Contraction analysis of nonlinear random dynamical systems

In order to bring contraction analysis into the very fruitful and topical fields of stochastic and Bayesian systems, we extend here the theory describes in [Lohmiller and Slotine, 1998] to random differential equations. We propose new definitions of contraction (almost sure contraction and contraction in mean square) which allow to master the evolution of a stochastic system in two manners. The first one guarantees eventual exponential convergence of the system for almost all draws, whereas the other guarantees the exponential convergence in L2 of to a unique trajectory. We then illustrate the relative simplicity of this extension by analyzing usual deterministic properties in the presence of noise. Specifically, we analyze stochastic gradient descent, impact of noise on oscillators synchronization and extensions of combination properties of contracting systems to the stochastic case. This is a first step towards combining the interesting and simplifying properties of contracting systems with the probabilistic approach. Key-words: random differential equations, contraction theory ∗ Inria, Nantes, France † Non Linear System Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139, USA ha l-0 08 64 07 9, v er si on 1 20 S ep 2 01 3 Analyse de la contraction pour les systèmes dynamique alétoires non-linéaires Résumé : In order to bring contraction analysis into the very fruitful and topical fields of stochastic and Bayesian systems, we extend here the theory describes in [Lohmiller and Slotine, 1998] to random differential equations. We propose new definitions of contraction (almost sure contraction and contraction in mean square) which allow to master the evolution of a stochastic system in two manners. The first one guarantees eventual exponential convergence of the system for almost all draws, whereas the other guarantees the exponential convergence in L2 of to a unique trajectory. We then illustrate the relative simplicity of this extension by analyzing usual deterministic properties in the presence of noise. Specifically, we analyze stochastic gradient descent, impact of noise on oscillators synchronization and extensions of combination properties of contracting systems to the stochastic case. This is a first step towards combining the interesting and simplifying properties of contracting systems with the probabilistic approach. Mots-clés : équations différentielles aléatoires, théorie de la contraction ha l-0 08 64 07 9, v er si on 1 20 S ep 2 01 3 Contraction analysis of nonlinear random dynamical systems 3