The renowned Freidlin-Wentzell theory, initiated by M. Freidlin and A. Wentzell in the late 1960s, focuses on the study of small random perturbations of deterministic flow $\dot{\phi}(t) = b(\phi(t))$. The authors examined random perturbations in the form of Brownian motion with a small diffusion parameter, described by the stochastic differential equation: $dX_t= b(X_t) dt + \sqrt{\epsilon} \Sigma(X_t) dW_t$.
A central problem they addressed was the first exit time from a positively invariant for the flow $\dot{\phi}(t) = b(\phi(t))$ set $D$. In particular, they analyzed the asymptotic behavior of the first exit time $\tau = \{t \geq 0: X_t \notin D\}$ and the exit position $X_\tau$ in the small noise regime ($\epsilon \to 0$).
In the first part of this talk, we will review the foundational results of Freidlin-Wentzell theory and highlight their significance in describing the metastable behavior of stochastic processes. In the second part, we will address the additional challenges that arise when the process under consideration is time-inhomogeneous. As an example, we will discuss the McKean-Vlasov process, presenting exit-time results for it and the techniques used to obtain them. Finally, we will explore what can be said about the exit-time problem in the general time-inhomogeneous case.
This talk is based on my recent work with Julian Tugaut and Stéphane Villeneuve.
