## On an Ergodic Property of Diffusion Semigroups on Euclidean Space

J. Math. Sci. Univ. Tokyo
Vol. 10 (2003), No. 3, Page 537--553.

Kusuoka, Shigeo ; Liang, Song
On an Ergodic Property of Diffusion Semigroups on Euclidean Space
Consider a class of uniform elliptic diffusion processes on Euclidean spaces. We estimate transition densities and their derivatives uniformly with respect to the starting points. We use these results to prove an ergodicity of $\nabla P_t^c$ (see (\ref{eqn:definitionofPtc}) and Theorem 1.4) under certain conditions. This is useful in precise estimate of large deviation principles.