Boundary value problems for statistics of diffusion in a randomly switching environment: PDE and SDE perspectives

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.Comment: 22 pages, 3 figure

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.Comment: 11 pages. Added comments about the deterministic problem. Typos in references fixe

Analysis of non-processive molecular motor transport using renewal reward theory

We propose and analyze a mathematical model of cargo transport by non-processive molecular motors. In our model, the motors change states by random discrete events (corresponding to stepping and binding/unbinding), while the cargo position follows a stochastic differential equation (SDE) that depends on the discrete states of the motors. The resulting system for the cargo position is consequently an SDE that randomly switches according to a Markov jump process governing motor dynamics. To study this system we (1) cast the cargo position in a renewal theory framework and generalize the renewal reward theorem and (2) decompose the continuous and discrete sources of stochasticity and exploit a resulting pair of disparate timescales. With these mathematical tools, we obtain explicit formulas for experimentally measurable quantities, such as cargo velocity and run length. Analyzing these formulas then yields some predictions regarding so-called non-processive clustering, the phenomenon that a single motor cannot transport cargo, but that two or more motors can. We find that having motor stepping, binding, and unbinding rates depend on the number of bound motors, due to geometric effects, is necessary and sufficient to explain recent experimental data on non-processive motors.Comment: updated to final journal versio

Extreme statistics of superdiffusive Levy flights and every other Levy subordinate Brownian motion

The search for hidden targets is a fundamental problem in many areas of science, engineering, and other fields. Studies of search processes often adopt a probabilistic framework, in which a searcher randomly explores a spatial domain for a randomly located target. There has been significant interest and controversy regarding optimal search strategies, especially for superdiffusive processes. The optimal search strategy is typically defined as the strategy that minimizes the time it takes a given single searcher to find a target, which is called a first hitting time (FHT). However, many systems involve multiple searchers and the important timescale is the time it takes the fastest searcher to find a target, which is called an extreme FHT. In this paper, we study extreme FHTs for any stochastic process that is a random time change of Brownian motion by a Levy subordinator. This class of stochastic processes includes superdiffusive Levy flights in any space dimension, which are processes described by a Fokker-Planck equation with a fractional Laplacian. We find the short-time distribution of a single FHT for any Levy subordinate Brownian motion and use this to find the full distribution and moments of extreme FHTs as the number of searchers grows. We illustrate these rigorous results in several examples and numerical simulations.Comment: 29 pages, 5 figure

The effects of fast inactivation on conditional first passage times of mortal diffusive searchers

The first time a searcher finds a target is called a first passage time (FPT). In many physical, chemical, and biological processes, the searcher is "mortal," which means that the searcher might become inactivated (degrade, die, etc.) before finding the target. In the context of intracellular signaling, an important recent work discovered that fast inactivation can drastically alter the conditional FPT distribution of a mortal diffusive searcher, if the searcher is conditioned to find the target before inactivation. In this paper, we prove a general theorem which yields an explicit formula for all the moments of such conditional FPTs in the fast inactivation limit. This formula is quite universal, as it holds under very general conditions on the diffusive searcher dynamics, the target, and the spatial domain. These results prove in significant generality that if inactivation is fast, then the conditional FPT compared to the FPT without inactivation is (i) much faster, (ii) much less affected by spatial heterogeneity, and (iii) much less variable. Our results agree with recent computational and theoretical analysis of a certain discrete intracellular diffusion model and confirm a conjecture related to the effect of spatial heterogeneity on intracellular signaling.Comment: 24 pages, 1 figur

Universal Formula for Extreme First Passage Statistics of Diffusion

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 work, 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 Riemannian manifolds, (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.Comment: 11 pages, 3 figure

Fractional reaction-subdiffusion equations: solution, stochastic paths, and applications

