Solution and Analysis of a One-Dimensional First-Passage Problem with a Nonzero Halting Probability

The first-passage problem for the one-dimensional Wiener process with drift in the presence of elastic boundaries is considered. We use the Kolmogorov backward equation with corresponding boundary conditions to derive explicit closed-form expressions for the expected value and the variance of the first-passage time. Special cases with pure absorbing and/or reflecting barriers arise for a certain choice of a parameter constellation.

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MULTIFRACTAL MEASURES ON SMALL-WORLD NETWORKS

Fractals ◽

10.1142/s0218348x0600312x ◽

2006 ◽

Vol 14 (02) ◽

pp. 119-123 ◽

Cited By ~ 1

Author(s):

K. H. CHANG ◽

B. C. CHOI ◽

SEONG-MIN YOON ◽

KYUNGSIK KIM

Keyword(s):

First Passage Time ◽

Small World ◽

Passage Time ◽

Small World Network ◽

Small World Networks ◽

First Passage ◽

One Dimensional ◽

Random Walker ◽

Regular Networks ◽

First Time

We investigate the multifractals of the first passage time on a one-dimensional small-world network with reflecting and absorbing barriers. The multifractals can be obtained from the distribution of the first passage time at which the random walker arrives for the first time at an absorbing barrier after starting from an arbitrary initial site. Our simulation is found to estimate the fractal dimension D0 = 0.920 ~ 0.930 for the different network sizes and random rewiring fractions. In particular, the multifractal structure breaks down into a small-world network, when the rewiring fraction p is larger than the critical value pc = 0.3. Our simulation results are compared with the numerical computations for regular networks.

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First Passage Problems for Asymmetric Wiener Processes

Journal of Applied Probability ◽

10.1239/jap/1143936251 ◽

2006 ◽

Vol 43 (1) ◽

pp. 175-184 ◽

Cited By ~ 7

Author(s):

Mario Lefebvre

Keyword(s):

Wiener Process ◽

Generating Function ◽

First Passage Time ◽

Moment Generating Function ◽

Passage Time ◽

Expected Value ◽

First Passage ◽

One Dimensional ◽

Wiener Processes ◽

The Moment

The problem of computing the moment generating function of the first passage time T to a > 0 or −b < 0 for a one-dimensional Wiener process {X(t), t ≥ 0} is generalized by assuming that the infinitesimal parameters of the process may depend on the sign of X(t). The probability that the process is absorbed at a is also computed explicitly, as is the expected value of T.

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On the Time Transition Between Short- and Long-Time Regimes of Colloidal Particles in External Periodic Potentials

Frontiers in Physics ◽

10.3389/fphy.2021.635269 ◽

2021 ◽

Vol 9 ◽

Author(s):

Daniela Pérez-Guerrero ◽

José Luis Arauz-Lara ◽

Erick Sarmiento-Gómez ◽

Guillermo Iván Guerrero-García

Keyword(s):

Colloidal Particles ◽

First Passage Time ◽

Critical Time ◽

Passage Time ◽

First Passage ◽

Escape Probability ◽

One Dimensional ◽

Random Walker ◽

Dimensional Array ◽

Long Time

The dynamics of colloidal particles at infinite dilution, under the influence of periodic external potentials, is studied here via experiments and numerical simulations for two representative potentials. From the experimental side, we analyzed the motion of a colloidal tracer in a one-dimensional array of fringes produced by the interference of two coherent laser beams, providing in this way an harmonic potential. The numerical analysis has been performed via Brownian dynamics (BD) simulations. The BD simulations correctly reproduced the experimental position- and time-dependent density of probability of the colloidal tracer in the short-times regime. The long-time diffusion coefficient has been obtained from the corresponding numerical mean square displacement (MSD). Similarly, a simulation of a random walker in a one dimensional array of adjacent cages with a probability of escaping from one cage to the next cage is one of the most simple models of a periodic potential, displaying two diffusive regimes separated by a dynamical caging period. The main result of this study is the observation that, in both potentials, it is seen that the critical time t*, defined as the specific time at which a change of curvature in the MSD is observed, remains approximately constant as a function of the height barrier U0 of the harmonic potential or the associated escape probability of the random walker. In order to understand this behavior, histograms of the first passage time of the tracer have been calculated for several height barriers U0 or escape probabilities. These histograms display a maximum at the most likely first passage time t′, which is approximately independent of the height barrier U0, or the associated escape probability, and it is located very close to the critical time t*. This behavior suggests that the critical time t*, defining the crossover between short- and long-time regimes, can be identified as the most likely first passage time t′ as a first approximation.

