Asymptotic results for independent wiener sausages. application to generalized intersection local times

Abstract

The aim of this work is to study the asymptotic behavior of the intersection of two independent localized Wiener sausages in R³, when their radius is decreasing to zero (a Wiener sausage with radius r is defined by letting a ball with radius r run along a Brownian path in R³). If a measure µ on R$sup:3$esup: satisfies some integrability conditions, a renormalization of the µ-measure of this random intersection leads to the definition of intersection local times of two independent Brownian motions in R³ with respect to µ. We show that, like the intersection local times with respect to Lebesgue measure, these processes satisfy a density of occupation formula and admit a stochastic expression, extending the "Tanaka-Rosen" formula. Then we give their Holder continuity with respect to the space variable and we finally show how they appear in the limit of a homogenization problem modeling a chemical reaction with catalysis

Keywords:

• Wiener sausage
• intersection local time
• additive functional
• equilibrium measure
• weak convergence