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

TL;DR

This paper investigates the scaling behavior of nonlinear functions of a long-range dependent random grain model, with applications to the Burgers' equation, using polynomial expansions and generalized formulas.

Contribution

It introduces a novel analysis of nonlinear transformations of the random grain model under scaling, extending previous work with new polynomial expansion techniques and applications.

Findings

Derived scaling limits for nonlinear functions of the model

Applied results to solutions of Burgers' equation with random initial data

Extended mathematical tools for long-range dependent spatial models

Abstract

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.