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

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.

The proof of Corollary 3.7 in the original submission was incomplete as only the continuity of the embeddings are verified in the proof but the denseness of the embeddings are not. In the present version we provide a new poof of Proposition 3.8 that is independent of Corollary 3.7. This way, we may use Proposition 3.8 to fill in the gap in the proof of Corollary 3.7