A class of Langevin equations with non Markovian gaussian noises

This seminar proposes a generalized Langevin equation for a classical mechanical system imbedded in a reservoir. Firstly, a finite number n of particles is considered in the reservoir and the action of this reservoir on the small system is described by a memory kernel and a zero-mean gaussian process of a very general nature, based on the gaussian transform introduced in [2]. This gives raise to an integrodifferential equation for the evolution of a generic particle in the main system. The existence of a unique solution X_n of the above equation is proved: it is also a gaussian process. A thermodynamics limit when n goes to infinity is considered and it is proved that X_n converges in distribution to the solution of a non-Markovian evolution equation. The above class of equations includes as a particular case the one studied by Kupferman in [1].

This report is based on a joint research with Carlos Lizama from the University of Santiago.

[1] R. Kupferman. Fractional kinetics in Kac--Zwanzig heat bath models. Journal of Statistical Physics, Jan 2004.

[2] C. Lizama and R. Rebolledo. Comm. on Stochastic Analysis, 4(4):541--551, 2010.