Role of the interpretation of stochastic calculus in systems with cross-correlated Gaussian white noises

We derive the Fokker-Planck equation for multivariable Langevin equations with cross-correlated Gaussian white noises for an arbitrary interpretation of the stochastic differential equation. We formulate the conditions when the solution of the Fokker-Planck equation does not depend on which stochastic calculus is adopted. Further, we derive an equivalent multivariable Ito stochastic differential equation for each possible interpretation of the multivariable Langevin equation. To demonstrate the usefulness and significance of these general results, we consider the motion of Brownian particles. We study in detail the stability conditions for harmonic oscillators with two white noises, one of which is additive, random forcing, and the other, which accounts for fluctuations of either the damping or the spring coefficient, is multiplicative. We analyze the role of cross correlation in terms of the different noise interpretations and confirm the theoretical predictions via numerical simulations. We stress the interest of our results for numerical simulations of stochastic differential equations with an arbitrary interpretation of the stochastic integrals.

Figure 1

Mean particle position in the steady state as a function of the parameter ${\lambda }_1$ for different values of the parameter $r$. The solid lines represent the theoretical result (4.4), and the symbols show the simulation data. Both theoretical and numerical results are obtained for $D_1=D_2=0.25$, $m=b=k=x_0=1$, and $v_0=0$.

Figure 2

Second moment of the particle position in the steady state as a function of the noise intensity $D_1$ for different values of the parameter ${\lambda }_1$. The solid lines are obtained from (4.9), and the symbols indicate the simulation results. The parameters are $D_2=0.25$, $b=0.5$, $m=k=x_0=1$, and $r=0$, $v_0=0$. According to (4.7), $D_1^{\left(2\right)}=\frac{4}{3},1$, and $\frac{2}{3}$ for ${\lambda }_1=0.25,0.5$, and 1, respectively.

Figure 3

Mean particle position as a function of time. The solid line represents the theoretical result (4.12), and the symbols show the results obtained from the numerical solution of (3.3) with $h=kx$ and the parameters $D_2=0.25$, $b=0.5$, $m=k=x_0=1$, and $r=v_0=0$.

Figure 4

Second moment of the particle position in the steady state as a function of the noise intensity $D_1$. The solid line represents the theoretical result (4.13), and the symbols indicate the simulation results obtained by numerical solution of (3.3) with $h=kx$. In accordance with our theoretical prediction, the latter do not depend on the parameter ${\lambda }_1$. The simulation parameters are chosen to be the same as in Fig. 3.