Mean Exit Times of Markov Processes
Gray and Pinsky  have investigated the geometric content of the mean exit time of Brownian motion Xt from a ball on a Riemannian manifold. That is, if m is the center of the geodesic ball Be of radius e, and if Te is the first time Xt exits Be, they obtain the asymptotic expansion of Pm[Te] as e goes to zero and identify the first three nonzero terms of the expression in terms of the geometry of the manifold. In view of the fact that pm[Te] contains so much geometric information, it seems natural to ask to what extent pm[Te] determines the Brownian motion itself. Could there be another process (Yt,QX) with pm[Te] = Qm[Te] ? Without additional information, this seems difficult to determine. Restricting to exit times of balls with center m limits us severely. Our purpose in this note is to indicate that if the hypotheses are strengthened somewhat, then mean exit times do determine the process. The proof naturally appears in two parts. First, we indicate that the mean exit times determine the geometric trajectories of the process, and then we show that the speed at which the processes move along these trajectories is also determined. In fact, if two processes are time changes of one another and if the mean exit, times of one process dominate the mean exit times of the other process, then one process runs more slowly than the other, path for path. This is not true if the mean last exit times of one process dominate the mean last exit times of a time change of the process as we show in Example (5).
KeywordsBrownian Motion Riemannian Manifold MARKOV Process Exit Time Potential Density
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