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The Radial Part of Brownian Motion with Respect to \(\mathcal L \)-Distance Under Ricci Flow

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Abstract

Let \(\{g_t\}_{t\in [0,T)}\) be a family of complete time-depending Riemannian metrics on a manifold, which evolves under backwards Ricci flow. The Itô formula is established for the \(\mathcal L \)-distance of the \(g_t\)-Brownian motion to a fixed reference point (\(\mathcal L \)-base). Furthermore, as an application, we construct a coupling by parallel displacement, which yields a new proof of some results of Topping.

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Acknowledgments

The author thanks Professor Feng-Yu Wang for valuable suggestions. This work is supported in part by NNSFC(11131003), SRFDP, and the Fundamental Research Funds for the Central Universities.

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  1. Laboratory of Mathematics and Complex Systems, Ministry of Education, School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Li-Juan Cheng

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  1. Li-Juan Cheng
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Correspondence to Li-Juan Cheng.

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Cheng, LJ. The Radial Part of Brownian Motion with Respect to \(\mathcal L \)-Distance Under Ricci Flow. J Theor Probab 28, 449–466 (2015). https://doi.org/10.1007/s10959-013-0512-1

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  • DOI: https://doi.org/10.1007/s10959-013-0512-1

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