Abstract.
Weak convergence of the laws of discrete time re-metrized stochastic processes derived from Brownian motions on compact Riemannian manifolds with heat kernels uniformly bounded by a constant on each compact set of the time parameter and bounded volumes to a stochastic process is given. With a weak condition, we also give weak convergence of those of Brownian motions themselves on manifolds in the same class. Several examples are given, which cover the cases when the manifolds collapse, the cases when the original Brownian motions converge to a non-local Markov process, and the cases when the Gromov-Hausdorff limit and the spectral limit by Kasue and Kumura are different.
Author information
Authors and Affiliations
Additional information
Received: 22 February 2000¶Published online: 9 March 2001
Rights and permissions
About this article
Cite this article
Ogura, Y. Weak convergence of laws of stochastic processes on Riemannian manifolds. Probab Theory Relat Fields 119, 529–557 (2001). https://doi.org/10.1007/PL00008770
Issue Date:
DOI: https://doi.org/10.1007/PL00008770
- Mathematics Subject Classification (2000): 58J65, 60F17