  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. 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.
