The investigator will examine the properties of diffusions on manifolds and diffusion corresponding to elliptic divergence form operators. Diffusion processes provide widely used models especially in Physics. Both the analytic and the probabilistic tools will be used to seek solutions to these important problems. On manifolds, the smoothness of the exit distribution of Brownian motion will be studied. On special structures, the investigator will study the asymptotic behavior of conditioned Brownian motion. For diffusions arising from elliptic divergence form operators, transformation of drift formulae, conditional gauge theorems and sample path properties will be studied.
