Asymptotics of the principal eigenvalue and expected hitting time for positive recurrent elliptic operators in a domain with a small puncture

Let X(t) be a positive recurrent diffusion process corresponding to an operator L on a domain Dsubset of or equal toR(d) with oblique reflection at partial derivativeD if Dnot equalR(d). For each xis an element ofD, we define a volume-preserving norm that depends on the diffusion matrix a(x). We calculate the asymptotic behavior as epsilon-->0 of the expected hitting time of the epsilon-ball centered at x and of the principal eigenvalue for L in the exterior domain formed by deleting the ball, with the oblique derivative boundary condition at partial derivativeD and the Dirichlet boundary condition on the boundary of the ball. This operator is non-self-adjoint in general. The behavior is described in terms of the invariant probability density at x and Det(a(x)). In the case of normally reflected Brownian motion, the results become isoperimetric-type equalities. (C) 2002 Elsevier Science (USA). All rights reserved.