Itô versus Stratonovich white-noise limits for systems with inertia and colored multiplicative noise

We consider the dynamics of systems in the presence of inertia and colored multiplicative noise. We study the limit where the particle relaxation time and the correlation time of the noise both tend to zero. We show that the limiting equation for the particle position depends on the magnitude of the particle relaxation time relative to the noise correlation time. In particular, the limiting equation should be interpreted either in the Itô or Stratonovich sense, with a crossover occurring when the two fast-time scales are of comparable magnitude. At the crossover the limiting stochastic differential equation is neither of Itô nor of Stratonovich type. This means that, after adiabatic elimination, the governing equations have different drift fields, leading to different physical behavior depending on the relative magnitude of the two fast-time scales. Our findings are supported by numerical simulations.

Figure 1

Difference between the solution of the limiting equation and the equation of motion for $\gamma =1$, for $ϵ=0.1$.

Figure 2

Difference between the solution of the limiting equation and the equation of motion for $\gamma =2$, for $ϵ=0.1$.

Figure 3

Difference between the solution of the limiting equation and the equation of motion for $\gamma =3$, for $ϵ=0.1$.

Figure 4

Convergence rate.