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 operator in this is generator of non-Gaussian Levy processes, either integrable or non-integrable jump kernels. examine limiting behavior as scaling parameter tends to zero, and derive reduced (local nonlocal) equation. In particular, work leads an data assimilation system noise, by examining corresponding Zakai show that probability density approximates original system.