A study of anomalous stochastic processes via generalizing fractional calculus

Due to the very importance of fractional calculus in studying anomalous stochastic processes, we systematically investigate the existing formulation of fractional calculus and generalize it to broader applied contexts. Specifically, based on the improved Riemann-Liouville fractional calculus operators and the modified Maruyama's notation for fractional Brownian motion, we develop the fractional Ito^'s calculus and derive a generalized Fokker-Planck equation corresponding to the Maruyama's process, along with which, the stochastic realizations of trajectories, both underdamped and overdamped, have been studied in terms of the stochastic dynamics equations newly formulated. This paves a way to study the path integrals and the stochastic thermodynamics of anomalous stochastic processes. We also explicitly derive several fundamental results in fractional calculus, including the relation between fractional and normal differentiation, the Laplace transform for fractional derivatives, the analytic solution of one type of generalized diffusion equations, and the fractional integration formulas. Our results advance the existing fractional calculus and provide practical references for studying anomalous diffusion, mechanics of memory materials in engineering, and stochastic analysis in fractional orders.