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 $[ {0,\infty } ),3$. Conditions for the asymptotic stability of $\dot x = AX$, where A is an $n \times n$ matrix-valued random process, are obtained.

[1]  M. Rosenblatt.  A CENTRAL LIMIT THEOREM AND A STRONG MIXING CONDITION. , 1956, Proceedings of the National Academy of Sciences of the United States of America.

[2]  N. G. Parke,et al.  Ordinary Differential Equations. , 1958 .

[3]  V. Volkonskii,et al.  Some Limit Theorems for Random Functions. II , 1959 .

[4]  Extension of the averaging method to stochastic equations , 1966 .

[5]  R. Has’minskiĭ.  On Stochastic Processes Defined by Differential Equations with a Small Parameter , 1966 .

[6]  R. Khas'minskii.  A Limit Theorem for the Solutions of Differential Equations with Random Right-Hand Sides , 1966 .

[7]  E. F. Infante,et al.  On the Stability of Some Linear Nonautonomous Random Systems , 1968 .

[8]  P. Billingsley,et al.  Convergence of Probability Measures , 1969 .

[9]  M. T. Wasan.  Stochastic Approximation , 1969 .

[10]  M. Sain.  Finite dimensional linear systems , 1972 .

[11]  Richard O. Duda,et al.  Pattern classification and scene analysis , 1974, A Wiley-Interscience publication.

[12]  Benjamin S. White.  Some limit theorems for stochastic delay-differential equations , 1976 .

[13]  H. Kushner.  Convergence of recursive adaptive and identification procedures via weak convergence theory , 1977 .

[14]  G. Blankenship,et al.  Stability of linear differential equations with random coefficients , 1977 .

[15]  L. Ljung.  Strong Convergence of a Stochastic Approximation Algorithm , 1978 .

[16]  G. Papanicolaou,et al.  Stability and Control of Stochastic Systems with Wide-band Noise Disturbances. I , 1978 .