Passage Time Probability Research Articles

First passage statistics of periodically driven models for neural dynamics

The dynamics of two simple neuron models subjected to periodic stimuli is studied at the level of the first passage time probability density. The results from analytic treatments of these models are presented, and a discussion of the important properties displayed by these models is given. Specifically, both models possess resonans phenomena in the first passage probability distribution, arising from of the interplay of characteristic time-scales in the system.

First passage time in a two-layer system

As a first step in the first passage problem for passive tracer in stratified porous media, we consider the case of a two-dimensional system consisting of two layers with different convection velocities. Using a lattice generating function formalism and a variety of analytic and numerical techniques, we calculate the asymptotic behavior of the first passage time probability distribution. We show analytically that the asymptotic distribution is a simple exponential in time for any choice of the velocities. The decay constant is given in terms of the largest eigenvalue of an operator related to a half-space Green's function. For the anti-symmetric case of opposite velocities in the layers, we show that the decay constant for system length $L$ crosses over from $L^{-2}$ behavior in diffusive limit to $L^{-1}$ behavior in the convective regime, where the crossover length $L^*$ is given in terms of the velocities. We also have formulated a general self-consistency relation, from which we have developed a recursive approach which is useful for studying the short time behavior.

Tracer dynamics in ocean sediments and the deciphering of past climates

An analysis of tracer dynamics in ocean sediments taking into account the disordered, fractal, character of geological materials is performed. The complexity of the medium is modeled by augmenting the equation of motion of the tracer through the addition of a stochastic forcing. In the one-dimensional version of the model, it is found both analytically and numerically that the dynamics is characterized by a large dispersion of arrival times at a given depth of the sediment as deduced from the first passage time probability distribution of the tracer. Moreover, the trajectories of two initially close tracer particles deviate at later times according to a power law, thereby raising the question of reliability of climatic trends deduced from geological data. Numerical simulations in a more realistic, two-dimensional setting show that this uncertainty becomes even more pronounced entailing the formation of a fractal precipitation front whose width increases with time. The implications of the results on the interpretation of geological records are discussed.

Reliability of Daniels systems subject to Gaussian load processes

A general method is developed for estimating reliability of Daniels systems subject to quasistatic and dynamic Gaussian load processes. The method is based on a generalization of the Slepian model, an identity involving first passage time probabilities and conditional mean crossing rates of response processes, and first/second order reliability methods. The use of the method is illustrated by an example.

Reliability of Daniels systems subject to quasistatic and dynamic nonstationary Gaussian load processes

A general method is developed for estimating reliability of Daniels systems subject to quasistatic and dynamic Gaussian load processes. The method is based on a generalization of the Slepian model, an identity involving first passage time probabilities and conditional mean crossing rates of response processes, and first/second order reliability estimates (FORM/SORM). The use of the method is illustrated by an example.

Mean first passage time for bound non-Markovian stochastic processes with superimposed shot noise

In a recent paper, Masoliver and Weiss reported exact analytical expressions for the first passage time (FPT) probability distribution to exit a given interval in the case of a free diffusion process subjected to the effect of superimposed shot noise. In the present note we study by computer simulation the more general case of a bound harmonic system driven by colored Gaussian noise and superimposed shot noise. Approximate expressions are derived for the mean first passage time (MFPT) to reach a given absorbing boundary; these theoretical results are found to be in a good agreement with the results of digital simulation.

Computer simulation of diffusion-controlled reactions in dispersions of spherical sinks

The diffusion-controlled reaction rate of dispersions of overlapping (fully penetrable) and nonoverlapping static sinks are determined through computer simulations. The diffusive or random-walk motions of the solutes are simulated in continuum space based on the first passage time probability distribution. This method yields accurate results for the reduced diffusion-controlled reaction rate k/ks where ks represents the dilute concentration Smoluchowski limit. Values of k/ks over a wide range of sink concentrations are obtained. These ‘‘exact’’ data are compared with a number of approximate theories. The merits and the regimes of validity of these theories are determined.

On the evaluation of first passage time densities for Gaussian processes

A detailed study of first- and second-order approximations to the first passage time conditional probability density function (p.d.f.) of a stationary Gaussian process with differentiable sample paths is provided both theoretically and numerically. An evaluation of mode and peak value of this function is also given and asymptotic expressions are derived for the functions W n ( t 1, t 2, …, t n |; x 0) ( n ⩾ 1) appearing in the series expansion of the first passage time probability density function.

EVALUATION OF FIRST PASSAGE TIME DENSITIES FOR DIFFUSION PROCESSE

Abstract Use of a Volterra second-kind integral equation is made to evaluate first passage time probability density functions through time varying boundaries for diffusion processes. The solutions are constructed in the form of infinite series whose terms are expressed as multidimensional integrals. An evaluation of such solutions is provided for the cases of Wiener and Ornstein-Uhlenbeck processes by standard numerical procedures, and by a Monte Carlo method. Results are discussed with reference to other existing computational methods.

Numerical solution of the Fokker-Planck equation via chebyschev polynomial approximations with reference to first passage time probability density functions

Chebyschev approximations are employed to solve the one-dimensional, time-dependent Fokker-Planck (forward Kolmogrov) equation in the presence of two barriers a finite “distance” apart. Solutions are presented for the fundamental intervals (−1, +1) and (0, +1). In order to speed up the calculations, sparse matrix routines are utilized. The first passage time probability density function is also evaluated. Illustrative numerical results are presented for the Wiener process with drift, and the Ornstein-Uhlenbeck process for a variety of combinations of boundary conditions.

