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