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

Abstract

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.

Rotational properties of ferromagnetic nanoparticles driven by a precessing magnetic field in a viscous fluid

“…(3.13) on λ 2 is rather surprising (because the noise µ 2 is multiplicative). We note in this context that, in contrast to the univariate case, there always exists a class of multivariate Langevin equations with multiplicative Gaussian white noises whose interpretation does not influence the corresponding Fokker-Planck equations [38]. The above results show that Eqs.…”

“…is a constant column vector with components U 1 and U 2 , the straightforward calculations [38] lead to the following Fokker-Planck equation:…”

Stochastic thermodynamics of holonomic systems

“…[81]). It is interesting to observe that the theory can be generalized to take into consideration the possible cross-correlation of the noises [82,83].…”

Hierarchies of Landau-Lifshitz-Bloch equations for nanomagnets: A functional integral framework

“…In practice we find that it is possible to generate a sufficient number of noise configurations so that the map induced by the stochastic equations, as the result of this averaging procedure, realizes the exact statistical average over the noise [40,41]. These averages define, therefore, our reference model.…”