Renormalized stochastic pressure equation with log-correlated Gaussian coefficients
Benny Avelin, Tuomo Kuusi, Patrik Nummi, Eero Saksman, Jonas M. Tölle, Lauri Viitasaari
TL;DR
This work analyzes a divergence-form SPDE with Wick-renormalized gradient on the $d$-torus driven by a log-correlated Gaussian field, under the constraint $0<\beta<\sqrt{d}$. The authors renormalize the problem via the Wick exponential and employ an $S$-transform adapted to Gaussian multiplicative chaos to reduce the stochastic equation to a family of deterministic weighted PDEs parameterized by $z\in\mathbb{T}^d$, with weights $w(y;z)=|y-z|^{\beta^2}e^{-\beta^2 g(y-z)}$. They prove existence and uniqueness of solutions in the $L^2_\beta(\Omega)$ framework and provide a representation of the stochastic solution as $U(y)=\int_{\mathbb{T}^d}\varphi(y;z)\,d\mu_\beta(z)$, where $\varphi$ solves the weighted PDE and $d\mu_\beta$ is a Gaussian multiplicative chaos measure. A key part of the analysis is establishing $z$-regularity of the deterministic PDE solutions; this is achieved by shifting the pole and constructing a corrector expansion, culminating in uniform $H^{d/2}$-regularity in $z$ and enabling inversion of the $S$-transform. The results connect stochastic renormalization, GMC, and weighted elliptic theory, offering a rigorous framework for the stochastic pressure equation with log-correlated random coefficients and suggesting avenues for extensions via rough-path or regularity-structure techniques.
Abstract
We study periodic solutions to the following divergence-form stochastic partial differential equation with Wick-renormalized gradient on the $d$-dimensional flat torus $\mathbb{T}^d$, \[ -\nabla\cdot\left(e^{\diamond (- βX) }\diamond\nabla U\right)=\nabla \cdot (e^{\diamond (- βX)} \diamond \mathbf{F}), \] where $X$ is the log-correlated Gaussian field, $\mathbf{F}$ is a random vector field representing the flux, the in/out-flow of fluid per unit area per unit time, and $\diamond$ denotes the Wick product. The problem is a variant of the stochastic pressure equation, in which $U$ is modeling the pressure of a creeping water-flow in crustal rock that occurs in enhanced geothermal heating. In the original model, the Wick exponential term $e^{\diamond(-βX)}$ is modeling the random permeability of the rock. The porosity field is given by a log-correlated Gaussian random field $βX$, where $β<\sqrt{d}$. We use elliptic regularity theory in order to define a notion of a solution to this (a priori very ill-posed) problem, via modifying the $S$-transform from Gaussian white noise analysis, and then establish the existence and uniqueness of solutions. Moreover, we show that the solution to the problem can be expressed in terms of the Gaussian multiplicative chaos measure.