Asymptotic phase for stochastic limit cycles

The asymptotic phase for a limit cycle (an isolated, closed periodic orbit of a nonlinear system of ordinary differential equations) is a map from the LC’s basin of attraction to the circle, that has a uniform rate of change under the flow of the ODE.  The phase function’s level sets, or isochrons, play a fundamental role in studies of synchronization and entrainment of coupled and driven oscillators.  The classical definition of the asymptotic phase breaks down when the dynamics are stochastic; moreover nonlinear systems that oscillate only when subjected to noise cannot easily be treated as small noise perturbations of a related deterministic LC system.  I will discuss two alternative definitions for the phase of an oscillator when the oscillator is noisy.  One, due to Schwabedal and Pikovsky, formulates isochrons as surfaces of constant mean first return time.  A second, due to Thomas and Lindner, identifies the asymptotic phase with the argument of the complex eigenfunction associated with the slowest decaying complex eigenvalues of the adjoint Kolmogorov operator.  The relation between these two generalizations of the phase remains an open question.