Boundary Value Problems for Statistics of Diffusion in a Randomly Switching Environment: PDE and SDE Perspectives

Abstract

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.keywords. random PDE, stochastic hybrid system, piecewise deterministic Markov process, randomly switching boundary, switched dynamical system.

Including Rebinding Reactions in Well-Mixed Models of Distributive Biochemical Reactions

“…1.3 and 1.4. A detailed proof can be found in (19), and similar derivations of exit statistics of diffusing particles in the presence of randomly switching boundary conditions can be found in (28)(29)(30).…”

“…By a direct application of Theorem 3 in (19), it follows that the probability that they react before becoming distance R > ε apart, given that they are initially distance r˛ðε; RÞ apart, is p 1 ðr; RÞ that satisfies the following: …”

Neurotransmitter concentrations in the presence of neural switching in one dimension

“…Assume that J(t) is initially distributed according to its invariant distribution. The following proposition follows from Theorem 1 in [14]. The PDE and boundary conditions satisfied by v i follow from interchanging differentiation with expectation.…”

“…The purpose of this paper is to use recently developed mathematical machinery on the stochastic switching of boundary conditions in PDEs [14,16] to understand certain aspects of volume transmission. Suppose that a large number of neurons with the same neurotransmitter project randomly to a distant volume where they release neurotransmitter into the extracellular space.…”

“…In Section 3 we consider the much more general stochastic process where u(x, t) satisfies (1) but now there are neurons at both x = 0 and x = L. Both neurons fire stochastically and independently with exponential rates that are different and release rates, c 0 , c L , that are different. Methods used in Section 2 and from [14] are used to show that lim t→∞ Eu(x, t) can be computed in terms of all the given parameters of the problem by solving a set of eight linear equations. Section 4 is devoted to extracting information about volume transmission from the explicit formulas for lim t→∞ Eu(x, t) derived in Sections 2 and 3.…”