Stochastic representation of subdiffusion processes with time-dependent drift

Abstract

In statistical physics, subdiffusion processes are characterized by certain power-law deviations from the classical Brownian linear time dependence of the mean square displacement. For the mathematical description of subdiffusion, one uses fractional Fokker–Planck equations. In this paper we construct a stochastic process, whose probability density function is the solution of the fractional Fokker–Planck equation with time-dependent drift. We propose a strongly and uniformly convergent approximation scheme which allows us to approximate solutions of the fractional Fokker–Planck equation using Monte Carlo methods. The obtained results for moments of stochastic integrals driven by the inverse α -stable subordinator play a crucial role in the proofs, but may be also of independent interest.