Multiplicative stochastic resonance in linear systems: Analytical solution

Abstract

The maximum of a signal as a function of the noise intensity (“stochastic resonance”) is found in a linear system subjected to multiplicative colour noise. This result is obtained for an arbitrary dichotomous noise and for a colour noise with a short autocorrelation time. Stochastic resonance does not occur for Gaussian white noise.

Linear systems with adiabatic fluctuations

“…So, to summarize, we point out that the implementation of Bourret's decoupling approximation is a major step for almost any treatment of multiplicative noise upto date [2,7,[11][12][13][14][15]. This is because of the fact that the average of a product of stochastic quantities does not factorize into the product of averages, although it has been argued that [2,7,[11][12][13][14][15][16][17][18][19][20][21] good approximations can be derived by assuming such factorization.…”

“…This is because of the fact that the average of a product of stochastic quantities does not factorize into the product of averages, although it has been argued that [2,7,[11][12][13][14][15][16][17][18][19][20][21] good approximations can be derived by assuming such factorization. In the case of fast fluctuations it has been justified if the driving stochastic noise has a fast correlation time such that Kubo number α 2 τ c is very small in the cumulant expansion scheme ( an expansion in ατ c ).…”

“…The slow fluctuations characterized by very long correlation time have also attracted a lot of attention of various workers over the years [2][3][4]7,8,11,14,15]. While the overwhelming majority of the treatment of stochastic differential equation with fast fluctuations is based on the assumption that there is a very short auto-correlation time τ c , such that one can adopt the scheme of expansion in ατ c , suitable simplifying approximation for dealing with very long correlation-time is relatively scarce.…”

“…In general, the problem of long correlation time is theoretically handled at the expense of severe restriction on the type of stochastic behavior. For instance, several authors [2][3][4]7,8,11,14,15] have tried the linear and nonlinear models within the framework of Markov processes of the type dichotomic processes, two-state Markov processes, random telegraphic processes etc. Our aim here is to explore a perturvative method for finding an equation for the average solution pertaining to the separation of timescale implied in the inequality…”

“…Linear multiplicative noise ( Eq. (2) ) has got wide application in studying the random Markov process [7], fluctuating barrier crossing [8], enzymatic kinetics in biology [9], nuclear magnetic resonance in physics [10] and stochastic resonance in linear system [11] and in many other context [12].…”