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]).

We analyze a piecewise deterministic PDE consisting of the diffusion equation on a finite interval Ω with randomly switching boundary conditions and diffusion coefficient. We proceed by spatially discretizing the diffusion equation using finite differences and constructing the Chapman-Kolmogorov (CK) equation for the resulting finite-dimensional stochastic hybrid system. We show how the CK equation can be used to generate a hierarchy of equations for the r-th moments of the stochastic field, which take the form of r-dimensional parabolic PDEs on Ω r that couple to lower order moments at the boundaries. We explicitly solve the first and second order moment equations (r = 2). We then describe how the r-th moment of the stochastic PDE can be interpreted in terms of the splitting probability that r non-interacting Brownian particles all exit at the same boundary; although the particles are non-interacting, statistical correlations arise due to the fact that they all move in the same randomly switching environment. Hence the stochastic diffusion equation describes two levels of randomness: Brownian motion at the individual particle level and a randomly switching environment. Finally, in the limit of fast switching, we use a quasi-steady state approximation to reduce the piecewise deterministic PDE to an SPDE with multiplicative Gaussian noise in the bulk and a stochastically-driven boundary.

Many events in biology are triggered when a diffusing searcher finds a target, which is called a first passage time (FPT). The overwhelming majority of FPT studies have analyzed the time it takes a single searcher to find a target. However, the more relevant timescale in many biological systems is the time it takes the fastest searcher(s) out of many searchers to find a target, which is called an extreme FPT. In this paper, we apply extreme value theory to find a tractable approximation for the full probability distribution of extreme FPTs of diffusion. This approximation can be easily applied in many diverse scenarios, as it depends on only a few properties of the short time behavior of the survival probability of a single FPT. We find this distribution by proving that a careful rescaling of extreme FPTs converges in distribution as the number of searchers grows. This limiting distribution is a type of Gumbel distribution and involves the LambertW function. This analysis yields new explicit formulas for approximations of statistics of extreme FPTs (mean, variance, moments, etc.) which are highly accurate and are accompanied by rigorous error estimates.

The timescales of many physical, chemical, and biological processes are determined by first passage times (FPTs) of diffusion. The overwhelming majority of FPT research studies the time it takes a single diffusive searcher to find a target. However, the more relevant quantity in many systems is the time it takes the fastest searcher to find a target from a large group of searchers. This fastest FPT depends on extremely rare events and has a drastically faster timescale than the FPT of a given single searcher. In this paper, we prove a simple explicit formula for every moment of the fastest FPT. The formula is remarkably universal, as it holds for d-dimensional diffusion processes (i) with general space-dependent diffusivities and force fields, (ii) on any smooth Riemannian manifold, (iii) in the presence of reflecting obstacles, and (iv) with partially absorbing targets. Our results rigorously confirm, generalize, correct, and unify various conjectures and heuristics about the fastest FPT.S l (t) ≈ 1 − e −l 2 /t for t 1.

Consider N independently diffusing particles that reversibly bind to a target. We study a problem recently introduced by Grebenkov of finding the first passage time (FPT) for K of the N particles to be simultaneously bound to the target. Since binding is reversible, bound particles may unbind before the requisite K particles bind to the target. This so-called “impatience” leads to a delicate temporal coupling between particles. Recent work found the mean of this FPT in the case that N = K = 2 in a one-dimensional spatial domain. In this paper, we approximate the full distribution of the FPT for any N ≥ K ≥ 1 in a broad class of domains in any space dimension. We prove that our approximation (i) is exact in the limit that the target and/or binding rate is small and (ii) is an upper bound in any parameter regime. Our approximation is analytically tractable and we give explicit formulas for its mean and distribution. These results reveal that the FPT can depend sensitively and nonlinearly on both K and N. The analysis is accompanied by detailed numerical simulations.