Invariant densities for noisy maps

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Abstract
The invariant density of discrete dynamical systems in the presence of weak Gaussian noise is studied both by means of path integrals and a WKB method. The leading order of the invariant density in the noise strength defines a generalized potential that is a Lyapunov function of the deterministic map. For this generalized potential an efficient functional recursion relation is established and applied to a variety of different maps. The typical singular behavior of the generalized potential in the neighborhood of an unstable fixed point is studied in detail. We show that noise-dependent corrections to the leading order restore the smoothness of the invariant density.