Long Asymptotic Correlation Time for Non-Linear Autonomous Itô's Stochastic Differential Equation

Itô's stochastic differential equations theory is a common approach to analysis of stochastic phenomena in various systems. In many applications, an important feature of the systems is the flicker effect. It is well known that it cannot be described with linear autonomous scalar equations of the above kind. The reason is that the flicker effect is usually associated with a correlation time which is much greater than the correlation time in the linear case. In the present work, we discuss modelling of the long correlation time with the help of non-linear autonomous scalar Itô's stochastic differential equation which includes non-linear drift. The expression for the asymptotic correlation time as time separation tends to zero is derived in terms of the equation. We formulate the condition for this time to be long in the above sense. It is pointed out that this condition can hold if the nonlinear damping is reduced compared to the linear case. These results are illustrated with an example of the equation with non-linear drift of a specific form.