The Porous Medium Equation: Multiscale Integrability in Large Deviations
Benjamin Gess, Daniel Heydecker
TL;DR
The article develops a robust multiscale framework to prove a large deviations principle for a zero-range process with unbounded and degenerate diffusion that converges to the porous medium equation $\partial_t u=\tfrac12\Delta(u^{\alpha})$ with $\alpha>1$. By introducing a scale-separated integrability mechanism, the authors bridge pathwise regularity and regularity of laws using a Feynman–Kac variational approach and Orlicz-norm control, enabling uniform integrability of $u^\alpha$ across all vanishing particle-size regimes $χ_N\to0$. The main technical feat is a multiscale recombination argument that propagates integrability gains from macroscopic to mesoscopic and microscopic scales, complemented by a novel pathwise-regularity lemma and a hierarchy of block-averaging estimates. Exponential tightness and a superexponential estimate are established, yielding the LDP with the same rate function $\mathcal{I}_ρ$ as in prior rescaled results, but now valid in any scaling regime, including regimes with highly degenerate noise. The work thus advances the macroscopic fluctuation theory for degenerate nonlinear diffusions and provides a versatile template for LDPs in SPDEs with unbounded nonlinearities and scale-dependent noise.
Abstract
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.