Effective reduction for a nonlocal Zakai stochastic partial differential equation in data assimilation

TL;DR

This paper investigates the limiting behavior of a nonlocal Zakai stochastic PDE with Lévy noise, deriving effective reduced equations that approximate the original system's probability density in data assimilation.

Contribution

It introduces a novel reduction method for nonlocal stochastic PDEs with Lévy processes, applicable to data assimilation systems with jump noise.

Findings

Derived effective local and nonlocal equations as limits of the original PDEs.

Proved the probability density of the reduced system approximates that of the original.

Established the reduction's validity for systems with integrable and non-integrable jump kernels.

Abstract

We study the effective reduction for a nonlocal stochastic partial differential equation with oscillating coefficients. The nonlocal operator in this stochastic partial differential equation is the generator of non-Gaussian L\'{e}vy processes, with either \textbf{integrable} or \textbf{non-integrable} jump kernels. We examine the limiting behavior of this equation as a scaling parameter tends to zero, and derive a reduced (local or nonlocal) effective equation. In particular, this work leads to an effective reduction for a data assimilation system with L\'{e}vy noise, by examining the corresponding nonlocal Zakai stochastic partial differential equation. We show that the probability density for the reduced data assimilation system approximates that for the original system.