scholarly journals Hybrid asymptotic-numerical approach for estimating first-passage-time densities of the two-dimensional narrow capture problem

2016 ◽  
Vol 94 (4) ◽  
Author(s):  
A. E. Lindsay ◽  
R. T. Spoonmore ◽  
J. C. Tzou
Keyword(s):  
First Passage Time ◽  
Numerical Approach ◽  
Passage Time ◽  
Two Dimensional ◽  
First Passage

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 a particular stationary periodic Gaussian process

1976 ◽  
Vol 13 (01) ◽  
pp. 27-38 ◽  
Author(s):  
L. A. Shepp ◽  
D. Slepian
Keyword(s):  
Gaussian Process ◽  
Infinite Series ◽  
First Passage Time ◽  
Passage Time ◽  
Numerical Calculations ◽  
Time Interval ◽  
Two Dimensional ◽  
First Passage ◽  
Gaussian Density ◽  
First Passage Probability

We find the first-passage probability that X(t) remains above a level a throughout a time interval of length T given X(0) = x 0 for the particular stationary Gaussian process X with mean zero and (sawtooth) covariance P(τ) = 1 – α | τ |, | τ | ≦ 1, with ρ(τ + 2) = ρ(τ), – ∞ < τ < ∞. The desired probability is explicitly found as an infinite series of integrals of a two-dimensional Gaussian density over sectors. Simpler expressions are found for the case a = 0 and also for the unconditioned probability that X(t) be non-negative throughout [0, T]. Results of some numerical calculations are given.

scholarly journals First excess levels of vector processes

1994 ◽  
Vol 7 (3) ◽  
pp. 457-464 ◽  
Author(s):  
Jewgeni H. Dshalalow
Keyword(s):  
Point Process ◽  
Renewal Process ◽  
Stochastic Models ◽  
First Passage Time ◽  
Passage Time ◽  
Two Dimensional ◽  
First Passage ◽  
Compound Process ◽  
First Time

This paper analyzes the behavior of a point process marked by a two-dimensional renewal process with dependent components about some fixed (two-dimensional) level. The compound process evolves until one of its marks hits (i.e. reaches or exceeds) its associated level for the first time. The author targets a joint transformation of the first excess level, first passage time, and the index of the point process which labels the first passage time. The cases when both marks are either discrete or continuous or mixed are treated. For each of them, an explicit and compact formula is derived. Various applications to stochastic models are discussed.

Remarks on renewal theory for percolation processes

1976 ◽  
Vol 13 (02) ◽  
pp. 290-300 ◽  
Author(s):  
R. T. Smythe
Keyword(s):  
Ergodic Theorem ◽  
First Passage Time ◽  
Renewal Theory ◽  
Passage Time ◽  
Integer Lattice ◽  
Two Dimensional ◽  
First Passage Percolation ◽  
First Passage ◽  
Underlying Distribution ◽  
Percolation Processes

We extend some results of Hammersley and Welsh concerning first-passage percolation on the two-dimensional integer lattice. Our results include: (i) weak renewal theorems for the unrestricted reach processes; (ii) an L 1-ergodic theorem for the unrestricted first-passage time from (0, 0) to the line X = n; and (iii) weakening of the boundedness restrictions on the underlying distribution in Hammersley and Welsh's weak renewal theorems for the cylinder reach processes.

First Passage Time Distribution of a Two-Dimensional Wiener Process with Drift

1993 ◽  
Vol 7 (4) ◽  
pp. 545-555 ◽  
Author(s):  
Marco Dominé ◽  
Volkmar Pieper
Keyword(s):  
Wiener Process ◽  
First Passage Time ◽  
Exit Time ◽  
Passage Time ◽  
Absorbing Boundary Conditions ◽  
Two Dimensional ◽  
First Exit Time ◽  
First Passage ◽  
Absorbing Boundary ◽  
Starting Point

The two-dimensional correlated Wiener process (or Brownian motion) with drift is considered. The Fokker-Planck (or Kolmogorov forward) equation for the Wiener process (X1(t), X2(t)) is solved under absorbing boundary conditions on the lines x1= h1 and x2 = h2 and a fixed starting point (x0,1, x0,2). The first passage (or first exit) time when the process leaves the domain G = ( −∞, h1) × ( −∞, h2) is derived.

First-passage time for a particular stationary periodic Gaussian process

1976 ◽  
Vol 13 (1) ◽  
pp. 27-38 ◽  
Author(s):  
L. A. Shepp ◽  
D. Slepian
Keyword(s):  
Gaussian Process ◽  
Infinite Series ◽  
First Passage Time ◽  
Passage Time ◽  
Numerical Calculations ◽  
Time Interval ◽  
Two Dimensional ◽  
First Passage ◽  
Gaussian Density ◽  
First Passage Probability

We find the first-passage probability that X(t) remains above a level a throughout a time interval of length T given X(0) = x0 for the particular stationary Gaussian process X with mean zero and (sawtooth) covariance P(τ) = 1 – α | τ |, | τ | ≦ 1, with ρ(τ + 2) = ρ(τ), – ∞ < τ < ∞. The desired probability is explicitly found as an infinite series of integrals of a two-dimensional Gaussian density over sectors. Simpler expressions are found for the case a = 0 and also for the unconditioned probability that X(t) be non-negative throughout [0, T]. Results of some numerical calculations are given.

Critical behavior of the two-dimensional first passage time

1986 ◽  
Vol 45 (5-6) ◽  
pp. 933-951 ◽  
Author(s):  
J. T. Chayes ◽  
L. Chayes ◽  
R. Durrett
Keyword(s):  
Critical Behavior ◽  
First Passage Time ◽  
Passage Time ◽  
Two Dimensional ◽  
First Passage

Remarks on renewal theory for percolation processes

1976 ◽  
Vol 13 (2) ◽  
pp. 290-300 ◽  
Author(s):  
R. T. Smythe
Keyword(s):  
Ergodic Theorem ◽  
First Passage Time ◽  
Renewal Theory ◽  
Passage Time ◽  
Integer Lattice ◽  
Two Dimensional ◽  
First Passage Percolation ◽  
First Passage ◽  
Underlying Distribution ◽  
Percolation Processes

We extend some results of Hammersley and Welsh concerning first-passage percolation on the two-dimensional integer lattice. Our results include: (i) weak renewal theorems for the unrestricted reach processes; (ii) an L1-ergodic theorem for the unrestricted first-passage time from (0, 0) to the line X = n; and (iii) weakening of the boundedness restrictions on the underlying distribution in Hammersley and Welsh's weak renewal theorems for the cylinder reach processes.

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