The Porous Medium Equation: Multiscale Integrability in Large Deviations

We consider a zero-range process ${\eta }_t^N(x)$ with superlinear local jump rate, which in a hydrodynamic-small particle rescaling converges to the porous medium equation ${\mathrm{\partial }}_{t}u=\frac{1}{2}\mathrm{\Delta }{u}^{\alpha },\alpha >1$. 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.