Short time behavior of logarithmic derivatives of the heat kernel
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.
scientific article; zbMATH DE number 1310255