Searching for a one-dimensional random walker

Let {xk } k ≧ − r be a simple Bernoulli random walk with x –r = 0. An integer valued threshold ϕ = {ϕ k } k≧1 is called a search plan if |ϕ k+1−ϕ k |≦1 for all k ≧ 1. If ϕ is a search plan let τϕ be the smallest integer k such that x and ϕ cross or touch at k. We show the existence of a search plan ϕ such that ϕ 1 = 0, the definition of ϕ does not depend on r, and the first crossing time τϕ has finite mean (and in fact finite moments of all orders). The analogous problem for the Wiener process is also solved.

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On the two-boundary first-crossing-time problem for diffusion processes

Journal of Applied Probability ◽

10.1017/s0021900200038456 ◽

1990 ◽

Vol 27 (01) ◽

pp. 102-114 ◽

Cited By ~ 7

Author(s):

A. Buonocore ◽

V. Giorno ◽

A. G. Nobile ◽

L. M. Ricciardi

Keyword(s):

Integral Equations ◽

Diffusion Processes ◽

Continuous Functions ◽

System Of Integral Equations ◽

One Dimensional ◽

Dimensional Diffusion ◽

Time Problem ◽

Crossing Time ◽

Single Integral ◽

First Crossing Time

The first-crossing-time problem through two time-dependent boundaries for one-dimensional diffusion processes is considered. The first-crossing p.d.f.'s from below and from above are proved to satisfy a new system of Volterra integral equations of the second kind involving two arbitrary continuous functions. By a suitable choice of such functions a system of continuous-kernel integral equations is obtained and an efficient algorithm for its solution is provided. Finally, conditions on the drift and infinitesimal variance of the diffusion process are given such that the system of integral equations reduces to a non-singular single integral equation for the first-crossing-time p.d.f.

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S-shooting: a Bennett–Chandler-like method for the computation of rate constants from committor trajectories

Faraday Discussions ◽

10.1039/c6fd00124f ◽

2016 ◽

Vol 195 ◽

pp. 345-364 ◽

Cited By ~ 4

Author(s):

Georg Menzl ◽

Andreas Singraber ◽

Christoph Dellago

Keyword(s):

Rate Constants ◽

Simulation Method ◽

Free Energy Calculations ◽

Stable States ◽

One Dimensional ◽

Random Walker ◽

Reaction Rate Constants ◽

Definition Of ◽

Double Well Potential ◽

Condensed Systems

Mechanisms of rare transitions between long-lived stable states are often analyzed in terms of commitment probabilities, determined from swarms of short molecular dynamics trajectories. Here, we present a computer simulation method to determine rate constants from such short trajectories combined with free energy calculations. The method, akin to the Bennett–Chandler approach for the calculation of reaction rate constants, requires the definition of a valid reaction coordinate and can be applied to both under- and overdamped dynamics. We verify the correctness of the algorithm using a one-dimensional random walker in a double-well potential and demonstrate its applicability to complex transitions in condensed systems by calculating cavitation rates for water at negative pressures.

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On the two-boundary first-crossing-time problem for diffusion processes

Journal of Applied Probability ◽

10.2307/3214598 ◽

1990 ◽

Vol 27 (1) ◽

pp. 102-114 ◽

Cited By ~ 28

Author(s):

A. Buonocore ◽

V. Giorno ◽

A. G. Nobile ◽

L. M. Ricciardi

Keyword(s):

Integral Equations ◽

Diffusion Processes ◽

Continuous Functions ◽

System Of Integral Equations ◽

One Dimensional ◽

Dimensional Diffusion ◽

Time Problem ◽

Crossing Time ◽

Single Integral ◽

First Crossing Time

The first-crossing-time problem through two time-dependent boundaries for one-dimensional diffusion processes is considered. The first-crossing p.d.f.'s from below and from above are proved to satisfy a new system of Volterra integral equations of the second kind involving two arbitrary continuous functions. By a suitable choice of such functions a system of continuous-kernel integral equations is obtained and an efficient algorithm for its solution is provided. Finally, conditions on the drift and infinitesimal variance of the diffusion process are given such that the system of integral equations reduces to a non-singular single integral equation for the first-crossing-time p.d.f.

