Stochastic reaction\u2013diffusion equations on networks

E Sikolya,M Kovács

doi:10.1007/s00028-021-00719-w

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.

Highlights

We consider a finite connected network, represented by a finite graph G with m edges e1, . . . , em and n vertices v1, . . . , vn
This operator can be regarded as maximal since no other boundary condition except continuity is imposed for the functions in its domain
We introduce the following assumptions for the operators in (SCP)

Summary

Introduction

We consider a finite connected network, represented by a finite graph G with m edges e1, . . . , em and n vertices v1, . . . , vn. In [16], the case of multiplicative Wiener noise is treated with bounded and globally Lipschitz continuous drift and diffusion coefficients and noise both on the edges and vertices In all these papers, the semigroup approach is utilized in a Hilbert space setting and the only work that treats non-globally Lipschitz continuous coefficients is [11], but the noise is there is additive and square-integrable. The semigroup approach is utilized in a Hilbert space setting and the only work that treats non-globally Lipschitz continuous coefficients is [11], but the noise is there is additive and square-integrable In this case, energy arguments are possible using the additive nature of the equation which does not carry over to the multiplicative case. We treat separately the models where stochastic noise is only present in the nodes

The system of equations

Spaces and operators

Well-posedness of the abstract Cauchy problem

Abstract results for a stochastic reaction–diffusion equation

Preparatory results

Main results