space of continuous functions on the graph. In order to do so we generalize existing results on abstract stochastic reaction-diffusion equations in Banach spaces.