Moments Of Passage Times Research Articles

Approximating the first passage time density from data using generalized Laguerre polynomials

This paper analyzes a method to approximate the first passage time probability density function which turns to be particularly useful if only sample data are available. The method relies on a Laguerre-Gamma polynomial approximation and iteratively looks for the best degree of the polynomial such that a normalization condition is preserved. The proposed iterative algorithm relies on simple and new recursion formulae involving first passage time moments. These moments can be computed recursively from cumulants, if they are known. In such a case, the approximated density can be used also for the maximum likelihood estimates of the parameters of the underlying stochastic process. If cumulants are not known, suitable unbiased estimators relying on κ-statistics might be employed. To check the feasibility of the method both in fitting the density and in estimating the parameters, the first passage time problem of a geometric Brownian motion is considered.

Asymptotic Analysis of Target Fluxes in the Three-Dimensional Narrow Capture Problem

We develop an asymptotic analysis of target fluxes in the three-dimensional (3D) narrow capture problem. The latter concerns a diffusive search process in which the targets are much smaller than the size of the search domain. The small target assumption allows us to use matched asymptotic expansions and Green's functions to solve the diffusion equation in Laplace space. In particular, we derive an asymptotic expansion of the Laplace transformed flux into each target in powers of the nondimensionalized target size $\epsilon$. One major advantage of working directly with fluxes is that one can generate statistical quantities, such as splitting probabilities and conditional first passage time moments, without having to solve a separate boundary value problem in each case. However, in order to derive asymptotic expansions of these quantities, it is necessary to eliminate Green's function singularities that arise in the limit $s\rightarrow 0$, where $s$ is the Laplace variable. We achieve this by considering a triple expansion in $\epsilon$, $s$, and $\Lambda\sim \epsilon /s$. This allows us to perform partial summations over infinite power series in $\Lambda$, which leads to multiplicative factors of the form $\Lambda^n/(1+\Lambda)^n $. Since $\Lambda^n/(1+\Lambda)^n \rightarrow 1$ as $s\rightarrow 0$, the singularities in $s$ are eliminated. We then show how corresponding asymptotic expansions of the splitting probabilities and conditional mean first passage times (MFPTs) can be derived in the small-$s$ limit. The resulting expressions agree with previous asymptotic expansions derived by solving a separate boundary value problem for each statistical quantity, although each expansion was only carried out to second order in the expansions. Here we also determine the third order contributions, which are $O(\epsilon^2)$ and $O(\epsilon)$ in the cases of the splitting probabilities and conditional (MFPTs), respectively. Finally, we illustrate the theory by considering a pair of targets in a spherical search domain, for which the Green's functions can be calculated explicitly.

First passage time moments of asymmetric Lévy flights

We investigate the first-passage dynamics of symmetric and asymmetric Lévy flights in semi-infinite and bounded intervals. By solving the space-fractional diffusion equation, we analyse the fractional-order moments of the first-passage time probability density function for different values of the index of stability and the skewness parameter. A comparison with results using the Langevin approach to Lévy flights is presented. For the semi-infinite domain, in certain special cases analytic results are derived explicitly, and in bounded intervals a general analytical expression for the mean first-passage time of Lévy flights with arbitrary skewness is presented. These results are complemented with extensive numerical analyses.

Landau-like expansion for phase transitions in stochastic resetting

We develop a Landau like theory to characterize the phase transitions in resetting systems. Restart can either accelerate or hinder the completion of a first passage process. The transition between these two phases is characterized by the behavioral change in the order parameter of the system namely the optimal restart rate. Like in the original theory of Landau, the optimal restart rate can undergo a first or second order transition depending on the details of the system. Nonetheless, there exists no unified framework which can capture the onset of such novel phenomena. We unravel this in a comprehensive manner and show how the transition can be understood by analyzing the first passage time moments. Power of our approach is demonstrated in two canonical paradigm setup namely the Michaelis Menten chemical reaction and diffusion under restart.

