Interacting Brownian particles and the Wigner law
type: Item
instance of: scholarly article
title: Interacting Brownian particles and the Wigner law
zbMATH Open document ID: 0794.60100
DOI: 10.1007/BF01196734
author: L. C. G. Rogers
author: Zhan Shi
publication date: 1994-08-15T00:00:00Z
review text: We study interacting diffusing particles governed by the stochastic differential equations \[ dX_ j (t)=\sigma_ n dB_ j (t)-D_ j \varphi_ n (X_ 1,\dots,X_ n) dt, \qquad j=1,2, \dots,n. \] Here the \(B_ j\) are independent Brownian motions in \(\mathbb{R}^ d\), and \[ \varphi_ n (x_ 1,\dots,x_ n)=\alpha_ n {\underset {i\neq j} {\sum \sum}} V(x_ i-x_ j)+\theta_ n \sum_ i U(x_ i). \] The potential \(V\) has a singularity at 0 strong enough to keep the particles apart, and the potential \(U\) serves to keep the particles from escaping to infinity. Our interest is in the behaviour as the number of particles increases without limit, which we study through the empirical measure process. We prove tightness of these processes in the case of \(d=1\), \(V(x)=-\log | x |\), \(U(x)= x^ 2/2\) where it is possible to prove uniqueness of the limiting evolution and deduce that a limiting measure-valued process exists. This process is deterministic, and converges to the Wigner law as \(t \to \infty\). Some information on the rates of convergence is derived, and the case of a Cauchy initial distribution is analysed completely.
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 62E20
zbMATH DE Number: 569078
zbMATH Keywords: interacting diffusing particles
zbMATH Keywords: stochastic differential equations
zbMATH Keywords: tightness
zbMATH Keywords: measure-valued process
zbMATH Keywords: Wigner law
zbMATH Keywords: Cauchy initial distribution
MaRDI profile type: Publication
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 569078