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MULTIFRACTALS ON SMALL-WORLD NETWORKS

International Journal of Modern Physics B ◽

10.1142/s0217979207045207 ◽

2007 ◽

Vol 21 (23n24) ◽

pp. 4059-4063

Author(s):

KYUNGSIK KIM ◽

K. H. CHANG ◽

DEOCK-HO HA

Keyword(s):

First Passage Time ◽

Small World ◽

Passage Time ◽

Small World Network ◽

Small World Networks ◽

First Passage ◽

One Dimensional ◽

Random Walker ◽

Regular Networks ◽

First Time

We investigate the multifractals of the first passage time on a one-dimensional small-world network with reflecting and absorbing barriers. We analyze numerically the distribution of the first passage time at which the random walker arrives for the first time at an absorbing barrier after starting from an arbitrary initial site. Our simulation is found to estimate the fractal dimension D0 = 0.920 ∼ 0.930 for the different network sizes and random rewiring fractions. In particular, our simulation results are compared with the numerical computations for regular networks.

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First Passage Problems for Asymmetric Wiener Processes

Journal of Applied Probability ◽

10.1017/s0021900200001443 ◽

2006 ◽

Vol 43 (01) ◽

pp. 175-184

Author(s):

Mario Lefebvre

Keyword(s):

Wiener Process ◽

Generating Function ◽

First Passage Time ◽

Moment Generating Function ◽

Passage Time ◽

Expected Value ◽

First Passage ◽

One Dimensional ◽

Wiener Processes ◽

The Moment

The problem of computing the moment generating function of the first passage time T to a > 0 or −b < 0 for a one-dimensional Wiener process {X(t), t ≥ 0} is generalized by assuming that the infinitesimal parameters of the process may depend on the sign of X(t). The probability that the process is absorbed at a is also computed explicitly, as is the expected value of T.

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Erratum to: “An inverse first-passage problem for one-dimensional diffusions with random starting point” [Statist. Probab. Lett. 82 (2012) 7–14]

Statistics & Probability Letters ◽

10.1016/j.spl.2011.11.020 ◽

2012 ◽

Vol 82 (3) ◽

pp. 705 ◽

Cited By ~ 3

Author(s):

Mario Abundo

Keyword(s):

First Passage ◽

One Dimensional ◽

Starting Point ◽

Random Starting Point ◽

First Passage Problem ◽

Inverse First Passage Problem

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Some remarks on the Rayleigh process

Journal of Applied Probability ◽

10.2307/3214182 ◽

1986 ◽

Vol 23 (2) ◽

pp. 398-408 ◽

Cited By ~ 34

Author(s):

V. Giorno ◽

A. G. Nobile ◽

L. M. Ricciardi ◽

L. Sacerdote

Keyword(s):

First Passage Time ◽

Moment Generating Function ◽

Passage Time ◽

First Passage ◽

One Dimensional ◽

Time Problem ◽

The Moment ◽

First Passage Time Problem ◽

Rayleigh Process ◽

Absorption Condition

The transition p.d.f. for a one-dimensional Rayleigh process in the presence of an absorption condition or a zero-flux condition in the origin is obtained in closed form. The first-passage-time problem through an arbitrary constant boundary is then considered and the moment-generating function is determined. In some particular cases the first-passage-time p.d.f. is explicitly derived. Use of some of these results is finally made to obtain the transition p.d.f. of the affine drift-linear infinitesimal-variance diffusion process when the origin is an entrance or a regular boundary in the presence of a reflection condition.

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Moments of the first-passage time of a Wiener process with drift between two elastic barriers

Journal of Applied Probability ◽

10.1017/s0021900200103481 ◽

1995 ◽

Vol 32 (04) ◽

pp. 1007-1013 ◽

Cited By ~ 3

Author(s):

Marco Dominé

Keyword(s):

Wiener Process ◽

First Passage Time ◽

Passage Time ◽

First Passage ◽

One Dimensional ◽

Special Cases ◽

Backward Equation ◽

The One ◽

First Passage Problem ◽

Reflecting Barriers

The first-passage problem for the one-dimensional Wiener process with drift in the presence of elastic boundaries is considered. We use the Kolmogorov backward equation with corresponding boundary conditions to derive explicit closed-form expressions for the expected value and the variance of the first-passage time. Special cases with pure absorbing and/or reflecting barriers arise for a certain choice of a parameter constellation.

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Some remarks on the Rayleigh process

Journal of Applied Probability ◽

10.1017/s0021900200029697 ◽

1986 ◽

Vol 23 (02) ◽

pp. 398-408 ◽

Cited By ~ 6

Author(s):

V. Giorno ◽

A. G. Nobile ◽

L. M. Ricciardi ◽

L. Sacerdote

Keyword(s):

First Passage Time ◽

Moment Generating Function ◽

Passage Time ◽

First Passage ◽

One Dimensional ◽

Time Problem ◽

The Moment ◽

First Passage Time Problem ◽

Rayleigh Process ◽

Absorption Condition

The transition p.d.f. for a one-dimensional Rayleigh process in the presence of an absorption condition or a zero-flux condition in the origin is obtained in closed form. The first-passage-time problem through an arbitrary constant boundary is then considered and the moment-generating function is determined. In some particular cases the first-passage-time p.d.f. is explicitly derived. Use of some of these results is finally made to obtain the transition p.d.f. of the affine drift-linear infinitesimal-variance diffusion process when the origin is an entrance or a regular boundary in the presence of a reflection condition.

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