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

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

2004-09-09

arxiv.org/abs/math/0409155v3

Mathematics

Probability

35 pages, revised version for publication, more detailed exposition

Scientific paper

Let (S(t)) be a one-parameter family S = (S(t)) of positive integral operators on a locally compact space L. For a possibly non-uniform partition of [0,1] define a measure on the path space C([0,1],L) by using a) S(dt) for the transition between cosecutive partition times of distance dt, and b) a suitable continuous interpolation scheme (e.g. Brownian bridges or geodesics). If necessary normalize to get a probability measure. We prove a version of Chernoff's theorem of semigroup theory and tighness results which together yield convergence in law of such measures as the partition gets finer. In particular let L be a closed smooth submanifold of a Riemannian 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 Radon-Nikodym density with repect to Brownian motion on L which contains scalar, mean and sectional curvature terms. Various approximation schemes for Brownian motion are also given. These results substantially extend earlier work by the authors and by Andersson and Driver.

Affiliated with

Also associated with

No associations

Rating

Chernoff's Theorem and Discrete Time Approximations of Brownian Motion on Manifolds does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Chernoff's Theorem and Discrete Time Approximations of Brownian Motion on Manifolds, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Chernoff's Theorem and Discrete Time Approximations of Brownian Motion on Manifolds will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-229561