Freidlin-Wentzell type exit-time estimates for time-inhomogeneous diffusions and their applications

Ashot Aleksian and Stéphane Villeneuve
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Ashot Aleksian: TSE-R - Toulouse School of Economics - UT Capitole - Université Toulouse Capitole - Comue de Toulouse - Communauté d'universités et établissements de Toulouse - EHESS - École des hautes études en sciences sociales - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement
Stéphane Villeneuve: TSE-R - Toulouse School of Economics - UT Capitole - Université Toulouse Capitole - Comue de Toulouse - Communauté d'universités et établissements de Toulouse - EHESS - École des hautes études en sciences sociales - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement, TSM - Toulouse School of Management Research - UT Capitole - Université Toulouse Capitole - Comue de Toulouse - Communauté d'universités et établissements de Toulouse - CNRS - Centre National de la Recherche Scientifique - TSM - Toulouse School of Management - UT Capitole - Université Toulouse Capitole - Comue de Toulouse - Communauté d'universités et établissements de Toulouse

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Abstract: This paper investigates the exit-time problem for time-inhomogeneous diffusion processes. The focus is on the small-noise behavior of the exit time from a bounded positively invariant domain. We demonstrate that, when the drift and diffusion terms are uniformly close to some time-independent functions, the exit time grows exponentially both in probability and in $L_1$ as a parameter that controls the noise tends to zero. We also characterize the exit position of the time-inhomogeneous process. Additionally, we investigate the impact of relaxing the uniform closeness condition on the exit-time behavior. As an application, we extend these results to the McKean-Vlasov process. Our findings improve upon existing results in the literature for the exit-time problem for this class of processes.

Keywords: Exit time; McKean-Vlasov process; Time-inhomogeneous diffusion; Freidlin-Wentzell theory (search for similar items in EconPapers)
Date: 2026-07
Note: View the original document on HAL open archive server: https://hal.science/hal-05547126v1
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Published in Stochastic Processes and their Applications, 2026, 197, pp.104932. ⟨10.1016/j.spa.2026.104932⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-05547126

DOI: 10.1016/j.spa.2026.104932

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