1.5

Let L be a uniformly elliptic operator on d dimensional space,

The matrix is assumed to satisfy the ellipticity bounds in the quadratic form sense. Suppose is the diffusion process associated with L. Let be the ball of radius R in and S a subset of . Then if denotes the characteristic function of S, the solution of the boundary value problem,

is the expected time the diffusion process , starting at , spends in S before exiting . According to the Alexandrov-Bakelman-Pucci (ABP) inequality (``Elliptic partial differential equations of second order", by Gilbarg and Trudinger, Springer, NewYork, 1983) there is a constant C, depending only on such that , where denotes Lebesgue measure.

Suppose is a neighborhood with radius r;SPMlt;R, of a d-1 dimensional plane . Then one can see by a straightforward application of the maximum principle that , which is sharper than the estimate given by the ABP inequality. On the other hand the ABP estimate continues to hold if is a neighborhood of a d-1 dimensional Lipschitz manifold .

In this paper it is shown that the estimate continues to hold if is a neighborhood of a d-1 dimensional manifold which is . The argument breaks down for Lipschitz manifolds. A result for Lipschitz manifolds is , however, obtained in the paper. For x a distance r from , let be the probability that the process , started at x, escapes to a distance R from before hitting . If is , it is shown that , for some constant c depending only on . This imples the estimate . For a Lipschitz manifold it is shown that

for constants C,c depending only on and the Lipschitz constant.

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