Abstract: During this talk, I will give a survey of studies of stochastic processes related to a random perturbation of a nonlinear ordinary differential equation (ODE) satisfying the Peano phenomenon. Random perturbation produces a stochastic differential equation (SDE) having a nonlinear drift term and driven by Brownian motion, more generally, by a stable symmetric Levy process (pure jump). A number of models in physics, biology, and finance are based on this kind of SDE, and I will give some examples of models, questions, and results that can be obtained: existence and uniqueness, behavior under small random perturbations, and a time-inhomogeneous case for ODE. One can also look to some kinetic models, more precisely, the velocity of a mobile satisfies the SDE and its position is studied, for small random perturbations of the velocity or in large time. One can even go further and try to extend the model to SDEs with random drift term, (possibly varying with the time) this last situation being the Brownian (or Levy) motion in random environment.
Bio: Mihai Gradinaru earned his Ph.D. from the University of Paris-Sud (Orsay) and then he is an assistant professor (tenure) at the University of Nancy. He works on probability theory (stochastic processes and their analysis, limit theorems and large deviations) and its connections with other domains of mathematics.