Stochastic Switching in Evolution Equations

Abstract

We consider stochastic hybrid systems that stem from evolution equations with right-hand sides that stochastically switch between a given set of right-hand sides. To begin our study, we consider a linear ordinary differential equation whose right-hand side stochastically switches between a collection of different matrices. Despite its apparent simplicity, we prove that this system can exhibit surprising behavior.Next, we construct mathematical machinery for analyzing general stochastic hybrid systems. This machinery combines techniques from various fields of mathematics to prove convergence to a steady state distribution and to analyze its structure.Finally, we apply the tools from our general framework to partial differential equations with randomly switching boundary conditions. There, we see that these tools yield explicit formulae for statistics of the process and make seemingly intractable problems amenable to analysis.