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

Let (S(t))t≥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 CL[0, 1] by using a) S(∆t) for the transition between any two consecutive partition times of distance ∆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 without boundary 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.