In this paper the main goal of the authors is to study parabolic partial differential equations (PDE) with randomly switching boundary conditions. The precise model is given by an elliptic differential operator \(L\) and the corresponding linear PDE \[ \partial_t u = Lu \] on some domain \(D \subset\mathbb{R}^d\), subject to boundary conditions that switch between two given deterministic boundary conditions, where the switching times are given by a random jump process \(J_t\). These problems are not only motivated by biological applications, but also show a significant difference to the behavior usually seen in PDEs forced by a Gaussian noise.
In order to analyze the random PDEs arising from this model, the authors study more general stochastic hybrid systems and prove convergence in time of the solution \(u(t,x)\) to a stationary distribution. Moreover the properties of these limiting distributions are studied.
The general results can not only be applied to switching PDEs, but also to many other types of stochastic hybrid systems, such as ordinary differential equations with randomly switching right-hand sides.
Applications of the general results to the PDE case are given by the heat equation with randomly switching boundary conditions on the interval, where one point on the boundary has a fixed Dirichlet condition, while the other switches between Dirichlet and Neumann, or in a second example between two different Dirichlet conditions. The authors present explicit formulas for various statistics of the solution and obtain almost sure results about its regularity and structure.