Stochastic Landau equation with time-dependent drift

The stochastic differential equation τ₀{{{∂}}}ₜA=ε(t)A-{g}₃A³+f{}(t), where f{}(t) is Gaussian white noise, is studied for arbitrary time dependence of {ε}(t). In particular, cases are considered where {ε}(t) goes through the bifurcation of the deterministic system, which occurs at {ε}=0. In the limit of weak noise an approximate analytic expression generalizing earlier work of Suzuki [Phys. Lett. A 67, 339 (1978); Prog. Theor. Phys. (Kyoto) Suppl. 64, 402 (1978)] is obtained for the time-dependent distribution function P(A,t). The results compare favorably with a numerical simulation of the stochastic equation for the case of a linear ramp (both increasing and decreasing) and for a periodic time dependence of {ε}(t). The procedure can be generalized to an arbitrary deterministic part ∂ₜA=D(A,t)+f{}(t), but the deterministic equation may then have to be solved numerically.