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

TL;DR

The paper advances Freidlin–Wentzell exit-time theory to time-inhomogeneous diffusions by showing that WF-type exponential exit-time estimates remain valid when drift and diffusion are uniformly close to time-independent counterparts. It formalizes a robust class $\mathfrak{X}(\varepsilon,\kappa,x)$ of processes and proves that, for compact $K\subset\mathcal{D}$, the exit time $\tau(X)$ satisfies $\mathbb{P}\big(\exp\{2(H-\eta)/\varepsilon\} \le \tau(X) \le \exp\{2(H+\eta)/\varepsilon\}\big) \to 1$ as $\varepsilon\to0$, with $H$ the height of the quasi-potential, and the exit distribution concentrates on the boundary subset minimizing $Q$. The authors also derive corresponding $L_1$ asymptotics and discuss relaxing uniform closeness, plus an application to McKean–Vlasov diffusions, where the law evolves near a Dirac mass at an attractor. The MV analysis shows that the same exponential exit-time behavior holds under Wasserstein-2 controls, thereby unifying WF-type results for a broad class of time-inhomogeneous and measure-dependent systems and linking to prior MV literature such as HIP2 and AT24. Overall, the work broadens the WF framework to non-autonomous settings and provides rigorous exit-time and exit-position results with clear applicability to interacting particle systems.

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.

Figures (4)

• Figure 1: An illustration of the stopping times $\theta_i$.
• Figure 2: Depiction of the relationship between $\widetilde{K}_1$, $\widetilde{K}_2$, $(\widetilde{K}(2\rho) - \kappa)$, and $\widetilde{K}$.
• Figure 3: The dependency between $\psi(t)$ and $\dot{\psi}(t)$.
• Figure 4: Convergence of $\psi$.

Theorems & Definitions (39)

• Proposition 1.5: M.I. Freidlin, A.D. Wentzell
• Definition 1.6
• Remark 1.7
• Theorem 1.8
• Corollary 1.9
• Remark 1.10
• Corollary 1.11
• Lemma 1.17
• Theorem 1.18
• Lemma 2.1