Markov chains with heavy-tailed increments and asymptotically zero drift

We study the recurrence/transience phase transition for Markov chains on $\mathbb{R}_+$, $\mathbb{R}$, and $\mathbb{R}^2$ whose increments have heavy tails with exponent in $(1,2)$ and asymptotically zero mean. This is the infinite-variance analogue of the classical Lamperti problem. On $\mathbb{R}_+$, for example, we show that if the tail of the positive increments is about $c y^{-\alpha}$ for an exponent $\alpha \in (1,2)$ and if the drift at $x$ is about $b x^{-\gamma}$, then the critical regime has $\gamma = \alpha -1$ and recurrence/transience is determined by the sign of $b + c\pi \textrm{cosec} (\pi \alpha)$. On $\mathbb{R}$ we classify whether transience is directional or oscillatory, and extend an example of Rogozin \& Foss to a class of transient martingales which oscillate between $\pm \infty$. In addition to our recurrence/transience results, we also give sharp results on the existence/non-existence of moments of passage times.

Moments of passage times and asymptotic behavior of increasing self-similar markov processes

By using Lamperti's bijection between self-similar Markov processes and Lévy processes, we prove finiteness of moments and asymptotic behavior of passage times for increasing self-similar Markov processes valued in (0, ∞). We also investigate the behavior of the process when it crosses a level. A limit theorem concerning the distribution of the process immediately before it crosses some level is proved. Some useful examples are given.

Integrate-and-fire neurons driven by asymmetric dichotomous noise.

We consider a general integrate-and-fire (IF) neuron driven by asymmetric dichotomous noise. In contrast to the Gaussian white noise usually used in the so-called diffusion approximation, this noise is colored, i.e., it exhibits temporal correlations. We give an analytical expression for the stationary voltage distribution of a neuron receiving such noise and derive recursive relations for the moments of the first passage time density, which allow us to calculate the firing rate and the coefficient of variation of interspike intervals. We study how correlations in the input affect the rate and regularity of firing under variation of the model's parameters for leaky and quadratic IF neurons. Further, we consider the limit of small correlation times and find lowest order corrections to the first passage time moments to be proportional to the square root of the correlation time. We show analytically that to this lowest order, correlations always lead to a decrease in firing rate for a leaky IF neuron. All theoretical expressions are compared to simulations of leaky and quadratic IF neurons.

Explosion, implosion, and moments of passage times for continuous-time Markov chains: A semimartingale approach

We establish general theorems quantifying the notion of recurrence–through an estimation of the moments of passage times–for irreducible continuous-time Markov chains on countably infinite state spaces. Sharp conditions of occurrence of the phenomenon of explosion are also obtained. A new phenomenon of implosion is introduced and sharp conditions for its occurrence are proven. The general results are illustrated by treating models having a difficult behaviour even in discrete time.

The residence time equation

The residence time of a diffusing particle in a prescribed spatial region recently finds an increasing number of applications in physics and chemistry. A partial differential equation for the residence time moments is derived, as a generalization of the ordinary differential equation for the first passage time moments. When one seeks to calculate only the mean residence time and not its full distribution, this “residence time equation” constitutes a significant simplification over the conventional Feynman–Kac approach. We demonstrate this for a ball in d-dimensional space with an infinite observation time. The present approach may be useful also for other Brownian functionals.

Stochastic dynamics and passage times for diffusion approximations

We discuss first passage time problems for a class of one dimensional master equations with separable kernels. For this class of master equations, the integral equations for first passage time moments (FPTM) have been transformed exactly into ordinary differential equations by Weiss and Szabo. There the boundary conditions for the first passage time moments have been introduced separately. In our study we use the imbedding method to derive the first passage time densities(FPTD) by imbedding the boundary conditions for crossing the barriers into the master equations. We also derived the results of Weiss and Szabo model from our results for first passage time densities and FPTM for crossing of the barriers. In our study we also derived the results the FPTD and FPTM by taking one of the barriers as absorbing and the other barrier as reflecting. When the separable kernel has only single term the equation for FPTD and FPTM obtained are exactly the same as that of simple diffusion. Generalisation of the model is studied to obtain the differential equations for FPTD and FPTM. The generalisation in the separable kernel leads to higher order differential equation for FPTD and FPTM. The differential equation for FPTM is identified with that of time homogeneous Fokker-Plank equation.

