Scaling limits of nonlinear functions of random grain model, with application to Burgers' equation

We study scaling limits of nonlinear functions $G$ of random grain model $X$ on ${\mathbb{R}}^d$ with long-range dependence and marginal Poisson distribution. Following Kaj et al (2007) we assume that the intensity $M$ of the underlying Poisson process of grains increases together with the scaling parameter $\lambda$ as $M={\lambda }^{\gamma }$, for some $\gamma >0$. The results are applicable to the Boolean model and exponential $G$ and rely on an expansion of $G$ in Charlier polynomials and a generalization of Mehler’s formula. Application to solution of Burgers’ equation with initial aggregated random grain data is discussed.