Contraction analysis of nonlinear random dynamical systems

Understand

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 \cite{Lohmiller98} 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 $L_2$ of to a unique trajectory.
• We then illustrate the relative simplicity of this extension by analyzing usual deterministic properties in the presence of noise.