The dynamics of Brownian particle in the nonlinear potential of the form $V(x)=\frac{1}{2}x^2+\frac{\beta}{3}x^3+\frac{\delta}{4}x^4$ driven by white noise is carried out by use of the matrix continued fraction expansion technique. We obtain the dephasing relaxation time τ of position correlation functions and their spectral density function. The results show that τ becomes smaller as the friction coefficient $\gamma$ increases for 0.5 ≤ $\gamma$ ≤ 2.0, which correspondes to underdamping region. The dependence of τ on the nonlinear parameter β and δ are presented, too. We next developed a numerical method which compute time dependent solutions of two-dimensional Fokker-Planck equation for the case of colored noise. The numerically obtained stationary state solutions show that they become shifted from the Boltzmann distribution function and the degree of the deviation increases as correlation time τ increases; such a property is in agreement with what the unified theory and Risken's numerical results using matrix continued fraction method have predicted.
