Expected Exit Time Research Articles

Extending and validating a theoretical model to predict the effectiveness of building evacuations

To predict the effectiveness of building evacuations is a very difficult task in the general case. In a previous work, the historical results of 47 evacuation drills in 15 different university buildings, both academic and residential, involving more than 19 000 persons, was analyzed, and a method based on dimensional analysis and statistical regression was proposed to give an estimation of the exit time in case of evacuation. Comparing this estimated exit time with the real values obtained in evacuation drills, more informed decisions on whether to invest in more training and/or preventive culture of the occupants or to invest in structural improvements of the buildings can be taken. In this work, we both propose a refinement of the method to calculate expected exit times, that leads to an even better adjustment between predictions and real-world results, and we use this refined model to predict the results of evacuations of a new building, whose use and characteristics are different from those previously studied, and whose data was provided by other authors in the bibliography. We show that there exists a correlation between the published results and the predictions generated by our model, both from a quantitative and qualitative point of view.

Integrals of incomplete beta functions, with applications to order statistics, random walks and string enumeration

We study the probability that one beta-distributed random variable exceeds the maximum of two others, allowing all three to have general parameters. This amounts to studying Euler transforms of products of two incomplete beta functions. We provide a closed form for the general problem in terms of Kampé de Fériet functions and a variety of simpler closed forms in special cases. The results are applied to derive the moments of the maximum of two independent beta-distributed random variables and to find inner products of incomplete beta functions. Restricted to positive integer parameters, our results are applied to determine an expected exit time for a conditioned random walk and also to a combinatorial problem of enumerating strings comprised of three different letters, subject to constraints.

Expected exit time for time-periodic stochastic differential equations and applications to stochastic resonance

In this paper, we derive a parabolic partial differential equation for the expected exit time of non-autonomous time-periodic non-degenerate stochastic differential equations. This establishes a Feynman–Kac duality between expected exit time of time-periodic stochastic differential equations and time-periodic solutions of parabolic partial differential equations. Casting the time-periodic solution of the parabolic partial differential equation as a fixed point problem and a convex optimisation problem, we give sufficient conditions in which the partial differential equation is well-posed in a weak and classical sense. With no known closed formulae for the expected exit time, we show our method can be readily implemented by standard numerical schemes. With relatively weak conditions (e.g. locally Lipschitz coefficients), the method in this paper is applicable to wide range of physical systems including weakly dissipative systems. Particular applications towards stochastic resonance will be discussed.

Return Probability and Recurrence for the Random Walk Driven by Two-Dimensional Gaussian Free Field

Given any $\gamma>0$ and for $\eta=\{\eta_v\}_{v\in \mathbb Z^2}$ denoting a sample of the two-dimensional discrete Gaussian free field on $\mathbb Z^2$ pinned at the origin, we consider the random walk on~$\mathbb Z^2$ among random conductances where the conductance of edge $(u, v)$ is given by $\mathrm{e}^{\gamma(\eta_u + \eta_v)}$. We show that, for almost every~$\eta$, this random walk is recurrent and that, with probability tending to~1 as $T\to \infty$, the return probability at time~$2T$ decays as $T^{-1+o(1)}$. In addition, we prove a version of subdiffusive behavior by showing that the expected exit time from a ball of radius~$N$ scales as $N^{\psi(\gamma)+o(1)}$ with $\psi(\gamma)>2$ for all~$\gamma>0$. Our results rely on delicate control of the effective resistance for this random network. In particular, we show that the effective resistance between two vertices at Euclidean distance~$N$ behaves as~$N^{o(1)}$.

