Theory of eigenvalues for periodic non-stationary Markov processes: the Kolmogorov operator and its applications

We present an eigenvalue theory to study the stochastic dynamics of non-stationary time-periodic Markov processes. The analysis is carried out by solving an integral operator of the Fredholm type, i.e. considering complex-valued functions fulfilling the Kolmogorov compatibility condition. We show that the asymptotic behaviour of the stochastic process is characterized by the smaller time-scale associated with the spectrum of the Kolmogorov operator. The presence of time-periodic elements in the evolution equation for the semigroup leads to a Floquet analysis. The first non-trivial Kolmogorov's eigenvalue is interpreted from a physical point of view. This non-trivial characteristic time-scale strongly depends on the interplay between the stochastic behaviour of the process and the time-periodic structure of the Fokker-Planck equation for continuous processes, or the periodically modulated master equation for discrete Markov processes. We present pedagogical examples in a finite-dimensional vector space to calculate the Kolmogorov characteristic time-scale for discrete Markov processes