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

Let A = (a) be a matrix-valued Borel mapping on a domain Ω ⊂ R, let b = (b) be a vector field on Ω, and let LA,bφ = a ∂xi∂xjφ + b ∂xiφ. We study Borel measures μ on Ω that satisfy the elliptic equation L∗A,bμ = 0 in the weak sense: ∫ LA,bφdμ = 0 for all φ ∈ C∞ 0 (Ω). We prove that, under mild conditions, μ has a density. If A is locally uniformly nondegenerate, A ∈ H loc and b ∈ L p loc for some p > d, then this density belongs to H loc . 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. AMS 1991 Subject classification: Primary: 35J15, 35K10, 60J35 Secondary: 35D10, 35B65, 60J60, 35R05