  In view of solving convex optimization problems with noisy gradient input, we
analyze the asymptotic behavior of gradient-like flows under stochastic
disturbances. Specifically, we focus on the widely studied class of mirror
descent schemes for convex programs with compact feasible regions, and we
examine the dynamics' convergence and concentration properties in the presence
of noise. In the vanishing noise limit, we show that the dynamics converge to
the solution set of the underlying problem (a.s.). Otherwise, when the noise is
persistent, we show that the dynamics are concentrated around interior
solutions in the long run, and they converge to boundary solutions that are
sufficiently "sharp". Finally, we show that a suitably rectified variant of the
method converges irrespective of the magnitude of the noise (or the structure
of the underlying convex program), and we derive an explicit estimate for its
rate of convergence.
