Local times on curves and uniform invariance principles

Abstract

Sufficient conditions are given for a family of local times |L t µ | ofd-dimensional Brownian motion to be jointly continuous as a function oft and μ. Then invariance principles are given for the weak convergence of local times of lattice valued random walks to the local times of Brownian motion, uniformly over a large family of measures. Applications included some new results for intersection local times for Brownian motions on ℝ2 and ℝ2.