title: 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.
authors.affiliation: Grup de Física Estadística, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain.
authors.forename: Vicenç
authors.initials: V
authors.lastname: Méndez
authors.affiliation: Department of General and Theoretical Physics, Sumy State University, Rimsky-Korsakov Street 2, UA-40007 Sumy, Ukraine.
authors.forename: S I
authors.initials: SI
authors.lastname: Denisov
authors.affiliation: Grup de Física Estadística, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain.
authors.forename: Daniel
authors.initials: D
authors.lastname: Campos
authors.affiliation: Department of Chemistry, Southern Methodist University, Dallas, Texas 75275-0314, USA.
authors.forename: Werner
authors.initials: W
authors.lastname: Horsthemke
mesh_terms: D008962:Models, Theoretical; D016011:Normal Distribution; D011789:Quantum Theory; D013269:Stochastic Processes