Linear response characteristics in time-dependent subdiffusive fractional Fokker-Planck equations

For the time-dependent fractional Fokker-Planck equations (FFPE), obtained from the time-independent FFPE by directly replacing the time-independent external force with a time-modulated one, we investigate its response characteristics within linear response range based on a viewpoint of global perturbation. Two general response formulas are derived in time domain and frequency domain for the subdiffusive process, and they can be reduced to the counterparts in normal diffusion when the diffusion exponent tends to 1. For two concrete examples, a Kramers-Kronig relation is also discussed and is found to have different meaning in both cases. We suggest the results may be important in exploring the phenomenon of stochastic resonance and calculating the spectral density of fluctuations.