Quantum fractal noise: Time-autocorrelation functions

Starting from a distribution of time scales a quantum time-autocorrelation function (TAF) of a fluctuating force acting in a fractal environment is proposed as well as its associated mode density. The assumption of a stochastic Ornstein-Uhlenbeck process for every time scale gives rise to power laws for both the mode density and the TAF at frequencies and times of interest in the problem. Asymptotic analysis of the mode density and the TAF are made for different frequency and time ranges. The temperature dependence of the quantum TAF is also studied. Characteristic long-time tails appear in the low-temperature regime.