Renormalized stochastic pressure equation with log-correlated Gaussian coefficients

TL;DR

This paper develops a mathematical framework for solving a stochastic pressure equation with log-correlated Gaussian coefficients, relevant for modeling fluid flow in geothermal rock, by establishing existence, uniqueness, and solution representation.

Contribution

It introduces a novel approach using elliptic regularity and modified S-transform to define solutions for a challenging stochastic PDE with Wick-renormalized coefficients, connecting solutions to Gaussian multiplicative chaos.

Findings

Established existence and uniqueness of solutions.

Linked solutions to Gaussian multiplicative chaos measure.

Applied elliptic regularity theory to a highly ill-posed problem.

Abstract

We study periodic solutions to the following divergence-form stochastic partial differential equation with Wick-renormalized gradient on the $d$-dimensional flat torus $\mathbb{T}^d$, \[ -\nabla\cdot\left(e^{\diamond (- \beta X) }\diamond\nabla U\right)=\nabla \cdot (e^{\diamond (- \beta X)} \diamond \mathbf{F}), \] 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 $e^{\diamond(-\beta X)}$ is modeling the random permeability of the rock. The porosity field is given by a log-correlated Gaussian random field $\beta X$, where $\beta<\sqrt{d}$. 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.