Precise Gaussian estimates of heat kernels on asymptotically flat Riemannian manifolds with poles

We give precise Gaussian upper and lower bound estimates on heat kernels on Riemannian manifolds with poles under assumptions that the Riemannian curvature tensor goes to 0 sufficiently fast at infinity. Under additional assumptions on the curvature, we give estimates on the logarithmic derivatives of the heat kernels. The proof relies on the Elworthy-Truman’s formula of heat kernels and Elworthy and Yor’s observation on the derivative process of certain stochastic flows. As an application of them, we prove logarithmic Sobolev inequalities on pinned path spaces over such Riemannian manifolds.