Power-law behaviors from the two-variable Langevin equation: Ito's and Stratonovich's Fokker-Planck equations

We study the power-law behaviors generated from the stochastically dynamical system governed by the well known two-variable Langevin equations. The stationary solutions of the corresponding Ito, Stratonovich and Zwanzig (backward Ito) Fokker-Planck equations are solved under a new fluctuation-dissipation relation; they are presented in a unified form of the power-law distributions with a power index containing two parameters kappa and sigma, where kappa measures the distance away from thermal equilibrium and sigma distinguishes the above three forms of the Fokker-Planck equations. The numerical calculations show that the Ito, the Stratonovich and the Zwanzig forms of the power-law distributions are all exactly the stationary solutions based on the two-variable Langevin equations.