"Langevin
equation with scale-dependent noise
M. V. Altaisky (IKI)
Abstract:
The Langevin equation ¶Y(x,t)
/ ¶t
= l
U(Y(x,t))
+ h(x,t)
is one of the most general approximations for a large variety of physical,
chemical, biological and other systems interacting with random environment.
The common way of the solution of the Langevin equation is the stochastic
perturbation theory, viz. a power expansion in small parameter
l
0
with the averaging over gaussian random force <
h(õ) h(x¢)
> = D(x-x¢)
d(t-t')
in each order of perturbation theory. This approach leads to loop
divergences, identical to those arising in quantum field theory, and
demands renormalization group technique to get rid of the divergences. In
the present contribution a new type of perturbation theory based on
wavelet transform is applied to the Langevin equation. In case of limited
band forcing h
the method directly gives finite results and no further renormalization is
required.