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

Content Provider	Semantic Scholar
Author	Bogachev, Vladimir I. Krylov, Nicolai V. Röckner, Michael
Copyright Year	2001
Abstract	Let A = (aij ) be a matrix-valued Borel mapping on a domain Ω ⊂ R d , let b = (bi ) be a vector field on Ω, and let LA, b ϕ = a ij ∂ x i ∂ xj ϕ + bi ∂ xi ϕ. We study Borel measures μ on Ω that satisfy the elliptic equation LA, b *μ = 0 in the weak sense: ∫ LA, b ϕ dμ = 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 p, 1 and b ∈ 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.
Starting Page	2037
Ending Page	2080
Page Count	44
File Format	PDF HTM / HTML
DOI	10.1081/PDE-100107815
Volume Number	26
Alternate Webpage(s)	http://www.mathematik.uni-bielefeld.de/sfb343/preprints/pr99141.ps.gz
Alternate Webpage(s)	https://doi.org/10.1081/PDE-100107815
Language	English
Access Restriction	Open
Content Type	Text
Resource Type	Article