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Wasserstein asymptotics for Brownian motion on the flat torus and Brownian interlacements

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  • Mariani, Mauro
  • Trevisan, Dario

Abstract

We study the large time behaviour of the optimal transportation cost towards the uniform distribution, for the occupation measure of a stationary Brownian motion on the flat torus in d dimensions, where the cost of transporting a unit of mass is given by a power of the flat distance. We establish a global upper bound, in terms of the limit for the analogue problem concerning the occupation measure of the Brownian interlacement on Rd. We conjecture that our bound is sharp and that our techniques may allow for similar studies on a larger variety of problems, e.g. general diffusion processes on weighted Riemannian manifolds.

Suggested Citation

  • Mariani, Mauro & Trevisan, Dario, 2025. "Wasserstein asymptotics for Brownian motion on the flat torus and Brownian interlacements," Stochastic Processes and their Applications, Elsevier, vol. 183(C).
    • Handle: RePEc:eee:spapps:v:183:y:2025:i:c:s0304414925000365
    DOI: 10.1016/j.spa.2025.104595
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    References listed on IDEAS

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    1. Ibragimov, R. & Sharakhmetov, Sh., 2001. "The best constant in the Rosenthal inequality for nonnegative random variables," Statistics & Probability Letters, Elsevier, vol. 55(4), pages 367-376, December.
    2. Huesmann, Martin & Mattesini, Francesco & Trevisan, Dario, 2023. "Wasserstein asymptotics for the empirical measure of fractional Brownian motion on a flat torus," Stochastic Processes and their Applications, Elsevier, vol. 155(C), pages 1-26.
    3. Wang, Feng-Yu, 2022. "Wasserstein convergence rate for empirical measures on noncompact manifolds," Stochastic Processes and their Applications, Elsevier, vol. 144(C), pages 271-287.
    4. Huaiqian Li & Bingyao Wu, 2023. "Wasserstein Convergence Rates for Empirical Measures of Subordinated Processes on Noncompact Manifolds," Journal of Theoretical Probability, Springer, vol. 36(2), pages 1243-1268, June.
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