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Potential Analysis

, Volume 50, Issue 2, pp 197–219 | Cite as

Convergence of Continuous Stochastic Processes on Compact Metric Spaces Converging in the Lipschitz Distance

  • Kohei SuzukiEmail author
Article
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Abstract

We introduce a new distance, a Lipschitz–Prokhorov distancedLP, on the set \(\mathcal {PM}\) of isomorphism classes of pairs (X, P) where X is a compact metric space and P is the law of a continuous stochastic process on X. We show that \((\mathcal {PM}, d_{LP})\) is a complete metric space. For Markov processes on Riemannian manifolds, we study relative compactness and convergence.

Keywords

Weak convergence Lipschitz convergence Markov processes Riemannian manifolds 

Mathematics Subject Classification (2010)

Primary 60F17 Secondary 53C23 

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Notes

Acknowledgments

The author thanks Prof. Kouji Yano for careful reading of his manuscript and successive encouragement. He thanks Prof. Takashi Kumagai for giving comments and detailed references in related fields. He thanks to Yohei Yamazaki for a lot of valuable and constructive comments and useful discussions. He also thanks to an anonymous referee for useful suggestions and references. This work was supported by Grant-in-Aid for JSPS Fellows Number 261798 and DAAD PAJAKO Number 57059240.

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für Angewandte MathematikUniversität BonnBonnGermany

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