On the sample paths of diagonal Brownian motions on the infinite dimensional torus
type: Item
instance of: scholarly article
title: On the sample paths of diagonal Brownian motions on the infinite dimensional torus
zbMATH Open document ID: 1056.60029
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 2004-08-04T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77808
review text: Let \(T=R/2\pi Z\) be the circle group and \(T^\infty=\prod_1^\infty T_i\) be the countable product of circles \(T_i\). The group \(T^\infty\) is equipped with the product topology and its normalized Haar measure \(\mu\). Let \(\mathcal C\) be the set of all smooth functions on \(T^\infty\) which depend only on a finite number of coordinates. On the circle \(T\) denote by \(\nu_t\) the standard heat kernel measure associated to the infinitesimal generator \((\frac d{dx})^2\). The convolution semigroup \((\nu_t)_{t>0}\) is associated with a stochastic process \(\xi=(\xi_t)_{t\geq0}\) which is simply Brownian motion runned at twice the usual speed in classic probabilistic notation and wrapped around the circle. For any fixed sequence \({\mathbf a}=(a_1,\ldots)\) of positive numbers let us consider the product measures \(\mu_t=\mu_t^{\mathbf a}=\bigotimes_1^\infty\nu_{a_it}\). The family \((\mu_t)_{t>0}\) forms a convolution semigroup of measures on \(T^\infty\) and \(\mu_t\) is the marginal at time \(t\) of a diffusion process \(X=X^{\mathbf a}=(X_t)_{t\geq0}\) which is simply the product of independent circle Brownian motions \(X^i=(X^i_t)_{t\geq0}\) where \(X^i_t=\xi_{a_it}\). The intrinsic distance is defined by \[ d(x,y)=d^{\mathbf a}(x,y)= \sup\left\{f(x)-f(y):f\in{\mathcal C},\;\sum_1^\infty a_i | \partial_if | ^2\leq1\right\} \text{ and } d(x)=d^{\mathbf a}(x)=d^{\mathbf a}(e,x), \] where \(e=(0,0,\ldots)\) is the neutral element in \(T^\infty\). \(d\) is continuous and defines the topology of \(T^\infty\) if and only if \(\sum_1^\infty \frac 1{a_i}<\infty\). The last condition is assumed to hold throughout the paper under review. Under this condition for any \(t>0\) the measure \(\mu_t=\mu_t^{\mathbf a}\) is absolutely continuous with respect to the Haar measure \(\mu\) and admits a continuous density \(\mu_t^{\mathbf a}(x)\) given by \(\mu_t^{\mathbf a}(x)=\prod_1^\infty \nu_{a_it}(x_i)\). Given a sequence \({\mathbf a}=(a_i)\) of positive numbers define \(N(…
reviewed by: Vakhtang V. Kvaratskhelia
Mathematics Subject Classification ID: 60G17
Mathematics Subject Classification ID: 60B15
Mathematics Subject Classification ID: 60J60
zbMATH DE Number: 2081278
zbMATH Keywords: sample paths
zbMATH Keywords: modulus of continuity
zbMATH Keywords: diagonal Brownian motion
zbMATH Keywords: infinite-dimensional torus
zbMATH Keywords: intrinsic distance
zbMATH Keywords: law of iterated logarithm
author: A. Bendikov
author: Laurent Saloff-Coste
MaRDI profile type: Publication
OpenAlex ID: W2092840309
DOI: 10.1016/J.ANIHPB.2003.02.001
description: scientific article; zbMATH DE number 2081278