About an Unsolved Problem of Stabilization by Noise for Difference Equations

Abstract

The paper is devoted to the effect of “stabilization by noise”. The essence of this effect is that an unstable deterministic system is stabilized by stochastic perturbations of sufficiently high intensity. The problem is that the effect of “stabilization by noise”, well-known already for more than 50 years for stochastic differential equations, still has no analogue for stochastic difference equations. Here, a corresponding hypothesis is formulated and discussed, the truth of which is illustrated and confirmed by numerical simulation of solutions of stochastic linear and nonlinear difference equations. However, a problem of a formal proof of this hypothesis remains open.

Keywords

• difference equation
• stochastic perturbations
• stabilization by noise
• stability in probability
• zero and nonzero equilibria
• numerical simulation