Formalism

where

is a measure of the strength of the noise, and are independent identically distributed random variables with unit variance,

(3) |

such as a normalized Gaussian distribution. The methods used here are equally applicable to and that depend on

Instead of the Langevin form (1) it is more convenient to consider
the discrete Fokker-Planck equation for a probability distribution
transported by the dynamics and diffusing due to the noise:

(4) |

where is the noise kernel, for example

(5) |

reducing to a Dirac in the deterministic limit.

Long time properties of the dynamics are obtained from the leading
eigenvalue(s) of the linear evolution operator ,
which are
(the inverses of) solutions of the characteristic equation

For example, the probability of a point initially in an open system remaining there after

(7) |

Dynamical averages and diffusion coefficients can be obtained from the leading zero of appropriately weighted evolution operators [1,2,3].

The spectral determinant (6) of an infinite dimensional operator
may be defined by its cumulant expansion in powers of *z*, using the
matrix relation
and Taylor expanding the logarithm:

(9) |

which leads to the recursive equation

The trace is straightforward to write down as an *n*-dimensional integral,
a discrete periodic chain reminiscent of a path integral, obtained in
Ref. [13],

where the index