# Random Perturbations of some Non-linear Dynamical Systems: Interpretation and Large Time Behavior

During this talk I will tempt to made a survey of some studies of stochastic processes related to a random perturbation of a non-linear ordinary differential equation satisfying the so called Peano phenomenon. Random perturbation produces a stochastic differential equation (SDE) having a non-linear drift term and driven by a Brownian motion, more general, by a stable symmetric Levy processes (pure jump). A number of models in physics, biology or finance are based on these kind of SDE’s and I will try to give some examples of models, of questions and of results which could be obtained: existence and uniqueness, behavior under small random perturbations, 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 SDE’s with random drift term, (possibly varying with the time) this last situation being the so called Brownian (or Levy) motion in random environment.

Mihai Gradinaru earned his PhD from the University of Paris-Sud (Orsay) in 1995 and then he has been Assistant professor (tenure) at the University of Nancy. Mihai Gradinaru arrived to the University of Rennes 1 as a Professor of Mathematics in 2007. Professor Gradinaru works on probability theory (stochastic processes and their analysis, limit theorems and large deviations) and its connections with other domains of mathematics (partial differential equations, dynamical systems). He is mainly interested in the time evolution of stochastic processes. Mihai Gradinaru was Ph.D. adviser of three students and he is currently the head of group of Stochastic Processes of the Institut de Recherches Mathematiques of Rennes.