Cores for Feller semigroups with an invariant measure

Let A = Sigma(N)(i)(j) = 1a(ij) (x) D-ij + Sigma(N)(i) =1 bi (x) D-i be an elliptic differential operator with unbounded coefficients on R-N and assume that the associated Feller semigroup (T-p(t))t >= 0 has an invariant measure mu. Then (T(t))1 >= 0 extends to a strongly continuous semigroup (T-p(t))(t)>= 0 on L-P(mu) = L-P (R-N. mu) for every 1 <= p < infinity. We prove that, under mild conditions on the coefficients of A, the space of test functions C-c(infinity) (R-N) is a core for the generator (A(p), D-p,) of (T-p(t))(t) >= 0 in L-p(mu) for I <= P < infinity. (c) 2005 Elsevier Inc. All rights reserved.