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We study power-law behaviors produced from the stochastically dynamicalsystem governed by the well-known two-variable Langevin equations. Thestationary solutions of the corresponding Ito's, Stratonovich's and theZwanzig's (the backward Ito's) Fokker-Planck equations are solved under a newfluctuation-dissipation relation, which are presented in a unified form of thepower-law distributions with a power index containing two parameter kappa andsigma, where kappa measures a distance away from the thermal equilibrium andsigma distinguishes the above three forms of the Fokker-Planck equations. Thenumerical calculations show that the Ito's, the Stratonovich's and theZwanzig's form of the power-law distributions are all exactly the stationarysolutions based on the two-variable Langevin equations.

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