  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.