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On a symmetry-based constructive approach to probability densities for two-dimensional diffusion processes

Journal of Applied Probability ◽

10.2307/3215291 ◽

1995 ◽

Vol 32 (2) ◽

pp. 316-336 ◽

Cited By ~ 4

Author(s):

A. G. Di Crescenzo ◽

V. Giorno ◽

A. G. Nobile ◽

L. M. Ricciardi

Keyword(s):

Diffusion Processes ◽

Linear Process ◽

Two Dimensional ◽

Constructive Approach ◽

One Dimensional ◽

Absorbing Boundaries ◽

Dimensional Diffusion ◽

Crossing Time ◽

Probability Densities ◽

First Crossing Time

The method earlier introduced for one-dimensional diffusion processes [6] is extended to obtain closed form expressions for the transition p.d.f.'s of two-dimensional diffusion processes in the presence of absorbing boundaries and for the first-crossing time p.d.f.'s through such boundaries. Use of such a method is finally made to analyse a two-dimensional linear process.

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An approximate formula for the first-crossing-time density of a Wiener process perturbed by random jumps

Statistics & Probability Letters ◽

10.1016/j.spl.2009.03.019 ◽

2009 ◽

Vol 79 (13) ◽

pp. 1559-1567 ◽

Cited By ~ 2

Author(s):

Maria Teresa Giraudo

Keyword(s):

Wiener Process ◽

Approximate Formula ◽

Crossing Time ◽

Random Jumps ◽

First Crossing Time ◽

Time Density

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On some first-crossing-time probabilities for a two-dimensional random walk with correlated components

Advances in Applied Probability ◽

10.2307/1427699 ◽

1992 ◽

Vol 24 (2) ◽

pp. 441-454 ◽

Cited By ~ 1

Author(s):

A. Di Crescenzo ◽

V. Giorno ◽

A. G. Nobile

Keyword(s):

Random Walk ◽

Algebraic Equation ◽

Two Dimensional ◽

Fourth Degree ◽

Unit Slope ◽

Crossing Time ◽

The Mean ◽

First Crossing Time ◽

Straight Lines ◽

Correlated Components

For a two-dimensional random walk {X (n) = (X(n)1, X(n)2)T, n ∈ ℕ0} with correlated components the first-crossing-time probability problem through unit-slope straight lines x2 = x1 - r(r = 0,1) is analysed. The p.g.f.'s for the first-crossing-time probabilities are expressed as solutions of a fourth-degree algebraic equation and are then exploited to obtain the first-crossing-time probabilities. Several additional results, including the mean first-crossing time and the probability of ultimate crossing, are also given.

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Local limit theorem for the first crossing time of a fixed level by a random walk

Siberian Advances in Mathematics ◽

10.3103/s1055134410030041 ◽

2010 ◽

Vol 20 (3) ◽

pp. 191-200

Author(s):

A. A. Mogul’skiĭ

Keyword(s):

Random Walk ◽

Limit Theorem ◽

Local Limit Theorem ◽

Local Limit ◽

Fixed Level ◽

Crossing Time ◽

First Crossing Time

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On an exit problem in a one-dimensional random walk in the diffusion approximation

Canadian Journal of Physics ◽

10.1139/p91-192 ◽

1991 ◽

Vol 69 (10) ◽

pp. 1284-1285

Author(s):

Amal K. Das

Keyword(s):

Diffusion Coefficient ◽

Random Walk ◽

Diffusion Approximation ◽

Brownian Particle ◽

Linear Chain ◽

One Dimensional ◽

Random Walker ◽

Exit Problem ◽

Traversal Time

We address a nonstandard random-walk problem in which the random walker, a Brownian particle, enters a linear chain at x = 0 and exits at x = L under the constraint that allows the particle to return to an arbitrarily close neighbourhood of the entrance point, but does not allow the entrance point to be touched back. In the diffusion approximation, the traversal time T*, calculated in an unconventional way, is found to be T/3, where T is the usual diffusion traversal time, L2/2D, D being the diffusion coefficient.

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On some first-crossing-time probabilities for a two-dimensional random walk with correlated components

Advances in Applied Probability ◽

10.1017/s0001867800047595 ◽

1992 ◽

Vol 24 (02) ◽

pp. 441-454

Author(s):

A. Di Crescenzo ◽

V. Giorno ◽

A. G. Nobile

Keyword(s):

Random Walk ◽

Algebraic Equation ◽

Two Dimensional ◽

Fourth Degree ◽

Unit Slope ◽

Crossing Time ◽

The Mean ◽

First Crossing Time ◽

Straight Lines ◽

Correlated Components

For a two-dimensional random walk {X (n) = (X(n) 1, X(n) 2 )T, n ∈ ℕ0} with correlated components the first-crossing-time probability problem through unit-slope straight lines x 2 = x 1 - r(r = 0,1) is analysed. The p.g.f.'s for the first-crossing-time probabilities are expressed as solutions of a fourth-degree algebraic equation and are then exploited to obtain the first-crossing-time probabilities. Several additional results, including the mean first-crossing time and the probability of ultimate crossing, are also given.

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