Boundary value problems for statistics of diffusion in a randomly switching environment: PDE and SDE perspectives

TL;DR

This paper develops mathematical tools to analyze the statistics of diffusion processes with randomly switching boundary conditions, connecting PDE and SDE perspectives, and revealing complex dynamics in such systems.

Contribution

It establishes a theoretical framework linking moments of solutions to boundary value problems with exit statistics of switching SDEs, providing new insights into diffusion in random environments.

Findings

Moments of switching PDE solutions satisfy coupled hierarchy of boundary value problems.

Exit statistics for switching SDEs follow a similar hierarchy, linking PDE and SDE approaches.

Examples demonstrate surprising dynamics in systems with randomly switching boundaries.

Abstract

Driven by diverse applications, several recent models impose randomly switching boundary conditions on either a PDE or SDE. The purpose of this paper is to provide tools for calculating statistics of these models and to establish a connection between these two perspectives on diffusion in a random environment. Under general conditions, we prove that the moments of a solution to a randomly switching PDE satisfy a hierarchy of BVPs with lower order moments coupling to higher order moments at the boundaries. Further, we prove that joint exit statistics for a set of particles following a randomly switching SDE satisfy a corresponding hierarchy of BVPs. In particular, the $M$-th moment of a solution to a switching PDE corresponds to exit statistics for $M$ particles following a switching SDE. We note that though the particles are non-interacting, they are nonetheless correlated because they all follow the same switching SDE. Finally, we give several examples of how our theorems reveal the sometimes surprising dynamics of these systems.