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

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 Lévy 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 Lévy 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.