Properties of the noise-induced ‘‘spurious’’ drift. I.

Abstract

The coefficients of Fokker–Planck equations associated to Langevin equations (LE) may be interrelated, since both the diffusion matrix D and the noise-induced drift a are derived from the same coefficients of the LE. If D is regular and if furthermore its dimension M equals the number of independent noise sources (conditions to be dropped in the subsequent paper II), a is uniquely determined by D if M=1 and independent of D if M?3. For M=2, a splits into a nontensor part which is uniquely determined by D and a vector field with given divergence. The result for M?3 means that to any LE with noise terms specified by their covariance matrix only, there exists another stochastically equivalent LE with a fully arbitrary deterministic part. As a byproduct it is shown that any given a can be removed by a nonlinear change of the state variables.