Incorporating memory effect into a fractional stochastic diffusion particle tracking model for suspended sediment using Malliavin-Calculus-based fractional Brownian Motion

Abstract

This study aims to develop a Fractional Stochastic Diffusion Particle Tracking Model (FSDPTM) to illustrate suspended sediment’s long-term dependence (LTD) trajectories immersed in fully developed turbulent flow. Carried by eddies in the open channel, individual suspended sediment behavior exhibits dependence on its past increments. The Fractional Brownian Motion (FBM) is incorporated into the FSDPTM to account for the randomness of turbulent flow and to include the time-dependent increments as a memory effect. Additionally, to enhance the capacity for differentiating FBM, this study introduces Malliavin Calculus, the stochastic calculus of variation, to construct the square -differentiable FBM. For the numerical realizations of square differentiable FBM, the integration of fractional Gaussian white noise and Wiener–Ito Chaos expansion with the Hermite sequence is employed. The FSDPTM offers probabilistic LTD trajectories as solutions to the Ito-type Langevin equation. It also provides particle velocity, acceleration, and ensemble statistics, collectively demonstrating the application of FBM to sediment transport modeling. In conclusion, the key feature of the FSDPTM is its ability to translate LTD derived from FBM into the trajectory of suspended sediment particles in fully developed turbulent flow. Additionally, it expands the solutions from non-differentiable to square-differentiable stochastic processes.