Stochastic Switching in Infinite Dimensions with Applications to Random Parabolic PDE

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.Key words. Random PDEs, hybrid dynamical systems, switched dynamical systems, piecewise deterministic Markov process, ergodicity AMS subject classifications. 35R60, 37H99, 46N20, 60H15, 92C30 1. Introduction. The primary motivation for this paper is to study parabolic partial differential equations (PDEs) with randomly switching boundary conditions. More precisely, given an elliptic differential operator, L, on a domain D ⊂ R d , we want to study the stochastic process u(t, x) that solves ∂ t u = Lu in D subject to boundary conditions that switch at random times between two given boundary conditions. This type of random PDE is an example of a stochastic hybrid system. The word "hybrid" is used because these stochastic processes involve both continuous dynamics and discrete events. In this example, the continuous dynamics are the different boundary value problems corresponding to the different boundary conditions for the given PDE, and the discrete events are when the boundary condition switches.In general, a stochastic hybrid system is a continuous-time stochastic process with two components: a continuous component (X t ) t≥0 and a jump component (J t ) t≥0 . The jump component, J t , is a jump process on a finite set, and for each element of its state space we assign some continuous dynamics to X t . In between jumps of J t , the component X t evolves according to the dynamics associated with the current state of J t . When J t jumps, the component X t switches to follow the dynamics associated with the new state of J t .An ordinary differential equation (ODE) with a switching right-hand side is the type of stochastic hybrid system that is most commonly used in applications. Such ODE switching systems have been used extensively in applied areas such as control theory, computer science, and engineering (for example, [46], [8], [3], and [31]). More recently, these systems have been used in diverse areas of biology (for example, molecular biology [10], [37], [9], ecology [47], and epidemiology [21]). Furthermore, such ODE switching systems have also recently been the subject of much mathematical study ([29], [12], [6], [5], [2], [24], [25], and [4]).

Diffusive flux in a model of stochastically gated oxygen transport in insect respiration

“…30, we take that the gate dynamics is Markovian and characterized by the rate constants r 0 and r 1 , where the subscripts 0 and 1 denote open and closed states of the gate, respectively. The probabilities P 0 (t) and P 1 (t) of finding the gate open and closed at time t satisfy the rate equations…”

“…As in Ref. 30 we assume that the left boundary of the interval, at x = 0, is perfectly absorbing. The boundary condition on the right boundary depends on the state of the gate.…”

“…30, if we identify the product of the tube cross section area and the particle concentration in reservoir A, πa 2 c A , with the constant b from Ref. 30.…”

“…[25][26][27][28][29] The functional significance of this regime is still unclear, but several of the proposed mechanisms require quantitative analysis of diffusive fluxes through the tracheal network. 25 A first step in this direction was made by Lawley et al 30 who proposed and analyzed a one-dimensional diffusion model, in which the absorbing boundary condition at the tube exit was used to describe oxygen consumption by the tissue, and randomly switching boundary conditions at the tube entrance were used to describe the spiracle dynamics. The main result of Ref.…”

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