On the singular control of exchange rates

Consider a central bank that wants to manage the exchange rate between its domestic currency and a foreign one. The central bank can purchase and sell the foreign currency, and each direct intervention on the exchange market leads to a proportional cost whose instantaneous marginal value depends on the current level of the exchange rate. The central bank aims at minimizing the total expected costs of interventions on the exchange market, plus a total expected running cost. We formulate this problem as an infinite time-horizon bounded-variation stochastic control problem. The exchange rate evolves as a general one-dimensional diffusion, and it is linearly controlled by two nondecreasing processes modeling the cumulative amount of foreign currency that has been purchased and sold by the central bank. We provide a complete solution to this problem by finding the explicit expression of the value function and a complete characterization of the optimal control. At each instant of time, the optimally controlled exchange rate is kept within a band whose size is endogenously determined as part of the solution to the problem. We also study the expected exit time from the band, and the sensitivity of the width of the band with respect to the model’s parameters in the case when the exchange rate evolves (in absence of any intervention) as an Ornstein–Uhlenbeck process, and the marginal proportional costs of controls are constant. The techniques employed in the paper are those of the theory of singular stochastic control and of one-dimensional diffusions.

Computing the quasipotential for nongradient SDEs in 3D

Nongradient SDEs with small white noise often arise when modeling biological and ecological time-irreversible processes. If the governing SDE were gradient, the maximum likelihood transition paths, transition rates, expected exit times, and the invariant probability distribution would be given in terms of its potential function. The quasipotential plays a similar role for nongradient SDEs. Unfortunately, the quasipotential is the solution of a functional minimization problem that can be obtained analytically only in some special cases. We propose a Dijkstra-like solver for computing the quasipotential on regular rectangular meshes in 3D. This solver results from a promotion and an upgrade of the previously introduced ordered line integral method with the midpoint quadrature rule for 2D SDEs. The key innovations that have allowed us to keep the CPU times reasonable while maintaining good accuracy are (i) a new hierarchical update strategy, (ii) the use of Karush–Kuhn–Tucker theory for rejecting unnecessary simplex updates, and (iii) pruning the number of admissible simplexes and a fast search for them. An extensive numerical study is conducted on a series of linear and nonlinear examples where the quasipotential is analytically available or can be found at transition states by other methods. In particular, the proposed solver is applied to Tao's examples where the transition states are hyperbolic periodic orbits, and to a genetic switch model by Lv et al. (2014) [21]. The C source code implementing the proposed algorithm is available at M. Cameron's web page.

Exit times for ARMA processes

Abstract We study the asymptotic behaviour of the expected exit time from an interval for the ARMA process, when the noise level approaches 0.

Multilevel Estimation of Expected Exit Times and Other Functionals of Stopped Diffusions

This paper proposes and analyzes a new multilevel Monte Carlo method for the estimation of mean exit times for multidimensional Brownian diffusions and associated functionals which correspond to so...

Robustness and relative stability of multiple attractors in\\ a stochastic Boolean network

Multiple phenotypic states and dynamic pathways always exist in biological networks, but their robustness against fluctuations and relative stability have not been fully recognized or carefully analyzed yet. Here we try to address these issues with a widely used stochastic Boolean network model. The stability of phenotypic states and dynamical pathways corresponds to the robustness and relative stability of attractors in such a stochastic Boolean network, which can be analyzed using the theory of exponentially perturbed Markov chains. It is already known that the logarithm of the expected exit time escaping from an attractor is proportional to the ``minimum activation energy barrier in the exponentially perturbed Markov chains. We first prove that within an exponentially perturbed Markov chain, all pathways between attractors with the minimum possible ``activation energy barrier have the same probability weights in the zero-noise limit, and therefore the relative stability just depends on the number of optimal paths. Another important implication of this theory is that, once the non-neglectable fluctuations are rather low, a phase transition phenomenon emerges: In one parameter region the probability weights of the coexisted phenotypic states and pathways are comparable with each other; whereas in some other parameter regions, the probability weight of certain phenotypic state or pathway can even dominate and become globally attractive. Finally, we apply the theory to an artificial model and the Siah-1/beta-catenin/p14/19 ARF loop of protein p53 dynamics. Our theory can also determine how the transition time and the number of optimal transition paths between the phenotypic states and parallel pathways depend on all the parameters, and help to identify those possibly more crucial nodes and the interactions between them in a biological network.

