On a Property of Non-Degenerate Diffusion Processes
Igor Vladimirovich Girsanov · Theory of Probability and Its Applications · 1959
Let X be a non-degenerate diffusion process in n-dimensional Euclidean space $R^n $, U a domain, $\tau _U $ the time at which the boundary of U is first attained. In the present article it is proved that. the course of the process X inside U is determined by the functions\[ m(x) = M_x \tau _x {\text{ and }}\pi (x,\Gamma ) = {\bf P}_x \left\{ {x_{\tau _u } \in \Gamma } \right\}. \]The proof is based on an analytic method of the theory of elliptic differential operators.