Trace Formula for Noisy Flows

A trace formula is derived for the Fokker–Planck equation associated with Itô stochastic differential equations describing noisy time-continuous nonlinear dynamical systems. In the weak-noise limit, the trace formula provides estimations of the eigenvalues of the Fokker–Planck operator on the basis of the Pollicott–Ruelle resonances of the noiseless deterministic system, which is assumed to be non-bifurcating. At first order in the noise amplitude, the effect of noise on a periodic orbit is given in terms of the period and the derivative of the period with respect to the pseudo-energy of the Onsager–Machlup–Freidlin–Wentzell scheme.