Fokker–Planck equation and subdiffusive fractional Fokker–Planck equation of bistable systems with sinks

We study the dynamics of stochastic systems obeying the one-dimensional diffusive Fokker–Planck equation (FPE), as well as systems described by the subdiffusive fractional Fokker–Planck equation (SFFPE), with a confining potential U(x) and in the presence of delta-function sinks. For the one-sink and two-sink problems, we obtain exact expressions for the Laplace transform of the propagator P, and derive several asymptotic results for the survival probability Sp and the survival-time distribution f. The decay rate constants of the diffusive system are also analyzed. We apply our method to a bistable system with a quartic double-well potential, and calculate P, Sp and f of the corresponding FPE and SFFPE, for different strengths and positions of the sink(s). Finally, we derive asymptotic expressions of Sp and f for the subdiffusive system, which are valid for a general U(x) and a general sink distribution.