Moments of passage times for Lévy processes

We give necessary and sufficient conditions, in terms of characteristics of the process, for finiteness of moments of passage times of general Lévy processes above horizontal, linear or certain curved boundaries. They apply in particular to processes which drift almost surely to infinity, and lead to estimates of the rate of growth of certain expectations, constituting generalised kinds of renewal theorems. Further results concern the inverse local time at the maximum and the ladder height process, the amount of time spent below a given level, and the overall minimum of the Lévy process. Des conditions nécessaires et suffisantes sont données pour la finitude des moments de certains temps de passage des processus de Lévy.

Moments of passage times for Lévy processes

We give necessary and sufficient conditions, in terms of characteristics of the process, for finiteness of moments of passage times of general Lévy processes above horizontal, linear or certain curved boundaries. They apply in particular to processes which drift almost surely to infinity, and lead to estimates of the rate of growth of certain expectations, constituting generalised kinds of renewal theorems. Further results concern the inverse local time at the maximum and the ladder height process, the amount of time spent below a given level, and the overall minimum of the Lévy process.

Finite-size scaling analysis of biased diffusion on fractals

Diffusion on a T fractal lattice under the influence of topological biasing fields is studied by finite size scaling methods. This allows to avoid proliferation and singularities which would arise in a renormalization group approach on infinite system as a consequence of logarithmic diffusion. Within the scheme, logarithmic diffusion is proved on the basis of an analysis of various temporal scales such as first passage time moments and survival probability characteristic time. This confirms and puts on firmer basis previous renormalization group results. A careful study of the asymptotic occupation probabilities of different kinds of lattice points allows to elucidate the mechanism of trapping into dangling ends, which is responsible of the logarithmic time dependence of average displacement.

Passage-time moments for continuous non-negative stochastic processes and applications

We give criteria for the finiteness or infiniteness of the passage-time moments for continuous non-negative stochastic processes in terms of sub/supermartingale inequalities for powers of these processes. We apply these results to one-dimensional diffusions and also reflected Brownian motion in a wedge. The discrete-time analogue of this problem was studied previously by Lamperti and more recently by Aspandiiarov, Iasnogorodski and Menshikov [2]. Our results are continuous analogues of those in [2], but our proofs are direct and do not rely on approximation by discrete-time processes.

Passage-time moments for continuous non-negative stochastic processes and applications

We give criteria for the finiteness or infiniteness of the passage-time moments for continuous non-negative stochastic processes in terms of sub/supermartingale inequalities for powers of these processes. We apply these results to one-dimensional diffusions and also reflected Brownian motion in a wedge. The discrete-time analogue of this problem was studied previously by Lamperti and more recently by Aspandiiarov, Iasnogorodski and Menshikov [2]. Our results are continuous analogues of those in [2], but our proofs are direct and do not rely on approximation by discrete-time processes.

Passage-time moments for nonnegative stochastic processes and an application to reflected random walks in a quadrant

In this paper we get some sufficient conditions for the finiteness or nonfiniteness of the passage-time moments for nonnegative discrete parameter processes. The developed criteria are closely connected with the well-known results of Foster for the ergodicity of Markov chains and are given in terms of sub(super)martingales. Then, as an application of the obtained results, we get explicit conditions for the finiteness or nonfiniteness of passage-time moments for reflected random walks in a quadrant with zero drift in the interior.

