The First Exit Time Problem in the small noise limit

26.04.2018, 16:15  –  Haus 9, Raum 2.09.014
Forschungsseminar Wahrscheinlichkeitstheorie

André de Oliveira Gomes

It is our intention to describe the first exit time problem for an ordinary differential equation (ODE) perturbed by additive noise. We consider a dynamical system described by the ODE <tex>$\dot X_t= - \nabla U(X_t)$</tex> in <tex>$R^d$</tex> with a unique asymptotically stable state, which means that, for any initial condition, the solution trajectories converge to the stable state. Now we perturb this differential equation, adding some small random noise, given by a stochastic process <tex>$ \varepsilon(\eta_t)_{t \geq 0}$</tex> and where <tex>$\varepsilon>0$</tex> is the small noise  intensity parameter that will be tuned to 0.  In this different situation, even when the source of noise vanishes, the stabe state becomes meta-stable and exits from its domain of attraction become possible. It is the intention  of our talk to present results concerning the law and the  expected value of the random variable that describes the first exit of the trajectories of the perturbed ODE in different scenarios. We will present the asymptotics, in the small noise limit, of the first exit time using large deviations estimates. This can be rephrased, in an informal way, as saying that the law and the expected value of the first exit time are exponentially small in the parameter that tunes the noise and that they are written in the small noise limit in terms of deterministic
quantities that depend on the geometry of the potential function <tex>$U$</tex>.

The first part of the talk will be introductory and it aims to describe well-known results in the literature of
this problem through motivating examples. We will focus in discussing simple examples of randomly perturbed dynamical systems. The second part of the talk is about research results of the PhD thesis of the speaker on this subject.