  This paper is motivated by the problem of quantitatively bounding the
convergence of adaptive control methods for stochastic systems to a stationary
distribution. Such bounds are useful for analyzing statistics of trajectories
and determining appropriate step sizes for simulations. To this end, we extend
a methodology from (unconstrained) stochastic differential equations (SDEs)
which provides contractions in a specially chosen Wasserstein distance. This
theory focuses on unconstrained SDEs with fairly restrictive assumptions on the
drift terms. Typical adaptive control schemes place constraints on the learned
parameters and their update rules violate the drift conditions. To this end, we
extend the contraction theory to the case of constrained systems represented by
reflected stochastic differential equations and generalize the allowable
drifts. We show how the general theory can be used to derive quantitative
contraction bounds on a nonlinear stochastic adaptive regulation problem.
