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

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## Abstract

We introduce a new distance, *a Lipschitz–Prokhorov distance**d*_{LP}, 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## Preview

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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.

## References

- 1.Aldous, D.: Stopping times and tightness. Ann. Probab.
**6**(2), 335–340 (1978)MathSciNetCrossRefzbMATHGoogle Scholar - 2.Athreya, S., Löhr, W., Winter, A.: The gap between Gromov–vague and Gromov–Hausdorff–vague topology. Stochastic Process. Appl.
**126**(9), 2527–2553 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - 3.Athreya, S., Löhr, W., Winter, A.: Invariance principle for variable speed random walks on trees. Ann. Probab.
**45**(2), 625–667 (2017)MathSciNetCrossRefzbMATHGoogle Scholar - 4.Berger, M.: A Panoramic View of Riemannian Geometry, 2nd edn. Springer-Verlag, Berlin (2007)Google Scholar
- 5.Billingsley, P.: Convergence of Probability Measures, vol. 493, 2nd edn. Wiley, New York (1999)Google Scholar
- 6.Blumenthal, R.M., Getoor, R.K.: Markov processes and Potential theory. Academic Press, New York (1968)zbMATHGoogle Scholar
- 7.Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry, Graduate Studies in Mathematics, vol. 33. American Mathematical Society, Providence, RI, 2001. Errata list available online at http://www.pdmi.ras.ru/svivanov/papers/bbi-errata.pdf
- 8.Chen, Z.Q., Kim, P., Kumagai, T.: Discrete approximation of symmetric jump processes on metric measure spaces. Probab. Theory Relat. Fields
**155**, 703–749 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 9.Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, 2nd revised and extended edn. de Gruyter (2011)Google Scholar
- 10.Gromov, M.: Metric Structures for Riemannian and Non-Riemannian Spaces. Birkhäuser Boston Inc, Boston. Based on the 1981 French original [MR 85e:53051], With appendices by M. Katz, P. Pansu, and S. Semmes, Translated from the French by Sean Michael Bates (1999)Google Scholar
- 11.Kasue, A., Kumura, H.: Spectral convergence of Riemannian manifolds II. Tôhoku Math. J.
**48**, 71–120 (1996)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Katsuda, A.: Gromov’s convergence theorem and its application. Nagoya Math. J.
**100**, 11–48 (1985)MathSciNetCrossRefzbMATHGoogle Scholar - 13.Kuwae, K., Shioya, T.: Convergence of spectral structures: a functional analytic theory and its applications to spectral geometry. Commun. Anal. Geom.
**11**, 599–673 (2003)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Mosco, U.: Approximation of the solutions of some variational inequalities. Ann. Scuola Normale Sup., Pisa
**21**, 373–394 (1967)MathSciNetzbMATHGoogle Scholar - 15.Mosco, U.: Composite media and asymptotic Dirichlet forms. J. Funct. Anal.
**123**, 368–421 (1994)MathSciNetCrossRefzbMATHGoogle Scholar - 16.Ogura, Y.: Weak convergence of laws of stochastic processes on Riemannian manifolds. Probab. Theory Relat. Fields
**119**, 529–557 (2001)MathSciNetCrossRefzbMATHGoogle Scholar - 17.Sharapov, S., Sharapov, V.: Dimensions of some generalized Cantor sets, preprint: http://classes.yale.edu/fractals/FracAndDim/cantorDims/CantorDims.html
- 18.Sturm, K.-T.: Diffusion processes and heat kernels on metric spaces. Ann. Probab.
**26**, 1–55 (1998)MathSciNetCrossRefzbMATHGoogle Scholar - 19.Suzuki, K.: Convergences and projection Markov property of Markov processes on ultrametric spaces. RIMS J.
**50**(3), 569–588 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - 20.K. Suzuki: Convergence of brownian motion on metric measure spaces under riemannian curvature-dimension conditions, Preprint Available at: arXiv:1703.07234
- 21.Suzuki, K.: Convergence of non-symmetric diffusion processes on RCD spaces, Preprint Available at: arXiv:1709.09536
- 22.Suzuki, K., Yamazaki, Y.: Non-separability of the Lipschitz distance. Pacific J. Math. Ind.
**7**(1), 1–7 (2015)MathSciNetCrossRefzbMATHGoogle Scholar