Some Averaging and Stability Results for Random Differential Equations

This paper concerns differential equations which contain strongly mixing random processes (processes for which the “past” and the “future” are asymptotically independent). When the “rate” of mixing is rapid relative to the rate of change of the solution process, information about the behavior of the solution is obtained. Roughly, the results fall into three categories: 1. Quite generally, the solution process is well approximated by a deterministic trajectory, over a finite time interval. 2. For more restricted systems, this approximation extends to the infinite interval [ 0,∞ ),3. Conditions for the asymptotic stability of x = AX, where A is an n × n matrix-valued random process, are obtained.