scholarly journals Fast Solutions for the First-Passage Distribution of Diffusion Models with Space-Time-Dependent Drift Functions and Time-Dependent Boundaries

2021 ◽  
Author(s):  
Udo Boehm ◽  
Sonja Cox ◽  
Gregor Gantner ◽  
Rob Stevenson
Keyword(s):  
First Passage Time ◽  
Numerical Algorithms ◽  
Planck Equation ◽  
Passage Time ◽  
Space Time ◽  
Diffusion Models ◽  
Time Dependent ◽  
First Passage ◽  
Theoretical Evaluation ◽  
Wide Range

Diffusion models with constant boundaries and constant drift function have been successfully applied to model phenomena in a wide range of areas in psychology. In recent years, more complex models with time-dependent boundaries and space time-dependent drift functions have gained popularity. One obstacle to the empirical and theoretical evaluation of these models is the lack of simple and efficient numerical algorithms for computing their first-passage time distributions. In the present work we use a known series expansion for the first-passage time distribution for models with constant drift function and constant boundaries to simplify the Fokker-Planck equation for models with time dependent boundaries and space-time-dependent drift functions. We show how a simple Crank--Nicolson scheme can be used to efficiently solve the simplified equation.

Hitting-Time Densities of a Two-Dimensional Markov Process

1992 ◽  
Vol 6 (4) ◽  
pp. 561-580
Author(s):  
C. H. Hesse
Keyword(s):  
Global Analysis ◽  
First Passage Time ◽  
Type Equation ◽  
Passage Time ◽  
Two Dimensional ◽  
First Passage ◽  
Wide Range ◽  
The Laplace Transform ◽  
Mean And Variance

This paper deals with the two-dimensional stochastic process (X(t), V(t)) where dX(t) = V(t)dt, V(t) = W(t) + ν for some constant ν and W(t) is a one-dimensional Wiener process with zero mean and variance parameter σ2= 1. We are interested in the first-passage time of (X(t), V(t)) to the plane X = 0 for a process starting from (X(0) = −x, V(0) = ν) with x > 0. The partial differential equation for the Laplace transform of the first-passage time density is transformed into a Schrödinger-type equation and, using methods of global analysis, such as the method of dominant balance, an approximation to the first-passage density is obtained. In a series of simulations, the quality of this approximation is checked. Over a wide range of x and ν it is found to perform well, globally in t. Some applications are mentioned.

First-Passage Time for Oscillators With Nonlinear Damping

1978 ◽  
Vol 45 (1) ◽  
pp. 175-180 ◽  
Author(s):  
J. B. Roberts
Keyword(s):  
First Passage Time ◽  
Digital Simulation ◽  
Planck Equation ◽  
Passage Time ◽  
Nonlinear Damping ◽  
First Passage ◽  
One Dimensional ◽  
Fokker Planck Equation ◽  
First Passage Failure ◽  
Fokker Planck

A simple numerical scheme is proposed for computing the probability of first passage failure, P(T), in an interval O-T, for oscillators with nonlinear damping. The method depends on the fact that, when the damping is light, the amplitude envelope, A(t), can be accurately approximated as a one-dimensional Markov process. Hence, estimates of P(T) are found, for both single and double-sided barriers, by solving the Fokker-Planck equation for A(t) with an appropriate absorbing barrier. The numerical solution of the Fokker-Planck equation is greatly simplified by using a discrete time random walk analog of A(t), with appropriate statistical properties. Results obtained by this method are compared with corresponding digital simulation estimates, in typical cases.

scholarly journals Thermal dynamic phase transition of Reissner-Nordström Anti-de Sitter black holes on free energy landscape

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Ran Li ◽  
Kun Zhang ◽  
Jin Wang
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Free Energy ◽  
First Passage Time ◽  
Energy Landscape ◽  
Planck Equation ◽  
Passage Time ◽  
Free Energy Landscape ◽  
De Sitter ◽  
First Passage

Abstract We explore the thermodynamics and the underlying kinetics of the van der Waals type phase transition of Reissner-Nordström anti-de Sitter (RNAdS) black holes based on the free energy landscape. We show that the thermodynamic stabilities of the three branches of the RNAdS black holes are determined by the underlying free energy landscape topography. We suggest that the large (small) RNAdS black hole can have the probability to switch to the small (large) black hole due to the thermal fluctuation. Such a state switching process under the thermal fluctuation is taken as a stochastic process and the associated kinetics can be described by the probabilistic Fokker-Planck equation. We obtained the time dependent solutions for the probabilistic evolution by numerically solving Fokker-Planck equation with the reflecting boundary conditions. We also investigated the first passage process which describes how fast a system undergoes a stochastic process for the first time. The distributions of the first passage time switching from small (large) to large (small) black hole and the corresponding mean first passage time as well as its fluctuations at different temperatures are studied in detail. We conclude that the mean first passage time and its fluctuations are related to the free energy landscape topography through barrier heights and temperatures.

First passage time distribution of a Wiener process with drift concerning two elastic barriers

1996 ◽  
Vol 33 (01) ◽  
pp. 164-175 ◽  
Author(s):  
Marco Dominé
Keyword(s):  
Explicit Expression ◽  
Wiener Process ◽  
Time Distribution ◽  
First Passage Time ◽  
Planck Equation ◽  
Passage Time ◽  
Special Choice ◽  
First Passage ◽  
Special Cases ◽  
Reflecting Barriers

We solve the Fokker-Planck equation for the Wiener process with drift in the presence of elastic boundaries and a fixed start point. An explicit expression is obtained for the first passage density. The cases with pure absorbing and/or reflecting barriers arise for a special choice of a parameter constellation. These special cases are compared with results in Darling and Siegert [5] and Sweet and Hardin [15].

On the evaluation of first-passage-time probability densities via non-singular integral equations

1989 ◽  
Vol 21 (1) ◽  
pp. 20-36 ◽  
Author(s):  
V. Giorno ◽  
A. G. Nobile ◽  
L. M. Ricciardi ◽  
S. Sato
Keyword(s):  
Diffusion Processes ◽  
First Passage Time ◽  
Singular Integral Equations ◽  
Passage Time ◽  
Time Dependent ◽  
Computational Results ◽  
First Passage ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Probability Densities

The algorithm given by Buonocore et al. [1] to evaluate first-passage-time p.d.f.’s for Wiener and Ornstein–Uhlenbeck processes through a time-dependent boundary is extended to a wide class of time-homogeneous one-dimensional diffusion processes. Several examples are thoroughly discussed along with some computational results.

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