Large deviation property for Riemannian Brownian motion on a complete manifold (Q1261762)

Let \(M\) be a connected, complete Riemannian manifold, \(p(t,x,y)\) be the transition density for the Riemannian Brownian motion, \((P_{\varepsilon,x})\) be the diffusion measure family associated with \(p(\varepsilon t,x,y)\). Previously \textit{R. Azencott} [Lect. Notes Math. 774, 1-176 (1980; Zbl 0435.60028)] showed that \((P_{\varepsilon,x})\) has large deviation property as \(\varepsilon\to 0\), and \textit{M. I. Frejdlin} and \textit{A. D. Venttsel} ' [Random perturbations of dynamical systems (1984; Zbl 0522.60055)] showed that \((P_{\varepsilon,x})\) has strongly large deviation property as \(\varepsilon\to 0\) and \(y\to x\) under some additional conditions leading to the hypothesis that \(M\) has bounded geometry and \(\Delta\) is locally uniformly elliptic. Here the author shows that \((P_{\varepsilon,x})\) has strongly large deviation property as \(\varepsilon\to 0\) and \(x\to y\) provided only that the Ricci curvature of \(M\) is bounded from below.