, Volume 75, Issue 1, pp 55-65

The lifetime of conditioned Brownian motion in certain Lipschitz domains

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Summary

For certain Lipschitz domains D we obtain a series expansion for the distribution of the lifetime τ D of conditioned Brownian motion on D. From this we determine $$\mathop {lim}\limits_{t \to \infty } {\text{ }}t^{ - 1} {\text{ }}log{\text{ }}P_x^h (\tau _D {\text{ > }}t) = \mathop {lim}\limits_{t \to \infty } {\text{ }}t^{ - 1} {\text{ }}log{\text{ }}P_x (\tau _D {\text{ > }}t) = {\text{--}}\lambda _D {\text{,}}$$ where λ D is the first eigenvalue of 1/2 Δ on D.

Supported in part by the National Science Foundation under grant DMS-830167