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Accelerating Brownian motion on N-torus

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  • Pai, Hui-Ming
  • Hwang, Chii-Ruey

Abstract

On N-torus, we consider antisymmetric perturbations of Laplacian of the form LC≐Δ+C⋅∇, where C is a divergence free vector field. The spectral gap, denoted by λ(C), of L(C) is defined by −sup{real part of μ,μ is in the spectrum ofLC, μ≠0}. We characterize for a certain class of C’s, the limit of λ(kC) as k goes to infinity and prove that sup{λ(C),C is divergence free}=∞. This problem is motivated by accelerating diffusions. By adding a weighted antisymmetric drift to a reversible diffusion, the convergence to the equilibrium is accelerated using the spectral gap as a comparison criterion. However, how good can the improvement be is yet to be answered. In this paper, we demonstrate that on N-torus the acceleration of Brownian motion could be infinitely fast.

Suggested Citation

  • Pai, Hui-Ming & Hwang, Chii-Ruey, 2013. "Accelerating Brownian motion on N-torus," Statistics & Probability Letters, Elsevier, vol. 83(5), pages 1443-1447.
    • Handle: RePEc:eee:stapro:v:83:y:2013:i:5:p:1443-1447
    DOI: 10.1016/j.spl.2013.02.009
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    1. Bhattacharya, Rabi & Denker, Manfred & Goswami, Alok, 1999. "Speed of convergence to equilibrium and to normality for diffusions with multiple periodic scales," Stochastic Processes and their Applications, Elsevier, vol. 80(1), pages 55-86, March.
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    Cited by:

    1. Belmabrouk, Nadia & Damak, Mondher & Yaakoubi, Nejib, 2022. "Dirichlet eigenvalue problems of irreversible Langevin diffusion," Statistics & Probability Letters, Elsevier, vol. 180(C).
    2. Hwang, Chii-Ruey & Normand, Raoul & Wu, Sheng-Jhih, 2015. "Variance reduction for diffusions," Stochastic Processes and their Applications, Elsevier, vol. 125(9), pages 3522-3540.

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