Brownian Occupation Measures on Compact Manifolds

Abstract

Let M be a compact C^∞ manifold of dimension d. A C^∞ metric g and a C^∞ vector field V on M determine a “Brownian motion” on M, i.e. a strong Markov process

with continuous sample paths and generator $L = \frac{1}{{2\,}} {\Delta_g} + V,$ where Δ _g stands for the Laplace-Beltrami operator, associated with g. (If necessary we indicate by the subscript g, that an object is associated with the metric g.) Notice that the metric and the vector field can be recovered from the Brownian motion, from its generator L, to be precise.