1D stochastic pressure equation with log-correlated Gaussian coefficients

TL;DR

This paper establishes the well-posedness of a one-dimensional stochastic pressure equation with diffusion coefficients derived from log-correlated Gaussian fields, using Wick renormalization and explicit solution representations.

Contribution

It introduces new methods for proving existence and uniqueness of solutions for stochastic PDEs with complex Gaussian coefficients, including explicit solution formulas.

Findings

Proved well-posedness for various boundary conditions.

Provided explicit solution representations using the $S$-transform.

Analyzed cases with Wick renormalization and point-wise multiplication.

Abstract

We study unique solvability for one dimensional stochastic pressure equation with diffusion coefficient given by the Wick exponential of log-correlated Gaussian fields. We prove well-posedness for Dirichlet, Neumann and periodic boundary data, and the initial value problem, covering the cases of both the Wick renormalization of the diffusion and of point-wise multiplication. We provide explicit representations for the solutions in both cases, characterized by the $S$-transform and the Gaussian multiplicative chaos measure.