First Passage Time Problem Research Articles

Crossover between Lévy and Gaussian regimes in first-passage processes

We propose an approach to the problem of the first-passage time. Our method is applicable not only to the Wiener process but also to the non-Gaussian Lévy flights or to more complicated stochastic processes whose distributions are stable. To show the usefulness of the method, we particularly focus on the first-passage time problems in the truncated Lévy flights (the so-called KoBoL processes from Koponen, Boyarchenko, and Levendorskii), in which the arbitrarily large tail of the Lévy distribution is cut off. We find that the asymptotic scaling law of the first-passage time t distribution changes from t(-(alpha+1)/alpha)-law (non-Gaussian Lévy regime) to t(-32)-law (Gaussian regime) at the crossover point. This result means that an ultraslow convergence from the non-Gaussian Lévy regime to the Gaussian regime is observed not only in the distribution of the real time step for the truncated Lévy flight but also in the first-passage time distribution of the flight. The nature of the crossover in the scaling laws and the scaling relation on the crossover point with respect to the effective cutoff length of the Lévy distribution are discussed.

Energy diffusion controlled reaction rate in dissipative Hamiltonian systems

In this paper the energy diffusion controlled reaction rate in dissipative Hamiltonian systems is investigated by using the stochastic averaging method for quasi Hamiltonian systems. The boundary value problem of mean first-passage time (MFPT) of averaged system is formulated and the energy diffusion controlled reaction rate is obtained as the inverse of MFPT. The energy diffusion controlled reaction rate in the classical Kramers bistable potential and in a two-dimensional bistable potential with a heat bath are obtained by using the proposed approach respectively. The obtained results are then compared with those from Monte Carlo simulation of original systems and from the classical Kramers theory. It is shown that the reaction rate obtained by using the proposed approach agrees well with that from Monte Carlo simulation and is more accurate than the classical Kramers rate.

Limit at Zero of the First-Passage Time Density and the Inverse Problem for One-Dimensional Diffusions

We study the limit at zero of the first-passage time density of a one-dimensional diffusion process over a moving boundary and we also deal with the inverse first-passage time problem, which consists of determining the boundary shape when the first-passage density is known. Our results generalize the analogous ones already known for Brownian motion. We illustrate some examples for which the results are obtained analytically and by a numerical procedure.

Gaussian processes and neuronal modeling

This work is a contribution towards the understanding of certain features of mathematical models of single neurons. Emphasis is set on neuronal firing, for which the first passage time (FPT) problem bears a fundamental relevance. We focus the attention on modeling the change of the neuron membrane potential between two consecutive spikes by Gaussian stochastic processes, both of Markov and of non-Markov types. Methods to solve the FPT problems, both of a theoretical and of a computational nature, are sketched, including the case of random initial values. Significant similarities or diversities between computational and theoretical results are pointed out, disclosing the role played by the correlation time that has been used to characterize the neuronal activity. It is highlighted that any conclusion on this matter is strongly model-dependent. In conclusion, an outline of the asymptotic behavior of FPT densities is provided, which is particularly useful to discuss neuronal firing under certain slow activity conditions.

Residence Time Statistics for Normal and Fractional Diffusion in a Force Field

We investigate statistics of occupation times for an over-damped Brownian particle in an external force field. A backward Fokker-Planck equation introduced by Majumdar and Comtet describing the distribution of occupation times is solved. The solution gives a general relation between occupation time statistics and probability currents which are found from solutions of the corresponding problem of first passage time. This general relationship between occupation times and first passage times, is valid for normal Markovian diffusion and for non-Markovian sub-diffusion, the latter modeled using the fractional Fokker-Planck equation. For binding potential fields we find in the long time limit ergodic behavior for normal diffusion, while for the fractional framework weak ergodicity breaking is found, in agreement with previous results of Bel and Barkai on the continuous time random walk on a lattice. For non-binding potential rich physical behaviors are obtained, and classification of occupation time statistics is made possible according to whether or not the underlying random walk is recurrent and the averaged first return time to the origin is finite. Our work establishes a link between fractional calculus and ergodicity breaking.

