Reinforced Brownian Motion

A Prototype

Jerome Percus, Ora E. Percus

Research output: Contribution to journalArticle

Abstract

We analyze the Brownian Motion limit of a prototypical unit step reinforced random-walk on the half-line. A reinfoced random walk is one which changes the weight of any edge (or vertex) visited to increase the frequency of return visits. The generating function for the discrete case is first derived for the joint probability distribution of SN (the location of the walker at the Nt step) and AN, the maximum location the walker achieved in N steps. Then the bulk of the analysis concerns the statistics of the limiting Brownian walker, and of its "environment", both parametrized by the amplitude δ of the reinforcement. The walker marginal distribution can be interpreted as that of free diffusion with a source serving as a diffusing soft confinement, details depending very much on the value of -1<δ < ∞.

Original languageEnglish (US)
Pages (from-to)917-931
Number of pages15
JournalJournal of Statistical Physics
Volume156
Issue number5
DOIs
StatePublished - 2014

Fingerprint

Brownian motion
prototypes
Reinforced Random Walk
Prototype
random walk
Reinforcement
Marginal Distribution
Joint Distribution
Generating Function
Half line
Random walk
Probability Distribution
Limiting
reinforcement
Statistics
Unit
apexes
Vertex of a graph
statistics

Keywords

  • Brownian Motion limit
  • Reinforced random walk
  • Walk on half-line

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Reinforced Brownian Motion : A Prototype. / Percus, Jerome; Percus, Ora E.

In: Journal of Statistical Physics, Vol. 156, No. 5, 2014, p. 917-931.

Research output: Contribution to journalArticle

Percus, Jerome ; Percus, Ora E. / Reinforced Brownian Motion : A Prototype. In: Journal of Statistical Physics. 2014 ; Vol. 156, No. 5. pp. 917-931.
@article{162a8fc011ab49e9965cfa1e0ded838e,
title = "Reinforced Brownian Motion: A Prototype",
abstract = "We analyze the Brownian Motion limit of a prototypical unit step reinforced random-walk on the half-line. A reinfoced random walk is one which changes the weight of any edge (or vertex) visited to increase the frequency of return visits. The generating function for the discrete case is first derived for the joint probability distribution of SN (the location of the walker at the Nt step) and AN, the maximum location the walker achieved in N steps. Then the bulk of the analysis concerns the statistics of the limiting Brownian walker, and of its {"}environment{"}, both parametrized by the amplitude δ of the reinforcement. The walker marginal distribution can be interpreted as that of free diffusion with a source serving as a diffusing soft confinement, details depending very much on the value of -1<δ < ∞.",
keywords = "Brownian Motion limit, Reinforced random walk, Walk on half-line",
author = "Jerome Percus and Percus, {Ora E.}",
year = "2014",
doi = "10.1007/s10955-014-1036-5",
language = "English (US)",
volume = "156",
pages = "917--931",
journal = "Journal of Statistical Physics",
issn = "0022-4715",
publisher = "Springer New York",
number = "5",

}

TY - JOUR

T1 - Reinforced Brownian Motion

T2 - A Prototype

AU - Percus, Jerome

AU - Percus, Ora E.

PY - 2014

Y1 - 2014

N2 - We analyze the Brownian Motion limit of a prototypical unit step reinforced random-walk on the half-line. A reinfoced random walk is one which changes the weight of any edge (or vertex) visited to increase the frequency of return visits. The generating function for the discrete case is first derived for the joint probability distribution of SN (the location of the walker at the Nt step) and AN, the maximum location the walker achieved in N steps. Then the bulk of the analysis concerns the statistics of the limiting Brownian walker, and of its "environment", both parametrized by the amplitude δ of the reinforcement. The walker marginal distribution can be interpreted as that of free diffusion with a source serving as a diffusing soft confinement, details depending very much on the value of -1<δ < ∞.

AB - We analyze the Brownian Motion limit of a prototypical unit step reinforced random-walk on the half-line. A reinfoced random walk is one which changes the weight of any edge (or vertex) visited to increase the frequency of return visits. The generating function for the discrete case is first derived for the joint probability distribution of SN (the location of the walker at the Nt step) and AN, the maximum location the walker achieved in N steps. Then the bulk of the analysis concerns the statistics of the limiting Brownian walker, and of its "environment", both parametrized by the amplitude δ of the reinforcement. The walker marginal distribution can be interpreted as that of free diffusion with a source serving as a diffusing soft confinement, details depending very much on the value of -1<δ < ∞.

KW - Brownian Motion limit

KW - Reinforced random walk

KW - Walk on half-line

UR - http://www.scopus.com/inward/record.url?scp=84904420510&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84904420510&partnerID=8YFLogxK

U2 - 10.1007/s10955-014-1036-5

DO - 10.1007/s10955-014-1036-5

M3 - Article

VL - 156

SP - 917

EP - 931

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 5

ER -