Time-dependent probability density function in cubic stochastic processes

We report time-dependent Probability Density Functions (PDFs) for a nonlinear stochastic process with a cubic force by novel analytical and computational studies. Analytically, a transition probability is formulated by using a path integral and is computed by the saddle-point solution (instanton method) and a new nonlinear transformation of time. The predicted PDF p(x, t) is in general given as a time integral and useful PDFs with explicit dependence on x and t are presented in certain limits (e.g. in the short and long time limits). Numerical simulations of the FokkerPlanck equation provide exact time evolution of the PDFs and confirm analytical predictions in the limit of weak noise. In particular, we show that non-equilibrium PDFs behave drastically differently from the stationary PDFs in regards to the asymmetry (skewness) and kurtosis. Specifically, while stationary PDFs are symmetric, transient PDFs are skewed; transient PDFs are much broader than stationary PDFs, with the kurtosis larger and smaller than 3, respectively. We elucidate the effect of nonlinear interaction on the strong fluctuations and intermittency in relaxation process.