Stationary probability distribution for a particle subject to coloured noise

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Abstract
The stationary probability distribution, Pst, for an overdamped particle moving in a one-dimensional potential and subject to exponentially correlated noise having correlation time tau , is determined using several different methods. Firstly, the evaluation of a path-integral representation for a conditional probability distribution for small D, where D is the noise strength, is performed to show that Pst(x) approximately D-12/p(x) exp(-S(x)/D). The function S(x) and the prefactor p(x) are evaluated for various values of tau in the case of the double-well potential V(x)=-x2/2+x4/4. Secondly, a numerical simulation of the stochastic process is carried out directly to determine the validity of the small-D approximation. Excellent agreement is found for d1. Finally, an investigation of this region is carried out directly using a two-dimensional Fokker-Planck equation, which shows that the small-D expansion breaks down for tau >1 when mod x mod 1( tau +1/.