Stochastic switching in infinite dimensions with applications to random parabolic PDEs

TL;DR

This paper studies parabolic PDEs with randomly switching boundary conditions, providing convergence results, explicit statistics, and regularity properties, with applications to biological systems and other stochastic hybrid systems.

Contribution

It introduces a framework for analyzing stochastic hybrid systems with switching boundary conditions and derives explicit formulas and almost sure properties for solutions.

Findings

Convergence to stationary distributions for random PDEs.

Explicit formulas for solution statistics.

Almost sure regularity and structural results.

Abstract

We consider parabolic PDEs with randomly switching boundary conditions. In order to analyze these random PDEs, we consider more general stochastic hybrid systems and prove convergence to, and properties of, a stationary distribution. Applying these general results to the heat equation with randomly switching boundary conditions, we find explicit formulae for various statistics of the solution and obtain almost sure results about its regularity and structure. These results are of particular interest for biological applications as well as for their significant departure from behavior seen in PDEs forced by disparate Gaussian noise. Our general results also have applications to other types of stochastic hybrid systems, such as ODEs with randomly switching right-hand sides.