Regularity of invariant measures: The case of non-constant diffusion part

V. I. Bogachev; N. Krylov; M. Röckner

doi:10.1006/jfan.1996.0063

Regularity of invariant measures: The case of non-constant diffusion part

V. I. Bogachev, N. Krylov, M. Röckner

School of Mathematics

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Abstract

We prove regularity (i.e., smoothness) of measures μ on R^d satisfying the equation L*μ = 0 where L is an operator of type Lu = tr(Au″) + B · ∇u. Here A is a Lipschitz continuous, uniformly elliptic matrix-valued map and B is merely μ-square integrable. We also treat a class of corresponding infinite dimensional cases where ℝ^d is replaced by a locally convex topological vector space X. In this cases μ is proved to be absolutely continuous w.r.t. a Gaussian measure on X and the square root of the Radon-Nikodym density belongs to the Malliavin test function space double-struck D sign ^{2, 1}.

Original language	English (US)
Pages (from-to)	223-242
Number of pages	20
Journal	Journal of Functional Analysis
Volume	138
Issue number	1
DOIs	https://doi.org/10.1006/jfan.1996.0063
State	Published - May 25 1996

Bibliographical note

Funding Information:
We thank Sergio Albeverio for fruitful discussions on the contents of this paper. Financial support of the Sonderforschungsbereiche 256 (Bonn) and 343 (Bielefeld), EC-Science Project SC1*CT92-0784, the International Science Foundation (Grant M 38000), and the Russian Foundation of Fundamental Research (Grant, 94-01-01556) is gratefully acknowledged.

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title = "Regularity of invariant measures: The case of non-constant diffusion part",

abstract = "We prove regularity (i.e., smoothness) of measures μ on Rd satisfying the equation L*μ = 0 where L is an operator of type Lu = tr(Au″) + B · ∇u. Here A is a Lipschitz continuous, uniformly elliptic matrix-valued map and B is merely μ-square integrable. We also treat a class of corresponding infinite dimensional cases where ℝd is replaced by a locally convex topological vector space X. In this cases μ is proved to be absolutely continuous w.r.t. a Gaussian measure on X and the square root of the Radon-Nikodym density belongs to the Malliavin test function space double-struck D sign 2, 1.",

author = "Bogachev, {V. I.} and N. Krylov and M. R{\"o}ckner",

note = "Funding Information: We thank Sergio Albeverio for fruitful discussions on the contents of this paper. Financial support of the Sonderforschungsbereiche 256 (Bonn) and 343 (Bielefeld), EC-Science Project SC1*CT92-0784, the International Science Foundation (Grant M 38000), and the Russian Foundation of Fundamental Research (Grant, 94-01-01556) is gratefully acknowledged.",

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AU - Bogachev, V. I.

AU - Krylov, N.

AU - Röckner, M.

N1 - Funding Information: We thank Sergio Albeverio for fruitful discussions on the contents of this paper. Financial support of the Sonderforschungsbereiche 256 (Bonn) and 343 (Bielefeld), EC-Science Project SC1*CT92-0784, the International Science Foundation (Grant M 38000), and the Russian Foundation of Fundamental Research (Grant, 94-01-01556) is gratefully acknowledged.

PY - 1996/5/25

Y1 - 1996/5/25

N2 - We prove regularity (i.e., smoothness) of measures μ on Rd satisfying the equation L*μ = 0 where L is an operator of type Lu = tr(Au″) + B · ∇u. Here A is a Lipschitz continuous, uniformly elliptic matrix-valued map and B is merely μ-square integrable. We also treat a class of corresponding infinite dimensional cases where ℝd is replaced by a locally convex topological vector space X. In this cases μ is proved to be absolutely continuous w.r.t. a Gaussian measure on X and the square root of the Radon-Nikodym density belongs to the Malliavin test function space double-struck D sign 2, 1.

AB - We prove regularity (i.e., smoothness) of measures μ on Rd satisfying the equation L*μ = 0 where L is an operator of type Lu = tr(Au″) + B · ∇u. Here A is a Lipschitz continuous, uniformly elliptic matrix-valued map and B is merely μ-square integrable. We also treat a class of corresponding infinite dimensional cases where ℝd is replaced by a locally convex topological vector space X. In this cases μ is proved to be absolutely continuous w.r.t. a Gaussian measure on X and the square root of the Radon-Nikodym density belongs to the Malliavin test function space double-struck D sign 2, 1.

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Regularity of invariant measures: The case of non-constant diffusion part

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