Seminar on Stochastic Processes, 1984 pp 99-107 | Cite as

# Mean Exit Times of Markov Processes

## Abstract

Gray and Pinsky [3] have investigated the geometric content of the mean exit time of Brownian motion X_{t} from a ball on a Riemannian manifold. That is, if m is the center of the geodesic ball B_{e} of radius e, and if T_{e} is the first time X_{t} exits B_{e}, they obtain the asymptotic expansion of P^{m}[T_{e}] 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 p^{m}[T_{e}] contains so much geometric information, it seems natural to ask to what extent p^{m}[T_{e}] determines the Brownian motion itself. Could there be another process (Y_{t},Q^{X}) with p^{m}[T_{e}] = Q^{m}[T_{e}] ? 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).

## Keywords

Brownian Motion Riemannian Manifold MARKOV Process Exit Time Potential Density## Preview

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## References

- 1.R.M. BLUMENTHAL and R.K. GETOOR.
*Markov Processes and Potential Theory*. Academic Press, New York, 1968.MATHGoogle Scholar - 2.J. GLOVER. Markov processes with identical hitting probabilities.
*Trans. Amer. Math. Soc. 275*(1983), 131–141.MathSciNetMATHCrossRefGoogle Scholar - 3.A. GRAY and M. PINSKY. The mean exit time from a small geodesic ball in a Riemannian manifold."
*Bull. So. Math. 107*(1983), 345–370.MathSciNetMATHGoogle Scholar