Method for Random Vibration of Hysteretic Systems

Based on a Markov-vector formulation and a Galerkin solution procedure, a new method of modeling and solution of a large class of hysteretic systems (softening or hardening, narrow or wide-band) under random excitation is proposed. The excitation is modeled as a filtered Gaussian shot noise allowing one to take the nonstationarity and spectral content of the excitation into consideration. The solutions include time histories of joint density, moments of all order, and threshold crossing rate; for the stationary case, autocorrelation, spectral density, and first passage time probability are also obtained. Comparison of results of numerical example with Monte-Carlo solutions indicates that the proposed method is a powerful and efficient tool.

On the inverse of the first passage time probability problem

Since the pioneering work of Siegert (1951), the problem of determining the first passage time distribution for a preassigned continuous and time homogeneous Markov process described by a diffusion equation has been deeply analyzed and satisfactorily solved. Here we discuss the “inverse problem” — of applicative interest — consisting in deciding whether a given function can be considered as the first passage time probability density function for some continuous and homogeneous Markov diffusion process. A constructive criterion is proposed, and some examples are provided. One of these leads to a singular diffusion equation representing a dynamical model for the genesis of the lognormal distribution.

On the inverse of the first passage time probability problem

Since the pioneering work of Siegert (1951), the problem of determining the first passage time distribution for a preassigned continuous and time homogeneous Markov process described by a diffusion equation has been deeply analyzed and satisfactorily solved. Here we discuss the “inverse problem” — of applicative interest — consisting in deciding whether a given function can be considered as the first passage time probability density function for some continuous and homogeneous Markov diffusion process. A constructive criterion is proposed, and some examples are provided. One of these leads to a singular diffusion equation representing a dynamical model for the genesis of the lognormal distribution.

On relations between moment properties and almost sure Lyapunov stability for linear stochastic systems

In this paper we shall be concerned with the implications of almost sure asymptotic Lyapunov stability in the large that are contained in properties of the moments of the system. Definitions for Lyapunov stability relative to the three common modes of convergence of probability theory can be found, for example, in Bertram and Sarachik [l], who were the first in this country to apply the second method of Lyapunov to study stability in the mean of stochastic systems. A somewhat more comprehensive study of this nature can be found in a later paper by Kats and Krasovskii [2]. They were concerned primarily with systems whose parameters are finite state Markov processes. Almost sure Lyapunov stability is certainly the desirable property to ascertain and establish when studying real systems that are subjected to random variations in their parameter values, or are operating within randomly perturbed environmental conditions. However, almost sure properties of random systems are not as immediately obtainable as are mean properties of the system. A few results for almost sure stability of continuous parameter systems have already appeared in the literature. Bogdanoff [3] studies systems whose coefficients are finite sums of sinusoidal terms with incommensurable frequencies and random phases; Khas’minskii [4] studies diffusion processes by Lyapunov’s second method using a clever combination of first passage time probabilities and the maximum principle for elliptic operators; Bharucha [5] studies linear systems whose coefficient processes are piecewise constant, independent or Markovian, and applies the Borel-Cantelli lemma to show that mean square asymptotic stability implies almost sure asymptotic stability for these systems; Kozin [6] studies linear systems with continuous, stationary, ergodic coefficient processes and obtains a sufficient condition in terms of the expected value of the norm of the random coefficient matrix.

The response of simple nonlinear systems to a random disturbance of the earthquake type

abstract Stationary properties of the response of simple nonlinear systems to earthquake type (nonstationary) random disturbances are determined. The method used consists in generating member functions of the input processes, integrating the equations of motion using these as inputs to obtain output member functions, and then, from this ensemble of outputs, determining the statistical properties of interest. The method is flexible in that it is easy to apply to simple as well as complex linear and nonlinear systems; it yields a variety of statistical properties of engineering significance; it provides a method for approaching the design of aseismic structures; and more generally, shock resistant structures on a practical basis. In addition to determining and displaying input and corresponding output member functions, we determine individual as well as average velocity spectra, first passage time probabilities, and extreme value distributions of relative displacements. Comments are also offered on the utility of these results in design.

Statistical Properties of the Integral of a Binary Random Process

Statistical properties of the output z(t) of a finite time integrator are discussed. The input considered is a binary random y(t) having successive axis-crossing intervals which are statistically independent. Transform expressions are derived for the first- and second-order transition probability densities of the integrated process, and it is shown how these results may be extended to three or more dimensions. Four processes are considered as examples. The integrated z(t) is shown to be a projection of a Markov in three dimensions. The other two components are the original binary y(t) , and an associated ramp process x(t) . Various statistical properties of this ramp are considered and it is shown that x(t) is Markovian in one dimension. The first-order probability density and the transition probability density are discussed. Also, the transition probability density for the joint [x(t), y(t), z(t)] is given. Finally, in Appendix II, results are given for the first passage and recurrence time probability densities of x(t) , together with a relation between these two density functions.

Letter to the Editor—Comment on the “Reliability of Aircraft Structures in Resisting Chance Failure”

The purpose of this brief paper is to point out an objection to the assumptions made in a recently published formula on the reliability of an aircraft in resisting chance failure, and to indicate how, with first passage time probabilities, it is possible to derive a more realistic expression for this formula.

On the First Passage Time Probability Problem

We have derived an exact solution for the first passage time probability of a stationary one-dimensional Markoffian random function from an integral equation. A recursion formula for the moments is given for the case that the conditional probability density describing the random function satisfies a Fokker-Planck equation. Various known solutions for special applications (noise, Brownian motion) are shown to be special cases of our solution. The Wiener-Rice series for the recurrence time probability density is derived from a generalization of Schr\"odinger's integral equation, for the case of a two-dimensional Markoffian random function.