Intersection Local Times for Infinite Systems of Brownian Motions and for the Brownian Density Process

Abstract

The Brownian density process is a distribution-valued process that arises either via a limiting operation on an infinite collection of Brownian motions or as the solution of a stochastic partial differential equation. It has a (self-) intersection local time, that is formally defined through an operation involving delta functions, much akin to the better studied intersection local time of measure-valued ("super") processes. Our main aim is to show that this formal definition not only makes sense mathematically, but can also be understood, at least in two and three dimensions, via the intersection local times of simple Brownian motions. To show how useful this way of looking at the Brownian density intersection local time can be, we also derive a Tanaka-like evolution equation for it in the two-dimensional case.