On Regularity of Transition Probabilities and Invariant Measures of Singular Diffusions Under Minimal Conditions

Let A = (a(ij)) be a matrix-valued Borel mapping on a domain Omega subset of R-d, let b = (b(i)) be a vector field on Omega, and let L-A,L-bphi = a(ij)partial derivative(xi)partial derivative(xj)phi + b(i)partial derivative(xi)phi. We study Borel measures mu on Omega that satisfy the elliptic equation L-A,L-bmu* = 0 in the weak sense: integral L(A,b)phidmu = 0 for all phi is an element of C-0(infinity)(Omega). We prove that, under mild conditions, mu has a density. If A is locally uniformly nondegenerate, A is an element of H-loc(p,1) and b is an element of L-loc(p) for some p > d, then this density belongs to H-loc(p,1). Actually, we prove Sobolev regularity for solutions of certain generalized nonlinear elliptic inequalities. Analogous results are obtained in the parabolic case. These results are applied to transition probabilities and invariant measures of diffusion processes.