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

Grégoire Véchambre ()
Additional contact information
Grégoire Véchambre: Chinese Academy of Sciences

Journal of Theoretical Probability, 2023, vol. 36, issue 2, 876-925

Abstract: 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.

Keywords: Diffusion; Random potential; Lévy process; Renewal process; Local time; Lévy process conditioned to stay positive; Exponential functional; 60K37; 60J55; 60G51 (search for similar items in EconPapers)
Date: 2023
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10959-022-01191-z Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jotpro:v:36:y:2023:i:2:d:10.1007_s10959-022-01191-z

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10959

DOI: 10.1007/s10959-022-01191-z

Access Statistics for this article

Journal of Theoretical Probability is currently edited by Andrea Monica

More articles in Journal of Theoretical Probability from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().