Martin boundary of unbounded sets for purely discontinuous Feller processes

Abstract In this paper, we study the Martin kernels of general open sets associated with inaccessible points for a large class of purely discontinuous Feller processes in metric measure spaces. Let D be an unbounded open set. Infinity is accessible from D if the expected exit time from D is infinite, and inaccessible otherwise. We prove that under suitable assumptions there is only one Martin boundary point associated with infinity, and that this point is minimal if and only if infinity is accessible from D. Similar results are also proved for finite boundary points of D.

Local trapping for elliptic random walks in random environments in $$\mathbb {Z}^d$$

We consider elliptic random walks in i.i.d. random environments on \(\mathbb {Z}^d\). The main goal of this paper is to study under which ellipticity conditions local trapping occurs. Our main result is to exhibit an ellipticity criterion for ballistic behavior which extends previously known results. We also show that if the annealed expected exit time of a unit hypercube is infinite then the walk has zero asymptotic velocity.

Quasipotential and exit time for 2D Stochastic Navier-Stokes equations driven by space time white noise

We are dealing with the Navier-Stokes equation in a bounded regular domain $$\mathcal {O}$$ of $$\mathbb {R}^2$$ , perturbed by an additive Gaussian noise $$\partial w^{Q_\delta }/\partial t$$ , which is white in time and colored in space. We assume that the correlation radius of the noise gets smaller and smaller as $$\delta \searrow 0$$ , so that the noise converges to the white noise in space and time. For every $$\delta >0$$ we introduce the large deviation action functional $$S^\delta _{T}$$ and the corresponding quasi-potential $$U_\delta $$ and, by using arguments from relaxation and $$\Gamma $$ -convergence we show that $$U_\delta $$ converges to $$U=U_0$$ , in spite of the fact that the Navier-Stokes equation has no meaning in the space of square integrable functions, when perturbed by space-time white noise. Moreover, in the case of periodic boundary conditions the limiting functional $$U$$ is explicitly computed. Finally, we apply these results to estimate of the asymptotics of the expected exit time of the solution of the stochastic Navier-Stokes equation from a basin of attraction of an asymptotically stable point for the unperturbed system.

Barriers, exit time and survival probability for unimodal Lévy processes

AbstractWe give superharmonic functions and derive sharp bounds for the expected exit time and probability of survival for isotropic unimodal Lévy processes in smooth domains.

Traveling waves and statistical distributions connected to systems of interacting populations

We discuss the following two issues from the dynamics of interacting populations: •(I) density waves for the case or negligible random fluctuations of the population densities,•(II) probability distributions connected to the model equations for spatially averaged population densities for the case of significant random fluctuations of the independent quantity that can be associated with the population density. For the case of issue (I) we consider model equations containing polynomial nonlinearities. Such nonlinearities arise as a consequence of interaction among the populations (for the case of large population densities) or as a result of a Taylor series expansion (for the case of small density of interacting populations). By means of the modified method of the simplest equation we obtain exact traveling-wave solutions of the model equations and these solution. For the case of issue (II) we discuss model equations of the Fokker–Planck kind for the evolution of the statistical distributions of population densities. We derive a few stationary distributions for the population density and calculate the expected exit time associated with the extinction of the studied population.

