Regularity of Invariant Measures on Finite and Infinite Dimensional Spaces and Applications
Regularity of Invariant Measures on Finite and Infinite Dimensional Spaces and Applications
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DOI：
10.1006/jfan.1995.1123
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发表时间：
1995-10
影响因子：
1.7
通讯作者：
V. Bogachev;M. Röckner
中科院分区：
文献类型：
--
作者：
V. Bogachev;M. Röckner
摘要在有限维和无限维状态空间E上，对于L = Δ + B·∇型算子，我们证明了测度μ求解方程L *μ = 0的正则性（即光滑性）的新结果。特别地，我们解决了I. Shigekawa在Δ = Δ H是gross - laplace， （E， H, γ）是抽象Wiener空间，B = - id E + v（其中v取Cameron-Martin空间H中的值）的一个猜想。利用高斯对数sobolev不等式，我们证明了μ在高斯测度γ下总是绝对连续的，并且密度的平方根在l2 （γ）中的1阶Malliavin检验函数空间中。进一步讨论了在无限维随机微分方程中的应用，并证明了L *μ = 0的一些新的存在性结果。这些包括关于“逆问题”的结果，即，我们给出了确保B是一个测度的（向量）对数导数的条件。我们还证明了μ对称的充分必要条件（即L在l2 （μ）上对称）。最后，本工作的很大一部分致力于研究L的对称测度的唯一性。我们通过相关（经典）狄利克雷形式的不可约性来描述具有唯一性的情况。
Abstract In this paper we prove new results on the regularity (i.e., smoothness) of measures μ solving the equation L *μ = 0 for operators of type L = Δ + B · ∇ on finite and infinite dimensional state spaces E . In particular, we settle a conjecture of I. Shigekawa in the situation where Δ = Δ H is the Gross-Laplacian, ( E , H , γ) is an abstract Wiener space and B = −id E + v where v takes values in the Cameron-Martin space H . Using Gross′ logarithmic Sobolev-inequality in an essential way we show that μ is always absolutely continuous w.r.t. the Gaussian measure γ and that the square root of the density is in the Malliavin test function space of order 1 in L 2 (γ). Furthermore, we discuss applications to infinite dimensional stochastic differential equations and prove some new existence results for L *μ = 0. These include results on the "inverse problem", i.e., we give conditions ensuring that B is the (vector) logarithmic derivative of a measure. We also prove necessary and sufficient conditions for μ to be symmetrizing (i.e., L is symmetric on L 2 (μ)). Finally, a substantial part of this work is devoted to the uniqueness of symmetrizing measures for L . We characterize the cases, where we have uniqueness, by the irreducibility of the associated (classical) Dirichlet forms.