Short time behavior of logarithmic derivatives of the heat kernel
type: Item
instance of: scholarly article
title: Short time behavior of logarithmic derivatives of the heat kernel
zbMATH Open document ID: 0930.37027
author: Daniel W. Stroock
author: James Turetsky
published in: The Asian Journal of Mathematics
publication date: 2000-02-15T00:00:00Z
review text: Consider Varadhan's formula, that is (1) \(\lim_{T\downarrow 0}T\log p_T(x,y)=- \frac{\text{dist} (x,y)^2}{2}\) uniformly in \((x,y)\in M\times M\), where \(M\) is a compact connected Riemannian manifold, and \(P_t(x,y)\) is the fundamental solution to Cauchy's initial value problem for the heat equation \(\frac{\partial u}{\partial t}= \frac 12 \Delta_Mu\) with Levi-Civita Laplacian \(\Delta_M\). The main goal of this paper is to examine the extent to which the limit (1) can be made to commute with derivatives. The authors extend previous results (first- and second-order logarithmic derivatives) to derivative of all orders. In particular, the authors show that derivatives of all order commute with the limit in (1) when \(x\not\in \text{Cut}(y)\), where \(\text{Cut}(y)\) denotes the cut-locus of \(y\). In the case \(x\in \text{Cut}(y)\), the authors show that \(n\)th order logarithmic derivatives of \(P_T(\cdot,y)\) at \(x\) can be as bad as \(T^{-n}\) in general. To this end, among others the authors express the logarithmic derivatives of the heat kernel in terms of conditional expectations on the Wiener space.
Mathematics Subject Classification ID: 37H10
Mathematics Subject Classification ID: 60H99
zbMATH DE Number: 1310255
zbMATH Keywords: Varadhan's formula
zbMATH Keywords: fundamental solution
zbMATH Keywords: conditional expectations
zbMATH Keywords: Wiener space
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
full work available at URL: ajm.1997.v1.n1.a2
OpenAlex ID: W44521033
DOI: 10.4310/AJM.1997.V1.N1.A2
description: scientific article; zbMATH DE number 1310255