Path integrals for fractional Brownian motion and fractional Gaussian noise

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.

Path integration method for stochastic responses of differential equations under Lévy white noise

A path integration (PI) approach that is progressive for studying the stochastic response driven by Lévy white noise is presented. First, a probability mapping is constructed, which decouples the domain of interest for the system state and the probability space derived from the randomness of Lévy white noise within a short time interval. Then, solving the probability mapping yields the short-time response of the system. Finally, the stochastic evolution of the system can be grasped in a stepwise manner based on the fundamental concept of the PI method. The applicability and effectiveness of our approach in addressing the transient and stationary responses under Lévy white noises are verified by Monte Carlo simulation results. Moreover, the advances in utilization of this method are that it eliminates the restriction of the previous PI method on the controlling parameter of Lévy white noises, and it is highly efficient for solving responses of systems under Lévy white noises.

First-passage area distribution and optimal fluctuations of fractional Brownian motion

We study the probability distribution P(A) of the area A=∫_{0}^{T}x(t)dt swept under fractional Brownian motion (fBm) x(t) until its first passage time T to the origin. The process starts at t=0 from a specified point x=L. We show that P(A) obeys exact scaling relation P(A)=D^{1/2H}/L^{1+1/H}Φ_{H}(D^{1/2H}A/L^{1+1/H}), where 0 Collapse

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Affiliation(s)

Alexander K Hartmann Institut für Physik, Universitåt Oldenburg - 26111 Oldenburg, Germany
Baruch Meerson Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel

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Anomalous diffusion, nonergodicity, non-Gaussianity, and aging of fractional Brownian motion with nonlinear clocks

How do nonlinear clocks in time and/or space affect the fundamental properties of a stochastic process? Specifically, how precisely may ergodic processes such as fractional Brownian motion (FBM) acquire predictable nonergodic and aging features being subjected to such conditions? We address these questions in the current study. To describe different types of non-Brownian motion of particles-including power-law anomalous, ultraslow or logarithmic, as well as superfast or exponential diffusion-we here develop and analyze a generalized stochastic process of scaled-fractional Brownian motion (SFBM). The time- and space-SFBM processes are, respectively, constructed based on FBM running with nonlinear time and space clocks. The fundamental statistical characteristics such as non-Gaussianity of particle displacements, nonergodicity, as well as aging are quantified for time- and space-SFBM by selecting different clocks. The latter parametrize power-law anomalous, ultraslow, and superfast diffusion. The results of our computer simulations are fully consistent with the analytical predictions for several functional forms of clocks. We thoroughly examine the behaviors of the probability-density function, the mean-squared displacement, the time-averaged mean-squared displacement, as well as the aging factor. Our results are applicable for rationalizing the impact of nonlinear time and space properties superimposed onto the FBM-type dynamics. SFBM offers a general framework for a universal and more precise model-based description of anomalous, nonergodic, non-Gaussian, and aging diffusion in single-molecule-tracking observations.