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Heat kernel fluctuations and quantitative homogenization for the one-dimensional Bouchaud trap model

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  • Andres, Sebastian
  • Croydon, David A.
  • Kumagai, Takashi

Abstract

We present on-diagonal heat kernel estimates and quantitative homogenization statements for the one-dimensional Bouchaud trap model. The heat kernel estimates are obtained using standard techniques, with key inputs coming from a careful analysis of the volume growth of the invariant measure of the process under study. As for the quantitative homogenization results, these include both quenched and annealed Berry–Esseen-type theorems, as well as a quantitative quenched local limit theorem. Whilst the model we study here is a particularly simple example of a random walk in a random environment, we believe the roadmap we provide for establishing the latter result in particular will be useful for deriving quantitative local limit theorems in other, more challenging, settings.

Suggested Citation

  • Andres, Sebastian & Croydon, David A. & Kumagai, Takashi, 2024. "Heat kernel fluctuations and quantitative homogenization for the one-dimensional Bouchaud trap model," Stochastic Processes and their Applications, Elsevier, vol. 172(C).
    • Handle: RePEc:eee:spapps:v:172:y:2024:i:c:s0304414924000425
    DOI: 10.1016/j.spa.2024.104336
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    References listed on IDEAS

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    1. Croydon, D.A. & Hambly, B.M. & Kumagai, T., 2019. "Heat kernel estimates for FIN processes associated with resistance forms," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 2991-3017.
    2. Chen, Zhen-Qing & Kumagai, Takashi, 2003. "Heat kernel estimates for stable-like processes on d-sets," Stochastic Processes and their Applications, Elsevier, vol. 108(1), pages 27-62, November.
    3. Croydon, David & Muirhead, Stephen, 2015. "Functional limit theorems for the Bouchaud trap model with slowly varying traps," Stochastic Processes and their Applications, Elsevier, vol. 125(5), pages 1980-2009.
    4. Deuschel, Jean-Dominique & Fukushima, Ryoki, 2019. "Quenched tail estimate for the random walk in random scenery and in random layered conductance," Stochastic Processes and their Applications, Elsevier, vol. 129(1), pages 102-128.
    5. Takashi Kumagai & Jun Misumi, 2008. "Heat Kernel Estimates for Strongly Recurrent Random Walk on Random Media," Journal of Theoretical Probability, Springer, vol. 21(4), pages 910-935, December.
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