Langevin equation with scale - dependent noise

A new wavelet based technique for the perturbative solution of the Langevin equation is proposed. It is shown that for the random force acting in a limited band of scales the proposed method directly leads to a finite result with no renormalization required. The one-loop contribution to the Kardar-Parisi-Zhang equation Green function for the interface growth is calculated as an example. The Langevin equation is one of the most general approximations for the evolution of a dy-namical system affected by fluctuating environment. It arises in the description of magnetic at the presence of magnetic field fluctuations, in the description of hydrodynamic turbulence , in stochastic quantization problems, in the description of interface growth and in a large variety of other problems [1, 2, 3, 4]. In the most general form the Langevin equation can be casted as ∂φ(x, t) ∂t = U[φ(x, t)] + η(x, t), η(x)η(x ′) = D(x, x ′), (1) where U[φ] is the nonlinear interaction potential, η(t, x) is the Gaussian random noise, which standss for the fluctuations of the environment.The Minkovski-like (d+1)-dimensional notation x ≡ (x, t), k ≡ (k, ω) is used hereafter. The standard way to solve the Langevin equation (1) is to introduce the small parameter λ in the interaction potential U, and then solve the system iteratively in each order of the perturbative expansion. The averaging over the Gaussian random force η consists in evaluation of the pair correlators ηη. The procedure is simplified by the assumption of the Gaussian statistics of the random noise which allows one to take into account only even order correlators of the random noise: all terms containing the odd number of η are equal 1