Finding the quasipotential for nongradient SDEs

The quasipotential with respect to a given attractor of a system, evolving according to a stochastic differential equation (SDE) with a small noise, is a function that allows us to find the expected exit time from its basin of attraction and to estimate the equilibrium probability density in its neighborhood. This work is devoted to the study of the quasipotential of nongradient SDEs with isotropic diffusion. The goals are: (i) to understand the theoretical behavior of the quasipotential, and (ii) to develop tools for its numerical study.This work is focused on 2D equations with asymptotically stable equilibria and limit cycles. In the theoretical part, we establish a number of properties of the quasipotential. For 2D linear equations, we derive an exact formula for the quasipotential. For 2D nonlinear equations, we establish how the quasipotential grows near the stable equilibria and the stable limit cycles. In the numerical part of this work we adjust the OUM for computing the quasipotential on a regular mesh. We show that the quasipotential satisfies a Hamilton–Jacobi–Bellman equation with an anisotropic and unbounded speed function. The unbounded speed function may lead to an additional consistency error in specific areas of the phase space. However, we conduct an error analysis and show that even if this error appears, it decays quadratically as we refine the mesh.We apply the proposed numerical algorithm to a number of examples: linear systems, the Maier–Stein model, a system with stochastic resonance, and a system with two stable limit cycles. Whenever it is appropriate, we compute the Minimum Action Paths by shooting characteristics. The Minimum Action Paths found in this way are necessarily the global minimizers of the Freidlin–Wentzell action.

Stochastic exit time problems arising in process control

This paper concerns the problem of controlling a stochastic system, with small noise parameter, to prevent it leaving a safe region of the state space. Such problems arise in flow control and other areas. We consider a formulation of the problem, in which a control is sought, to maximize a cost which is related to the expected exit time, but modified to reduce the probability of an early exit, according to a specified level of risk aversion (‘risk sensitive’ stochastic control). Formally letting the noise parameter tend to zero, we find that the optimal control strategy for this problem coincides with the optimal feedback control strategy for a differential game. We identify a class of differential games arising in this way, the so called decomposable differential games, for which the optimal control strategy can be easily obtained and illustrate the proposed solution technique by applying it to a flow control problem arising in process systems engineering.

Exit times for a class of random walks exact distribution results

For a random walk with both downward and upward jumps (increments), the joint distribution of the exit time across a given level and the undershoot or overshoot at crossing is determined through its generating function, when assuming that the distribution of the jump in the direction making the exit possible has a Laplace transform which is a rational function. The expected exit time is also determined and the paper concludes with exact distribution results concerning exits from bounded intervals. The proofs use simple martingale techniques together with some classical expansions of polynomials and Rouché's theorem from complex function theory.

Exit times for a class of random walks exact distribution results

For a random walk with both downward and upward jumps (increments), the joint distribution of the exit time across a given level and the undershoot or overshoot at crossing is determined through its generating function, when assuming that the distribution of the jump in the direction making the exit possible has a Laplace transform which is a rational function. The expected exit time is also determined and the paper concludes with exact distribution results concerning exits from bounded intervals. The proofs use simple martingale techniques together with some classical expansions of polynomials and Rouché's theorem from complex function theory.

On the expected exit time of planar Brownian motion from simply connected domains

In this note, we explore applications of a known lemma which relates the expected exit time of Brownian motion from a simply connected domain with the power series of a conformal map into that domain. We use the lemma to calculate the expected exit time from a number of domains, and in the process describe a probabilistic method for summing certain series. In particular, we give a proof of Euler's classical result that $\zeta(2) = \pi^2/6$. We also show how the relationship between the power series and the Brownian exit time gives several immediate consequences when teamed with a deep result of de Branges concerning the coefficients of power series of normalized conformal maps. We conclude by stating an extension of the lemma in question to domains which are not simply connected.

Asymptotic first exit times of the Chafee-Infante equation with small heavy-tailed Lévy noise

This article studies the behavior of stochastic reaction-diffusion equations driven by additive regularly varying pure jump L'evy noise in the limit of small noise intensity. It is shown that the law of the suitably renormalized first exit times from the domain of attraction of a stable state converges to an exponential law of parameter 1 in a strong sense of Laplace transforms, including exponential moments. As a consequence, the expected exit times increase polynomially in the inverse intensity, in contrast to Gaussian perturbations, where this growth is known to be of exponential rate.