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Random walks among time increasing conductances: heat kernel estimates

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Abstract

For any graph having a suitable uniform Poincaré inequality and volume growth regularity, we establish two-sided Gaussian transition density estimates and parabolic Harnack inequality, for constant speed continuous time random walks evolving via time varying, uniformly elliptic conductances, provided the vertex conductances (i.e. reversing measures), increase in time. Such transition density upper bounds apply for discrete time uniformly lazy walks, with the matching lower bounds holding once the parabolic Harnack inequality is proved.

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Acknowledgements

We thank Takashi Kumagai for proposing to pursue the ghk in our aim to verify [1, Conj. 7.1]. This work further benefited from discussions (of A.D.) with Martin Barlow and (of T.Z.) with Laurent Saloff-Coste.

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Authors and Affiliations

  1. Department of Mathematics, Stanford University, Building 380, Sloan Hall, Stanford, CA, 94305, USA

    Amir Dembo

  2. Department of Statistics, Stanford University, Sequoia Hall, 390 Serra Mall, Stanford, CA, 94305, USA

    Amir Dembo & Ruojun Huang

  3. Department of Mathematics, University of California, San Diego, 9500 Gilman Dr, La Jolla, CA, 92093, USA

    Tianyi Zheng

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  1. Amir Dembo
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  2. Ruojun Huang
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  3. Tianyi Zheng
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Correspondence to Ruojun Huang.

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This research was supported in part by NSF Grant DMS-1613091.

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Dembo, A., Huang, R. & Zheng, T. Random walks among time increasing conductances: heat kernel estimates. Probab. Theory Relat. Fields 175, 397–445 (2019). https://doi.org/10.1007/s00440-018-0894-1

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