Chernoff's Theorem and Discrete Time Approximations of Brownian Motion on Manifolds

Abstract

Let $(S(t))_{t \ge 0}$ be a one-parameter family of positive integral operators on a locally compact space $L$. For a possibly non-uniform partition of $[0,1]$ define a finite measure on the path space $C_L[0,1]$ by using a) $S(\Delta t)$ for the transition between any two consecutive partition times of distance $\Delta t$ and b) a suitable continuous interpolation scheme (e.g. Brownian bridges or geodesics). If necessary normalize the result to get a probability measure. We prove a version of Chernoff's theorem of semigroup theory and tightness results which yield convergence in law of such measures as the partition gets finer. In particular let $L$ be a closed smooth submanifold of a manifold $M$. We prove convergence of Brownian motion on $M$, conditioned to visit $L$ at all partition times, to a process on $L$ whose law has a density with respect to Brownian motion on $L$ which contains scalar, mean and sectional curvatures terms. Various approximation schemes for Brownian motion on $L$ are also given.