ON REGULARITY OF TRANSITION PROBABILITIES AND INVARIANT MEASURES OF SINGULAR DIFFUSIONS UNDER MINIMAL CONDITIONS
ON REGULARITY OF TRANSITION PROBABILITIES AND INVARIANT MEASURES OF SINGULAR DIFFUSIONS UNDER MINIMAL CONDITIONS
复制标题
DOI：
10.1081/pde-100107815
复制
发表时间：
2001-11
影响因子：
1.9
通讯作者：
V. Bogachev;Nicolai V Krylov;M. Röckner
中科院分区：
文献类型：
--
作者：
V. Bogachev;Nicolai V Krylov;M. Röckner
设A =（aij）是域Ω <$Rd上的矩阵值Borel映射，B =（bi）是Ω上的向量场，LA，B = aij <$x i <$xj <$+ bi <$xi <$.本文研究了Ω上满足弱意义下椭圆方程LA，B *μ = 0的Borel测度μ：对任意的ε ∈ C 0 ∞（Ω），有ε LA，B dμ = 0.我们证明，在温和的条件下，μ有一个密度。如果A是局部一致非退化的，A ∈ Hlocp，1，且B ∈ Llocp，其中p > d，则这个密度属于Hlocp，1.实际上，我们证明了某些广义非线性椭圆型不等式解的Sobolev正则性。在抛物情形下得到了类似的结果。这些结果适用于转移概率和不变的措施的扩散过程。
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.