We discuss the simple, randomly driven system dx/dt =-..mu..x-yx/sup 3/+f(t), where f(t) is a Gaussian random function or stirring force with =F delta(t-t'). We show how to obtain approximately the coefficients of the expansion of the equal-time Green's function as power series in (1/R)/sup n/, where R is the internal Reynolds number (F..gamma..)/sup 1/2//..mu.., by using a new expansion for the path integral representation of the generating functional for the correlation functions. Exploiting the fact that the action for the randomly driven system is related to that of a quantum mechanical anharmonic oscillator with Hamiltonian p/sup 2//2+m/sup 2/x/sup 2//2+..nu..x/sup 4/+lambdax/sup 6//2, we evaluate the path integral on a lattice by assuming that the lambdax/sup 6/ term dominates the action. This gives an expansion of the lattice theory Green's functions as power series in 1/(lambdaa)/sup 1/3/, where a is the lattice spacing. Using Pade approximants to extrapolate to a=0, we obtain the desired large-Reynolds-number expansion of the two-point function.