# Stochastic reaction-diffusion equations on networks with dynamic time-delayed boundary conditions

## TL;DR

This paper studies stochastic reaction-diffusion equations on networks with dynamic, time-delayed boundary conditions and noise, proving existence and uniqueness of solutions and exploring an optimal control application.

## Contribution

It introduces a novel framework for analyzing stochastic reaction-diffusion equations with dynamic boundary conditions and delays on networks, including an optimal control aspect.

## Key findings

- Proved existence and uniqueness of mild solutions.
- Reformulated the problem as an abstract stochastic PDE.
- Applied the theory to a stochastic optimal control problem.

## Abstract

We consider a reaction-diffusion equation on a network subjected to dynamic boundary conditions, with time delayed behaviour, also allowing for multiplicative Gaussian noise perturbations. Exploiting semigroup theory, we rewrite the aforementioned stochastic problem as an abstract stochastic partial differential equation taking values in a suitable product Hilbert space, for which we prove the existence and uniqueness of a mild solution. Eventually, a stochastic optimal control application is studied.