Some measure-valued Markov processes attached to occupation times of Brownian motion (Q1975191)

Let \(B= \{B_t, t\geq 0\}\) be a one-dimensional Brownian motion starting at \(0\) and \(\{l^y, t\geq 0, y\in R\}\) its family of local times. The authors study the positive random measure \(\Pi_t(.,dy)= l^{B-t- y}_tdy\) which integrates positive functions \(f\) as \(\Pi_t(f)= \int^t_0 f(B_t- B_s) ds\). They prove that the measure-valued process \(\Pi= \{\Pi_t, t\geq 0\}\) is a continuous homogeneous Markov process, and they characterize \(\Pi\) as a solution of some stochastic differential equation, by its semigroup and by its generator. This generalizes earlier results by \textit{L. Alili}, \textit{D. Dufresne} and \textit{M. Yor} [in: Exponential functionals and principal values related to Brownian motion, 3-14 (1997; Zbl 0905.60059)] and \textit{J. B. Walsh} [in: Séminaire de probabilités XXVII. Lect. Notes Math. 1557, 173-176 (1993; Zbl 0793.60087)] who considered \(\Pi_t(f)\) for \(f(x)= \exp\{ax\}\) and \(f(x)= I\{X\geq 0\}\), respectively. In a second part of the paper the process \(\{\Pi^C_t, t\geq 0\}\), defined by the relation \(\Pi^C_t(f)= \int^t_0 dC_s f(B_t- B_s)\), is studied: first for the case of two independent Brownian motions \(B\), \(C\) and, secondly, for \(C= B\).