On- and off-diagonal heat kernel behaviors on certain infinite dimensional local Dirichlet spaces
On- and off-diagonal heat kernel behaviors on certain infinite dimensional local Dirichlet spaces
复制标题
某些无限维局部狄利克雷空间上的对角线和非对角线热核行为
DOI：
--
复制
发表时间：
2000
期刊：
影响因子：
--
通讯作者：
L. Saloff‐Coste
中科院分区：
文献类型：
--
作者：
A. Bendikov;L. Saloff‐Coste
本文研究了具有连续核的对称扩散半群的对角和非对角上界，在这种情况下，其基础空间通常是无限维的。对于无限维环面上的不变扩散，证明了某个对角估计与某个非对角行为之间的等价性。这给出了一个充分必要条件，在相关的无限对称矩阵，椭圆Harnack不等式得到满足。因此，椭圆和抛物Harnack不等式实际上是等价的性质在此设置。在位势理论中，这给出了一个相关调和层是布雷洛调和层的充分必要条件。还引入了与某些Dirichlet空间相关的一个新的距离。它在将对角线行为与非对角线行为联系起来方面起着至关重要的作用，在这种情况下，内在距离几乎处处都是无限的。
This paper studies on-diagonal and off-diagonal bounds for symmetric diffusion semi-groups that admit a continuous kernel, in the case where the underlying space is typically infinite dimensional. For invariant diffusions on the infinite dimensional torus, the equivalence between a certain on-diagonal estimate and a certain off-diagonal behavior is proved. This gives a necessary and sufficient condition, in terms of the associated infinite symmetric matrix, for an elliptic Harnack inequality to be satisfied. As a consequence, elliptic and parabolic Harnack inequalities are in fact equivalent properties in this setting. In terms of potential theory, this gives a necessary and sufficient condition for the associated harmonic sheaf to be a Brelot harmonic sheaf. A new distance associated to certain Dirichlet spaces is also introduced. It plays a crucial role in relating on-diagonal behavior to off-diagonal behavior in cases where the intrinsic distance is infinite almost everywhere.