A first-passage time problem for many random walkers

The passage of ions through membrane channels plays an important role in many fields of biology. An earlier paper [M. Boguñá, A.M. Berezhkovskii, G.H. Weiss, Phys. Rev. E 62 (2000) 3250] developed a toy model for statistical properties of the occupancy of a single site by different numbers of lattice random walkers chosen from an infinite set. It was assumed there that the residence time in one sojourn at the origin differed from the residence time of points elsewhere. In this paper we derive some properties of the corresponding first-passage time to the occupancy of the special site by k ( > 1 ) random walkers in one or two dimensions. Results of our study were obtained from an extensive set of simulations.

Diffusion in fluctuating media: first passage time problem

We study the actual and important problem of Mean First Passage Time (MFPT) for diffusion in fluctuating media. We exploit van Kampen's technique of composite stochastic processes, obtaining analytical expressions for the MFPT for a general system, and focus on the two state case where the transitions between the states are modelled introducing both Markovian and non-Markovian processes. The comparison between the analytical and simulations results show an excellent agreement.

Feedback minimization of first-passage failure of quasi non-integrable Hamiltonian systems

The non-linear stochastic optimal control of quasi non-integrable Hamiltonian systems for minimizing their first-passage failure is investigated. A controlled quasi non-integrable Hamiltonian system is reduced to an one-dimensional controlled diffusion process of averaged Hamiltonian by using the stochastic averaging method for quasi non-integrable Hamiltonian systems. The dynamical programming equations and their associated boundary and final time conditions for the problems of maximization of reliability and of maximization of mean first-passage time are formulated. The optimal control law is derived from the dynamical programming equations and the control constraints. The dynamical programming equations for maximum reliability problem and for maximum mean first-passage time problem are finalized and their relationships to the backward Kolmogorov equation for the reliability function and the Pontryagin equation for mean first-passage time, respectively, are pointed out. The boundary condition at zero Hamiltonian is discussed. Two examples are worked out to illustrate the application and effectiveness of the proposed procedure.

A First-Passage Time Problem in Reliability Theory

A common replacement policy for technical systems consists in replacing a system by a new one after its economic lifetime, i.e. at that moment, when its long-run maintenance cost rate is minimal. The strict application of the economic lifetime does not take into account individual deviations of maintenance cost rates of single systems from the average cost development. To avoid this disadvantage, paper [Beichelt, Economic Quality Control 12: 173-181, 1997] proposes the total repair cost limit replacement policy: The system is replaced as soon as its total repair cost reaches or exceeds a given level. In that paper, the repair cost development is modeled by functions of the Wiener process. Here the same policy is considered assuming a repair cost process with nondecreasing samples paths and given one-dimensional probability distribution. Examples show that applying the total repair cost limit replacement policy instead of the economic lifetime may lead to cost savings between 4% and 30%. Finally, it is illustrated how to include the reliability aspect into the policy.

Universal equivalence of mean first-passage time and Kramers rate.

We prove that for an arbitrary time-homogeneous stochastic process, Kramers's flux-over-population rate is identical to the inverse of the associated mean first-passage time. In this way the mean first-passage time problem can be treated without making use of the adjoint equation in conjunction with cumbersome boundary conditions.

Chord-distribution functions of three-dimensional random media: approximate first-passage times of Gaussian processes.

The main result of this paper is a semianalytic approximation for the chord-distribution functions of three-dimensional models of microstructure derived from Gaussian random fields. In the simplest case the chord functions are equivalent to a standard first-passage time problem, i.e., the probability density governing the time taken by a Gaussian random process to first exceed a threshold. We obtain an approximation based on the assumption that successive chords are independent. The result is a generalization of the independent interval approximation recently used to determine the exponent of persistence time decay in coarsening. The approximation is easily extended to more general models based on the intersection and union sets of models generated from the isosurfaces of random fields. The chord-distribution functions play an important role in the characterization of random composite and porous materials. Our results are compared with experimental data obtained from a three-dimensional image of a porous Fontainebleau sandstone and a two-dimensional image of a tungsten-silver composite alloy.

