Wasserstein Contraction of Stochastic Nonlinear Systems

Abstract

We suggest that the tools of contraction analysis for deterministic systems can be applied towards studying the convergence behavior of stochastic dynamical systems in the Wasserstein metric. In particular, we consider the case of Ito diffusions with identical dynamics but different initial condition distributions. If the drift term of the diffusion is contracting, then we show that the Wasserstein distance between the laws of any two solutions can be bounded by the Wasserstein distance between the initial condition distributions. Dependence on initial conditions exponentially decays in time at a rate governed by the contraction rate of the noise-free dynamics. The choice of the Wasserstein metric affords several advantages: it captures the underlying geometry of the space, can be efficiently estimated from samples, and advances a viewpoint which begins to bridge the gap between somewhat distinct areas of the literature.