Stability and approximations of symmetric diffusion semigroups and kernels

Let (P-t)(t greater than or equal to 0) and ((P) over tilde(t))(t greater than or equal to 0) be two diffusion semigroups on R-d (d greater than or equal to 2) associated with uniformly elliptic operators L = del.(A del) and (L) over tilde = del.((A) over tilde del) with measurable coefficients A = (a(ij)) and (A) over tilde = (a(ij)), respectively. The corresponding diffusion kernels are denoted by p(t)(x, y) and (p) over tilde(t)(x, y). We derive a pointwise estimate on \p(t)(x, y)-(p) over tilde(t)(x, y)\ as well as an L-p-operator norm bound, where p is an element of [1, infinity], for P-t-(P) over tilde(t) in terms of the local L-2-distance between a(ij) and (a) over tilde(ij). This implies in particular that \p(t)(x, y)-(p) over tilde(t)(x, y)\ converges to zero uniformly in (x, y) is an element of R-d x R-d and that the L-p-operator norm of P-t-(P) over tilde(t) converges to zero uniformly in p is an element of [1, infinity] when a(ij)-(a) over tilde(ij) goes to zero in the local L-2-norm for each 1 less than or equal to i, j less than or equal to n. (C) 1998 Academic Press.