Brownian motion on foliations: Entropy, invariant measures, mixing
Brownian motion on foliations: Entropy, invariant measures, mixing
复制标题
叶子上的布朗运动：熵、不变测度、混合
DOI：
10.1007/bf01077429
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发表时间：
1988
影响因子：
0.4
通讯作者：
V. Kaimanovich
中科院分区：
文献类型：
--
作者：
V. Kaimanovich
I.若~是紧致黎曼流形M上具有足够光滑叶的连续叶化，则~的叶上布朗运动的马氏过程与诱导度量对应的算子半群@/（x）=% I（g）pt（x，g）dy，其中p是叶L~ x上热方程的基本解. M上相对于半群{@ t}不变的测度称为调和测度。一个测度的调和性等价于它在叶上的条件测度关于叶的黎曼体积是绝对连续的，并且密度是叶上的调和函数。如果对于ae叶，密度是恒定的，那么测度被称为完全不变的[I]。叶上的黎曼体积在断面上的积分给出了叶理的完整不变测度和完全不变测度的锥的同构[2，3]。对于我们的目的，紧性和M上存在整体黎曼结构的条件是多余的。下面我们将考虑~是空间M的一个可测黎曼叶理[3]，具有遍历概率调和测度m~，并且~的叶具有有界几何。
I. If~ is a continuous foliation of a compact Riemannian manifold M with sufficient smooth leaves, then to the Markov process of Brownian motion on the leaves of~ with the induced metric, corresponds the semigroup of operators@~/(x)=% I (g) pt (x, g) dy, where p is the fundamental solution of the heat equation on the leaf L~ x. Measures on M, invariant relative to the semigroup {@ t}, are called harmonic. Harmonicity of a measure is equivalent to its conditional measures on ae leaves being absolutely continuous with respect to the Riemannian volume of the leaf, and the density is a harmonic function on the leaf. If for ae leaves, the densities are constant, then the measure is termed fully invariant [I]. Integration of the Riemannian volumes on the leaves over the transversals gives an isomorphism of the cones of the holonomically invariant and fully invariant measures of the foliation [2, 3].For our purposes, the conditions of compactness and of the presence of a global Riemannian structure on M are superfluous. We will consider below that~ is a measurable Riemannian foliation [3] of the space M with an ergodic probabilistic harmonic measure m~ and the leaves of~ have bounded geometry.