The Porous Medium Equation: Multiscale Integrability in Large Deviations

TL;DR

This paper establishes a large deviation principle for a zero-range process converging to the porous medium equation, introducing a multiscale method to handle nonlinearities without standard regularity assumptions.

Contribution

It develops a novel multiscale approach to prove large deviations for nonlinear PDE limits of particle systems, overcoming regularity challenges.

Findings

Large deviation principle established for the zero-range process.

Multiscale argument exploits pathwise regularity across scales.

Uniform integrability estimates obtained for nonlinearities.

Abstract

We consider a zero-range process $\eta^N_t(x)$ with superlinear local jump rate, which in a hydrodynamic-small particle rescaling converges to the porous medium equation $\partial_t u=\frac12\Delta u^\alpha, \alpha>1$. As a main result we obtain a large deviation principle in any scaling regime of vanishing particle size $\chi_N\to 0$. The key challenge is to develop uniform integrability estimate on the nonlinearity $(\eta^N(x))^\alpha$ in a situation where neither pathwise regularity nor Dirichlet-form based regularity is readily available. We resolve this by introducing a novel multiscale argument exploiting the appearance of pathwise regularity across scales.