Gaussian perturbations of circle maps: a spectral approach (Q2389606)

Let \(f\) be a sufficiently smooth circle map with a finite number of periodic orbits that attract all others orbits of the deterministic DS generated by \(f\), i.e. \(x_{n+1}=f(x_n)\). The author consider the associated Gaussian perturbation system, i.e. \[ X_{n+1}^{\varepsilon}= f(X_{n}^{\varepsilon})+\varepsilon \sigma(X_{n}^{\varepsilon})\chi_n \mod {2\pi}, \] where \((\chi_n)\) is sequence of i.i.d. standard normal random variables and \(\sigma\) is sufficiently smooth and positive. Let \(T^{\varepsilon}\) the transition operator of the Markov chain \((X_{n}^{\varepsilon})\). The author proves that the limiting eigenvalues (as \(\varepsilon\to 0\)) of \(T^{\varepsilon}\) are determined by the derivative of \(f\) along periodic orbits and the corresponding limiting eigenvectors are related to the Hermite functions. Moreover, the author gives a detailed analysis of the case when \(f\) has one stable fixed point and one unstable fixed point.