PUB - Publikationen an der Universität Bielefeld

Let A = (A(ij)) be a mapping with values in the space of the nonnegative symmetric operators on R-n and let B = (B-i) be a Borel vector field on R-n such that A is locally uniformly nondegenerate, A(ij) is an element of H-loc(p,1)(R-n), B-i is an element of L-loc(p)(R-n), where p > n. We show that the existence of a Lyapunov function for the operator L-A,L-B f = Sigma A(ij) partial derivative (xi) partial derivative (xj) f+Sigma B-i partial derivative (xi) f is sufficient for the existence of a probability measure mu with a strictly positive continuous density in the class H-loc(p,1)(R-n) such that mu satisfies L-A,L-B* mu = 0 in the weak sense and is an invariant measure for the diffusion with the generator L-A,L-B on domain C-o(infinity)(R-n). For arbitrary continuous nondegenerate A and locally bounded B, we prove the existence of absolutely continuous solutions. An analogous generalization of Khasminskii's theorem is obtained for manifolds.