SEMINAIRE LABO - Ashot Aleksian

Résumé : In this presentation, we begin by introducing the exit-time problem for solutions of the homogeneous SDE \[dX_t = b(X_t)dt + \sqrt{\epsilon} dW_t \]. Specifically, we study the stopping time \[\tau(X) := \inf{t \geq 0 : X_t \notin D}\], where \[D\] is a positively invariant set. We will review the foundational results of Freidlin-Wentzell theory, which describe the asymptotic behavior of \[\tau(X)\] as the noise parameter \[\epsilon\] tends to zero. In the second part of the talk, we present new results, obtained in collaboration with S. Villeneuve, addressing the exit-time problem for time-inhomogeneous diffusions, i.e. such that the drift or diffusion terms depend on time \[t\]. Finally, we discuss a specific application to time-inhomogeneous diffusions in the form of McKean-Vlasov processes. The presented results will be accessible to a wide audience familiar with stochastic processes and differential equations.