A Non-local Fokker-Planck Equation with Application to Probabilistic Evaluation of Sediment Replenishment Projects

We analyze a new mathematical model to evaluate performance of recent sediment-based environmental restoration projects from a stochastic viewpoint along with applications. We focus on unique jump-driven non-smooth dynamics governing streamflows and sediment storage subject to impulsive human interventions to replenish the sediment. We derive a non-local Fokker-Planck equation (FPE) governing the probability density of the coupled streamflow-sediment dynamics, which is a unique hyperbolic integro-partial differential equation subject to non-smooth coefficients and non-local boundary conditions. It admits measure-valued solutions. We propose a simple conservative discretization method of the FPE and verify it against Monte-Carlo simulation. The stationary probability density turns out to be singular along a boundary due to the non-smoothness and non-locality, which are effectively captured by the proposed numerical scheme. Based on public and experimental data, we finally apply the proposed model to numerical evaluation of replenishment strategies from a viewpoint of nuisance algae bloom.