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

TL;DR

This paper studies the small-noise exit-time behavior of time-inhomogeneous diffusions, showing exponential growth under certain conditions, characterizing exit positions, and extending results to McKean-Vlasov processes.

Contribution

It provides new exponential estimates for exit times of time-inhomogeneous diffusions and extends these results to McKean-Vlasov processes, improving upon existing literature.

Findings

Exit time grows exponentially as noise tends to zero.

Characterization of the exit position for the process.

Extension of results to McKean-Vlasov processes.

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.