An exactly solvable linear model giving indication of stochastic resonance

The linear stochastic equation dx(beta)(t)/dt + [1 + f(beta)(t)]x(beta)(t) = Asin(Omega t) is discussed. The function f(beta)(t) is defined as a Poissonian noise dependent on a parameter beta > 0, f beta(t) = beta Sigma(j)[delta(t - t(j)(+)) - delta(t - t(j)(-))]. The mean frequency of the delta-pulses is chosen as beta-dependent in the form lambda(beta) = 2 gamma(beta(-2) + 1) exp(-beta) where gamma is a constant from the interval (0, 0.974). With the stochastic function f(beta)(t) defined in this way, attention is paid on the oscillational term of the averaged function [x(t)], [x(t)](osc) = (A) over bar sin(Omega t - alpha). It is found that the dependence (A) over bar = (A) over bar(beta) exhibits one maximum and one minimum. The occurrence of these extrema seems to affirm the presence of stochastic resonance.