  We discuss here in detail a new analytical random walk approach to
calculating the phase-diagram for spatially extended systems with
multiplicative noise. We use the Anderson localization problem as an example.
The transition from delocalized to localized states is treated as a generalized
diffusion with a noise-induced first-order phase transition. The generalized
diffusion manifests itself in the divergence of averages of wavefunctions
(correlators). This divergence is controlled by the Lyapunov exponent $\gamma$,
which is the inverse of the localization length, $\xi=1/\gamma$. The appearance
of the generalized diffusion arises due to the instability of a fundamental
mode corresponding to correlators. The generalized diffusion can be described
in terms of signal theory, which operates with the concepts of input and output
signals and the filter function. Delocalized states correspond to bounded
output signals, and localized states to unbounded output signals, respectively.
The transition from bounded to unbounded signals is defined uniquely by the
filter function $H(z)$.
