Phase descriptions of a multidimensional Ornstein-Uhlenbeck process

Abstract

Stochastic oscillators play a prominent role in different fields of science. Their simplified description in terms of a phase has been advocated by different authors using distinct phase definitions in the stochastic case. One notion of phase that we put forward previously, the \emph{asymptotic phase of a stochastic oscillator}, is based on the eigenfunction expansion of its probability density. More specifically, it is given by the complex argument of the eigenfunction of the backward operator corresponding to the least negative eigenvalue. Formally, besides the `backward' phase, one can also define the `forward' phase as the complex argument of the eigenfunction of the forward Kolomogorov operator corresponding to the least negative eigenvalue. Until now, the intuition about these phase descriptions has been limited. Here we study these definitions for a process that is analytically tractable, the two-dimensional Ornstein-Uhlenbeck process with complex eigenvalues. For this process, (i) we give explicit expressions for the two phases; (ii) we demonstrate that the isochrons are always the spokes of a wheel, but that (iii) the spacing of these isochrons (their angular density) is different for backward and forward phases; (iv) we show that the isochrons of the backward phase are completely determined by the deterministic part of the vector field, whereas the forward phase also depends on the noise matrix; and (v) we demonstrate that the mean progression of the backward phase in time is always uniform, whereas this is not true for the forward phase except in the rotationally symmetric case. We illustrate our analytical results for a number of qualitatively different cases.