On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions. (English) Zbl 0997.35012

Bogachev, V. I.; Krylov, N. V.; Röckner, M.

Commun. Partial Differ. Equations 26, No. 11-12, 2037-2080 (2001).

Summary: Let \(A = (a^{ij})\) be a matrix-valued Borel mapping on a domain \(\Omega\subset\mathbb{R}^d\), let \(b = (b^i)\) be a vector field on \(\Omega\), and let \(L_{A,b}\varphi = a^{ij}\partial_{x_i}\partial_{x_j}\varphi+b^i\partial_{x_i}\varphi\). We study Borel measures \(\mu\) on \(\Omega\) that satisfy the elliptic equation \(L^*_{A,b}\mu = 0\) in the weak sense: \(\int L_{A,b}\varphi d\mu=0\) for all \(\varphi\in C^\infty_0(\Omega)\). We prove that, under mild conditions, \(\mu\) has a density. If \(A\) is locally uniformly nondegenerate, \(A\in H^{p,1}_{\text{loc}}\) and \(b\in L^p_{\text{loc}}\) for some \(p > d\), then this density belongs to \(H^{p,1}_{\text{loc}}\). 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.

MSC:

35J15	Second-order elliptic equations
60J60	Diffusion processes
60H15	Stochastic partial differential equations (aspects of stochastic analysis)
60J35	Transition functions, generators and resolvents
35D10	Regularity of generalized solutions of PDE (MSC2000)
35B65	Smoothness and regularity of solutions to PDEs

Keywords:

elliptic regularity; parabolic regularity; invariant measure; sub-Markovian semigroup; singular diffusion