Computation of first passage time moments for stochastic diffusion processes modelling nerve membrane depolarization

For further understanding of neural coding, stochastic variability of interspike intervals has been investigated by both experimental and theoretical neuroscientists. In stochastic neuronal models, the interspike interval corresponds to the time period during which the process imitating the membrane potential reaches a threshold for the first time from a reset depolarization. For neurons belonging to complex networks in the brain, stochastic diffusion processes are often used to approximate the time course of the membrane potential. The interspike interval is then viewed as the first passage time for the employed diffusion process. Due to a lack of analytical solution for the related first passage time problem for most diffusion neuronal models, a numerical integration method, which serves to compute first passage time moments on the basis of the Siegert recursive formula, is presented in this paper. For their neurobiological plausibility, the method here is associated with diffusion processes whose state spaces are restricted to finite intervals, but it can also be applied to other diffusion processes and in other (non-neuronal) contexts. The capability of the method is demonstrated in numerical examples and the relation between the integration step, accuracy of calculation and amount of computing time required is discussed.

First passage time problems for a class of master equations with separable kernels

Abstract We discuss first passage time problems for a class of one-dimensional master equations with separable kernels. For this class of master equations the integral equation for first passage time moments can be transformed exactly into ordinary differential equations. When the separable kernel has only a single term the equation for the mean first passage time obtained is exactly that for simple diffusion. The boundary conditions, however, differ from those appropriate to simple diffusion. The equations for higher moments differ slightly from those for simple diffusion. Analysis is presented, of a generalization of a model of a random walk with long-range jumps first investigated by Lindenberg and Shuler. Since the equations can be solved exactly one can study the behavior of boundary conditions in the continuum limit. The generalization to a larger number of terms in the separable kernel leads to higher order equations for the first passage time moments. In each case, boundary conditions can be found directly from the original master equation.

First passage times and the kinetics of unimolecular dissociation

Approximate solutions for multistep master equations describing the time evolution of product formation in multiphoton or thermal unimolecular reactions are investigated. In particular, a method based on fitting the first few moments of the passage time distribution associated with the given stochastic process to proposed simple expressions for the product yield function is studied. It is shown that reasonable agreement with the exact numerical solution of the corresponding master equation is obtained with a two parameter fit (using two passage time moments) and an excellent agreement is obtained with a three parameter fit (using three passage time moments). In no case studied does a need arise for more than a three-moment description and the quality of available experimental results makes the simpler two-moment description sufficient in most cases. Analytical solutions for the first and second passage time moments are obtained for simple discrete and continuous master equation models. Expressions for the incubation time and the reaction rate are obtained in terms of these solutions. The validity of discretizing a continuous master equation (which is an important simplifying step in evaluating the time evolution associated with multiphoton dissociations in the presence of collisions, or with thermal unimolecular reactions involving large molecules) is studied using both the approximate two-moment solutions and exact numerical solutions. It is concluded that a proper discretization of a continuous master equation may be carried out provided ε≪kBT, where ε is the discretization energy step, kB the Boltzmann constant, and T the effective (density of states weighted) temperature. A larger discretization step can be used if only the incubation time is required. Using the approximately discretized master equation, we next calculate the effect of collisions on the incubation time and the rate of multiphoton dissociation using a model constructed to correspond to the unimolecular dissociation of tetramethyldioxethane. Incubation times are found to be less sensitive to collisions then the reaction rates. Finally, we investigate the applicability of the passage time moments method to describe the time evolution of product formation in a system whose dynamics is determined by a quantum mechanical Liouville equation. Again the two-moment description provides a reasonable and the three-moment approximation a good approximation to the exact solution. The three-moment approximations, however, cannot be used when the pressure (i.e., the dephasing rate) is too low.

Incubation times in the multiphoton dissociation of polyatomic molecules

The incubation period revealed in the multiphoton dissociation of molecules by intense infrared lasers is discussed. It is found experimentally that large excess of added foreign gas affects the incubation period to a much smaller degree than the overall yield. A rate equations model is presented, including both the laser intensity and collisional effects. Exact numerical solution is compared with a simple analytical approximation, based on the passage time moments method. Agreement with experimental results is quite satisfactory, indicating that the role of collisions in the case discussed (tetramethyldioxetane dissociation) is primarily vibrational relaxation of excited molecules.