A recursive integration method for approximate solution of stochastic differential equations

A deterministic approach for solving stochastic differential equations is described and numerically tested. In this approach, probability distributions of the sample paths at successive time steps in a numerical procedure are shown to satisfy a recursive integral equation. These probability distributions are approximated by solving the integral equation numerically. The advantage of this approach is the elimination of the need for computing sample paths of the solutions. This approach is useful, for example, in approximating the solution of first-passage time problems.

Mean First-Passage Time for Non-Markovian Processes Driven by Continuous Noise11The project supported by National Natural Science Foundation of China

In this article, we discuss the mean first-passage time for non-Markovian processes driven by continuous noise. Applying the iterative method to the Volterra integral equation, we obtain for the first time the series solution to the mean first-passage time problem and present the exact expression for the mean first-passage time in both the cases of rectangular distribution and long-tail distribution.

First-passage study and stationary response analysis of a BWB hysteresis model using quasi-conservative stochastic averaging method

The quasi-conservative stochastic averaging (QCSA) method is applied to a Bouc-Wen-Baber hysteretic system (BWB) under Gaussian white noise excitations. The stationary probability density of the system's response amplitude and energy is obtained for different excitation levels and different damping ratios. These results are compared with the studies presented by Cai and Lin in A New Solution Technique for Randomly Excited Hysteretic Structures, Technical Report on Grant No. NCEER-88-0012, State University of New York, Buffalo, NY, 1988. The first-passage time problem for the hysteretic system is also studied using the method of QCSA. The results for the expected time to failure as a function of the initial total energy are shown, for different excitation levels and for different threshold values of energy. The relationship between the expected time to failure when initial total energy is zero and excitation level is shown for a wide range of hysteresis shape parameters.

First-Passage Time With Random Excitation of Time-Varying Mean

Approximate procedures for solving the first-passage time problem for a linear oscillator subjected to random excitation with a time-varying mean are presented. Numerical results are given for the example of the oscillator subjected to a model representing sonic boom excitation. These results are compared with results obtained from digital simulation. The methods are applicable for lightly damped (narrow-band) oscillators. To the author's knowledge, there is no previous work on oscillator first-passage time approximations where the excitation is assumed to have a time-varying mean.

Moment approximations for first-passage time problems

Moment approximations for first-passage time problems

First-passage solutions in fatigue crack propagation

The statistical characteristics of the time required by the crack size to reach a specified length are sought. This time is treated as the random variable time-to-failure and the analysis is cast into a first-passage time problem. The fatigue crack propagation growth equation is randomized by employing the pulse train stochastic process model. The resulting equation is stochastically averaged so that the crack size can be approximately modelled as Markov process. Choosing the appropriate transition density function for this process and setting the proper initial and boundary conditions it becomes possible to solve the associated forward Kolmogorov equation expressing the solution in the form of an infinite series. Next, the survival probability of a component, the cumulative distribution function and the probability density function of the first-passage time are determined in a series form as well. Corresponding expressions are also derived for its mean and mean square. Verification of the theoretical results is attempted through comparisons with actual experimental data and numerical simulation studies.

Numerical simulations of stochastic differential equations

A simple, very accurate algorithm for numerical simulation of stochastic differential equations is described. Its relationship to colored noise is elucidated and exhibited by explicit results. The especially delicate problem of mean first passage times is highlighted and highly accurate agreement between the numerical simulations and analytic results are shown.

Representations for multivariate reciprocal Gaussian processes

Multivariate reciprocal Gaussian processes are represented as a sum of two independent processes: a piecewise Markov process, which is also represented in terms of a Wiener-type process, and a time-dependent linear transformation of a normally distributed random vector. This result is then applied to the first-passage time problem. >

Bistability driven by weakly colored Gaussian noise: The Fokker-Planck boundary layer and mean first-passage times.

We develop a singular perturbation approach to the problem of first-passage times for non-Markovian processes driven by Gaussian colored noise near the white-noise limit. In particular we treat the problem of overdamped tunneling in a bistable quartic potential in the presence of Gaussian noise of short correlation time \ensuremath{\tau}. The correct treatment of the absorbing boundary yields a lowest-order correction proportional to ${\mathrm{\ensuremath{\tau}}}^{1/2}$ with a proportionality constant involving the Milne extrapolation length for the Fokker-Planck equation, given in terms of the Riemann \ensuremath{\zeta} function.