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Almost Sure Behavior for the Local Time of a Diffusion in a Spectrally Negative Lévy Environment

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  • Grégoire Véchambre

    (Chinese Academy of Sciences)

Abstract

We study the almost sure asymptotic behavior of the supremum of the local time for a transient sub-ballistic diffusion in a spectrally negative Lévy environment. More precisely, we provide the proper renormalizations for the extremely large values of the supremum of the local time. This is done by establishing a—so far unknown—connection between the latter and the exponential functional of a Lévy process conditioned to stay positive, which allows us to use the properties of such exponential functionals to characterize the sought behavior. It appears from our results that the renormalization of the extremely large values of the supremum of the local time is determined by the asymptotic behavior of the Laplace exponent of the Lévy environment and is surprisingly greater than the renormalization that was previously known for the recurrent case. Our results show moreover a rich variety of behaviors, which is a new phenomenon, as it does not occur in the discrete setting.

Suggested Citation

  • Grégoire Véchambre, 2023. "Almost Sure Behavior for the Local Time of a Diffusion in a Spectrally Negative Lévy Environment," Journal of Theoretical Probability, Springer, vol. 36(2), pages 876-925, June.
    • Handle: RePEc:spr:jotpro:v:36:y:2023:i:2:d:10.1007_s10959-022-01191-z
    DOI: 10.1007/s10959-022-01191-z
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    References listed on IDEAS

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    1. Diel, Roland, 2011. "Almost sure asymptotics for the local time of a diffusion in Brownian environment," Stochastic Processes and their Applications, Elsevier, vol. 121(10), pages 2303-2330, October.
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    8. Comets, Francis & Falconnet, Mikael & Loukianov, Oleg & Loukianova, Dasha & Matias, Catherine, 2014. "Maximum likelihood estimator consistency for a ballistic random walk in a parametric random environment," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 268-288.
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    10. Gantert, Nina & Shi, Zhan, 2002. "Many visits to a single site by a transient random walk in random environment," Stochastic Processes and their Applications, Elsevier, vol. 99(2), pages 159-176, June.
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