We study the non-local Poisson equation with a parameter and in the whole space \(\mathbb {R}^{d_1}\) . Regularities of the solution are obtained under mild assumptions on the coefficients. The result is then used to study the asymptotic behavior of the solution of a singularly perturbed non-local partial differential equation. As an application, we obtain the averaging principle for a fast-slow stochastic dynamical system with Hölder continuous coefficients and driven by multiplicative jump noise. Furthermore, the rate of convergence is also obtained.
