Mean first passage times for piecewise deterministic Markov processes and the effects of critical points

In this paper, we use probabilistic methods to determine the mean first passage time (MFPT) for a two-state piecewise deterministic Markov process (PDMP), also known as a dichotomous noise process, to escape from a finite interval. In particular, we consider the case where the set of functions generating the piecewise deterministic dynamics have one or more critical points. In order to solve this type of problem, we partition the domain into a set of subintervals that contain no critical points and impose conditions at the critical points separating these regions. Our analysis exploits the fact that a PDMP satisfies the strong Markov property. We prove that in the absence of common critical points, the MFPT is finite. Through specific examples, we also explore how the MFPT depends on the number of critical points and prove that the MFPT can be infinite if there are common critical points

Escape from a potential well with a randomly switching boundary

We consider diffusion in a potential well with a boundary that randomly switches between absorbing and reflecting and show how the switching boundary affects the classical escape theory. Using the theory of stochastic hybrid systems, we derive boundary value problems for the mean first passage time and splitting probability and find explicit solutions in terms of the spectral decomposition of the associated differential operator. Further, using a more probabilistic approach, we prove asymptotic formulae for these statistics in the small diffusion limit. In particular, we show that the statistical behavior depends critically on the gradient of the potential near the switching boundary and we derive corrections to Kramers' reaction rate theory

Moment equations for a piecewise deterministic PDE

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 rth moments of the stochastic field, which take the form of r-dimensional parabolic PDEs on Ωr 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 rth 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 boundar

Diffusion on a tree with stochastically gated nodes

We consider diffusion on a tree with nodes that randomly switch between allowing and prohibiting particles to pass. We find exact expressions for various splitting probabilities and mean first passage times for a single diffusing particle and show how the many parameters in the problem, such as the node gating statistics and tree topology, contribute to these exit statistics. We also consider a concentration of particles that can always pass through interior branch nodes and determine how an intermittent source at one end of the tree affects the flux at the other end. The latter problem is motivated by applications to insect respiration

Residence times of a Brownian particle with temporal heterogeneity

We consider a diffusing particle that randomly switches conformational state. Motivated by various scenarios in cell biology, we suppose that (a) the diffusion coefficient depends on the conformational state and/or (b) the particle can only pass through a series of gates in the domain when it is in a particular conformational state. We develop probabilistic methods to analyze this case of diffusion with temporal heterogeneity, and use these methods to calculate the expected residence time in portions of the domain before absorption at a boundary. We find several new phenomena not seen in recent studies of diffusion with spatial heterogeneity, some of which are counterintuitive. In particular, the expected residence times can be non-monotonic functions of (i) the initial distance from the absorbing boundary and (ii) the diffusion coefficients. We focus on one-dimensional intervals, but show how the analysis can be extended to spherically symmetric d-dimensional domains

Stochastic switching in infinite dimensions with applications to random parabolic PDE

© 2015 Society for Industrial and Applied Mathematics.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

Sensitivity to switching rates in stochastically switched ODEs

We consider a stochastic process driven by a linear ordinary differential equation whose right-hand side switches at exponential times between a collection of different matrices. We construct planar examples that switch between two matrices where the individual matrices and the average of the two matrices are all Hurwitz (all eigenvalues have strictly negative real part), but nonetheless the process goes to infinity at large time for certain values of the switching rate. We further construct examples in higher dimensions where again the two individual matrices and their averages are all Hurwitz, but the process has arbitrarily many transitions between going to zero and going to infinity at large time as the switching rate varies. In order to construct these examples, we first prove in general that if each of the individual matrices is Hurwitz, then the process goes to zero at large time for sufficiently slow switching rate and if the average matrix is Hurwitz, then the process goes to zero at large time for sufficiently fast switching rate. We also give simple conditions that ensure the process goes to zero at large time for all switching rates. © 2014 International Press

Diffusion in an age-structured randomly switching environment

Age-structured processes are well-established in population biology, where birth and death rates often depend on the age of the underlying populations. Recently, however, different examples of age-structured processes have been considered in the context of cell motility or certain types of stochastically gated ion channels, where the state of the system is determined by a switching process with age-dependent transition rates. In this paper we consider the particular problem of diffusion on a finite interval, with randomly switching boundary conditions due to the presence of an age-structured stochastic gate at one end of the interval. When the gate is closed the particles are reflected, whereas when it is open the domain is in contact with a particle bath. We use a moments method to derive a partial differential equation for the expectations of the stochastic concentration, conditioned on the state of the gate. We then use transform methods to eliminate the residence time of the age-structured switching, resulting in non-Markovian equations for the expectations, and determine the effective steady-state concentration gradient. Our analytical results are shown to match those obtained using Monte Carlo simulations

Hybrid colored noise process with space-dependent switching rates

A fundamental issue in the theory of continuous stochastic process is the interpretation of multiplicative white noise, which is often referred to as the Itô-Stratonovich dilemma. From a physical perspective, this reflects the need to introduce additional constraints in order to specify the nature of the noise, whereas from a mathematical perspective it reflects an ambiguity in the formulation of stochastic differential equations (SDEs). Recently, we have identified a mechanism for obtaining an Itô SDE based on a form of temporal disorder. Motivated by switching processes in molecular biology, we considered a Brownian particle that randomly switches between two distinct conformational states with different diffusivities. In each state, the particle undergoes normal diffusion (additive noise) so there is no ambiguity in the interpretation of the noise. However, if the switching rates depend on position, then in the fast switching limit one obtains Brownian motion with a space-dependent diffusivity of the Itô form. In this paper, we extend our theory to include colored additive noise. We show that the nature of the effective multiplicative noise process obtained by taking both the white-noise limit ( κ → 0 ) and fast switching limit ( ε → 0 ) depends on the order the two limits are taken. If the white-noise limit is taken first, then we obtain Itô, and if the fast switching limit is taken first, then we obtain Stratonovich. Moreover, the form of the effective diffusion coefficient differs in the two cases. The latter result holds even in the case of space-independent transition rates, where one obtains additive noise processes with different diffusion coefficients. Finally, we show that yet another form of multiplicative noise is obtained in the simultaneous limit ε , κ → 0 with ε / κ 2 fixed

Dynamically active compartments coupled by a stochastically gated gap junction

We analyze a one-dimensional PDE-ODE system representing the diffusion of signaling molecules between two cells coupled by a stochastically gated gap junction. We assume that signaling molecules diffuse within the cytoplasm of each cell and then either bind to some active region of the cell’s membrane (treated as a well-mixed compartment) or pass through the gap junction to the interior of the other cell. We treat the gap junction as a randomly fluctuating gate that switches between an open and a closed state according to a two-state Markov process. This means that the resulting PDE-ODE is stochastic due to the presence of a randomly switching boundary in the interior of the domain. It is assumed that each membrane compartment acts as a conditional oscillator, that is, it sits below a supercritical Hopf bifurcation. In the ungated case (gap junction always open), the system supports diffusion-induced oscillations, in which the concentration of signaling molecules within the two compartments is either in-phase or anti-phase. The presence of a reflection symmetry (for identical cells) means that the stochastic gate only affects the existence of anti-phase oscillations. In particular, there exist parameter choices where the gated system supports oscillations, but the ungated system does not, and vice versa. The existence of oscillations is investigated by solving a spectral problem obtained by averaging over realizations of the stochastic gate