A two-dimensional random walk in the presence of a partially reflecting barrier

1974 ◽  
Vol 11 (1) ◽  
pp. 199-205 ◽  
Author(s):  
Noel Cressie
Keyword(s):  
Random Walk ◽  
Two Dimensional ◽  
Absorption Probabilities

A general two-dimensional random walk is considered with a barrier along the y-axis. Absorption probabilities are derived when the barrier is absorbing, and when it is semi-reflecting.

A two-dimensional random walk in the presence of a partially reflecting barrier

1974 ◽  
Vol 11 (01) ◽  
pp. 199-205
Author(s):  
Noel Cressie
Keyword(s):  
Random Walk ◽  
Two Dimensional ◽  
Absorption Probabilities

A general two-dimensional random walk is considered with a barrier along the y-axis. Absorption probabilities are derived when the barrier is absorbing, and when it is semi-reflecting.

Wald's identity and absorption probabilities for two-dimensional random walks

1965 ◽  
Vol 61 (3) ◽  
pp. 747-762 ◽  
Author(s):  
V. D. Barnett
Keyword(s):  
Random Walk ◽  
Double Series ◽  
Moment Generating Function ◽  
Square Lattice ◽  
Two Dimensional ◽  
Finite Region ◽  
Discrete Random Variables ◽  
Image Position ◽  
Wald's Identity ◽  
Absorption Probabilities

SummarySuppose a particle executes a random walk on a two-dimensional square lattice, starting at the origin. The position of the particle after n steps of the walk is Xn = (Xl, n, X2n), whereand we will assume that the Yi are independent bivariate discrete random variables with common moment generating function (m.g.f.)where a, b, c and d are non-negative. We assume further that (i) pi, j is non-zero for some finite positive and negative i, and some finite positive and negative j (− a ≤ i ≤ b, − c ≤ jd), such values of i and j including – a, b and – c, d, respectively, whenever a, b, c or d is finite, and (ii) the double series defining Φ(α, β) is convergent at least in some finite region D, of the real (α, β) plane, which includes the origin.

Some explicit results for an asymmetric two-dimensional random walk

1963 ◽  
Vol 59 (2) ◽  
pp. 451-462 ◽  
Author(s):  
V. D. Barnett
Keyword(s):  
Random Walk ◽  
Lattice Point ◽  
General Interest ◽  
Two Dimensional ◽  
General Lattice ◽  
Methods Of Analysis ◽  
Starting Point ◽  
Accessible Point ◽  
Absorption Probabilities

AbstractThree distinct methods are used to obtain exact expressions for various characteristics of a particular asymmetric two-dimensional random walk. The results obtained include, for the transient unrestricted walk, the probability of return to the starting-point and the average number of arrivals at the general lattice point; and, for a walk restricted within a rectangular absorbing barrier, the average number of arrivals at any accessible point and the absorption probabilities on the boundary. Whilst there is some duplication of results by using the three different methods of analysis, this is not extensive and provides a useful check on the results. Also the methods are of some general interest in themselves.

On the absorption probabilities and mean time for absorption for discrete Markov chains

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nikolaos Halidias
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Generating Function ◽  
Discrete Time ◽  
Probability Generating Function ◽  
The Mean ◽  
Mean Time ◽  
Absorption Probabilities

Abstract In this note we study the probability and the mean time for absorption for discrete time Markov chains. In particular, we are interested in estimating the mean time for absorption when absorption is not certain and connect it with some other known results. Computing a suitable probability generating function, we are able to estimate the mean time for absorption when absorption is not certain giving some applications concerning the random walk. Furthermore, we investigate the probability for a Markov chain to reach a set A before reach B generalizing this result for a sequence of sets A 1 , A 2 , … , A k {A_{1},A_{2},\dots,A_{k}} .

A non-Wiener random walk in a two-dimensional Bernoulli environment

1988 ◽  
Vol 50 (3-4) ◽  
pp. 599-609
Author(s):  
A. Kr�mli ◽  
P. Luk�cs ◽  
D. Sz�sz
Keyword(s):  
Random Walk ◽  
Two Dimensional

scholarly journals Quantum walks and elliptic integrals

2010 ◽  
Vol 20 (6) ◽  
pp. 1091-1098 ◽  
Author(s):  
NORIO KONNO
Keyword(s):  
Random Walk ◽  
Generating Function ◽  
Elliptic Integral ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Elliptic Integrals ◽  
Two Dimensional ◽  
One Dimensional ◽  
Similar Expression ◽  
Return Probability

Pólya showed in his 1921 paper that the generating function of the return probability for a two-dimensional random walk can be written in terms of an elliptic integral. In this paper we present a similar expression for a one-dimensional quantum walk.

Sign in / Sign up

Export Citation Format

Share Document