  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.
