The Porous Medium Equation: Multiscale Integrability in Large Deviations

We consider a zero-range process $η^N_t(x)$ with superlinear local jump rate, which in a hydrodynamic-small particle rescaling converges to the porous medium equation $\partial_t u=\frac12Δu^α, α>1$. As a main result we obtain a large deviation principle in any scaling regime of vanishing particle size $χ_N\to 0$. The key challenge is to develop uniform integrability estimate on the nonlinearity $(η^N(x))^α$ 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.

💡 Research Summary

The paper addresses the large‑deviation principle (LDP) for a zero‑range interacting particle system whose macroscopic limit is the porous‑medium equation (PME) $\partial_t u=\frac12\Delta u^{\alpha}$ with $\alpha>1$. The microscopic dynamics is a zero‑range process $\eta^N_t(x)$ on the discrete torus $T^d_N$, with a super‑linear local jump rate $g(k)=dk^{\alpha}$. After a hydrodynamic‑small‑particle rescaling introduced in earlier works, the empirical density converges to the PME. Existing large‑deviation results for this model required a restrictive scaling relation between the particle size $χ_N$ and the lattice size $N$, namely $χ_N N^2\lesssim1$, which guarantees that the variance of discrete gradients stays bounded. This restriction excludes the regime where $χ_N$ decays slowly relative to $N$, a regime that is physically relevant and mathematically challenging because both degeneracy at $u=0$ and super‑linear growth of the diffusion coefficient destroy the usual one‑ and two‑block arguments.

The main contribution of the paper is to remove any scaling restriction on $χ_N\to0$ and to prove that the same rate function $I_{\rho}$ (as defined in earlier works) governs the LDP for all admissible sequences $χ_N$. The authors achieve this by developing two intertwined technical tools:

1. Multiscale regularity – The authors introduce a hierarchy of spatial scales. For a large box of side length $e\ell_N$ they consider a finer partition into boxes of side length $\ell_N\ll e\ell_N$. On each small box $B$ they define a coarse‑grained quantity \