Some measure-valued Markov processes attached to occupation times of Brownian motion

Abstract

We study the positive random measure ${\Pi }_t(\omega ,dy)=l_t^{B_t-y}dy$ , where $(l_t^a;a\in R,t>0)$ denotes the family of local times of the one-dimensional Brownian motion B. We prove that the measure-valued process $({\Pi }_t;t\ge 0)$ is a Markov process. We give two examples of functions $(f_i{)}_{i=1,...,n}$ for which the process $\left({\Pi }_t\right(f_i{)}_{i=1,...,n};t\ge 0)$ is a Markov process.