Stochastic methods in molecular dynamics.

Summary This thesis presents two independent research topics concerning the application of stochastic methods to problems arising from molecular dynamics. In the first part, we present work related to the probabilistic interpretation of the Poisson-Boltzmann equation which is used to describe the electrostatic potential of a molecular system. After introducing the Poisson-Boltzmann equation and the main mathematical tools used, we focus on the linear parabolic Poisson-Boltzmann equation. Before stating the main result of the thesis, we extend the existence and uniqueness results of stochastic backward differential equations. We then give a probabilistic interpretation of the nonlinear Poisson-Boltzmann equation in the form of the solution of a backward stochastic differential equation. Finally, in a second prospective part, we start the study of a method proposed by Paul Malliavin for the detection of slow and fast variables of a molecular dynamics.