In contrast to normal diffusion, there is no canonical model for reactions between chemical species which move by anomalous subdiffusion. Indeed, the type of mesoscopic equation describing reaction-subdiffusion depends on subtle assumptions about the microscopic behavior of individual molecules. Furthermore, the correspondence between mesoscopic and microscopic models is not well understood. In this paper, we study the subdiffusion-limited model, which is defined by mesoscopic equations with fractional derivatives applied to both the movement and the reaction terms. Assuming that the reaction terms are affine functions, we show that the solution to the fractional system is the expectation of a random time change of the solution to the corresponding integer order system. This result yields a simple and explicit algebraic relationship between the fractional and integer order solutions in Laplace space. We then find the microscopic Langevin description of individual molecules that corresponds to such mesoscopic equations and give a computer simulation method to generate their stochastic trajectories. This analysis identifies some precise microscopic conditions that dictate when this type of mesoscopic model is or is not appropriate. We apply our results to several scenarios in cell biology which, despite the ubiquity of subdiffusion in cellular environments, have been modeled almost exclusively by normal diffusion. Specifically, we consider subdiffusive models of morphogen gradient formation, fluctuating mobility, and fluorescence recovery after photobleaching (FRAP) experiments. We also apply our results to fractional ordinary differential equations.Comment: 29 pages, 3 figure

Anomalous reaction-diffusion equations for linear reactions

Deriving evolution equations accounting for both anomalous diffusion and reactions is notoriously difficult, even in the simplest cases. In contrast to normal diffusion, reaction kinetics cannot be incorporated into evolution equations modeling subdiffusion by merely adding reaction terms to the equations describing spatial movement. A series of previous works derived fractional reaction-diffusion equations for the spatiotemporal evolution of particles undergoing subdiffusion in one space dimension with linear reactions between a finite number of discrete states. In this paper, we first give a short and elementary proof of these previous results. We then show how this argument gives the evolution equations for more general cases, including subdiffusion following any fractional Fokker-Planck equation in an arbitrary $d$-dimensional spatial domain with time-dependent reactions between infinitely many discrete states. In contrast to previous works which employed a variety of technical mathematical methods, our analysis reveals that the evolution equations follow from (i) the probabilistic independence of the stochastic spatial and discrete processes describing a single particle and (ii) the linearity of the integro-differential operators describing spatial movement. We also apply our results to systems combining reactions with superdiffusion.Comment: 7 page

Extreme first passage times for random walks on networks

Many biological, social, and communication systems can be modeled by ``searchers'' moving through a complex network. For example, intracellular cargo is transported on tubular networks, news and rumors spread through online social networks, and the rapid global spread of infectious diseases occurs through passengers traveling on the airport network. To understand the timescale of search/transport/spread, one commonly studies the first passage time (FPT) of a single searcher (or ``spreader'') to a target. However, in many scenarios the relevant timescale is not the FPT of a single searcher to a target, but rather the FPT of the fastest searcher to a target out of many searchers. For example, many processes in cell biology are triggered by the first molecule to find a target out of many, and the time it takes an infectious disease to reach a particular city depends on the first infected traveler to arrive out of potentially many infected travelers. Such fastest FPTs are called extreme FPTs. In this paper, we study extreme FPTs for a general class of continuous-time random walks on networks (which includes continuous-time Markov chains). In the limit of many searchers, we find explicit formulas for the probability distribution and all the moments of the $k$th fastest FPT. These rigorous formulas depend only on network parameters along a certain geodesic path(s) from the starting location to the target since the fastest searchers take a direct route to the target. Furthermore, our results allow one to estimate if a particular system is in a regime characterized by fast extreme FPTs. We also prove similar results for mortal searchers on a network that are conditioned to find the target before a fast inactivation time. We illustrate our results with numerical simulations and uncover potential pitfalls of modeling diffusive or subdiffusive processes involving extreme statistics.Comment: 33 pages, 4 figure

Smooth invariant densities for random switching on the torus

We consider a random dynamical system obtained by switching between the flows generated by two smooth vector fields on the 2d-torus, with the random switchings happening according to a Poisson process. Assuming that the driving vector fields are transversal to each other at all points of the torus and that each of them allows for a smooth invariant density and no periodic orbits, we prove that the switched system also has a smooth invariant density, for every switching rate. Our approach is based on an integration by parts formula inspired by techniques from Malliavin calculus.Comment: 19 page