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1D 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 the existence and uniqueness of solutions to a one-dimensional stochastic pressure equation with diffusion given by Wick-exponentiated log-correlated fields. It develops parallel theories for non-Wick (pointwise) and Wick multiplications, providing explicit solution representations and convergence proofs via mollification, Gaussian multiplicative chaos, and S-transform techniques. The authors establish well-posedness across Dirichlet, Neumann, periodic boundary data, and initial value problems, and extend the framework by introducing projections onto GMC-based subspaces using the S-transform at a point of the field. Collectively, the results bridge renormalized and pathwise formulations, offering a robust, transform-based approach to singular stochastic boundary-value problems in 1D with log-correlated inputs.

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.

1D stochastic pressure equation with log-correlated Gaussian coefficients

TL;DR

This work analyzes the existence and uniqueness of solutions to a one-dimensional stochastic pressure equation with diffusion given by Wick-exponentiated log-correlated fields. It develops parallel theories for non-Wick (pointwise) and Wick multiplications, providing explicit solution representations and convergence proofs via mollification, Gaussian multiplicative chaos, and S-transform techniques. The authors establish well-posedness across Dirichlet, Neumann, periodic boundary data, and initial value problems, and extend the framework by introducing projections onto GMC-based subspaces using the S-transform at a point of the field. Collectively, the results bridge renormalized and pathwise formulations, offering a robust, transform-based approach to singular stochastic boundary-value problems in 1D with log-correlated inputs.

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 -transform and the Gaussian multiplicative chaos measure.
Paper Structure (6 sections, 16 theorems, 154 equations, 2 tables)

This paper contains 6 sections, 16 theorems, 154 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Log-correlated Gaussian field
  4. Unique solvability of non-Wick equation
  5. Unique solvability of Wick equation
  6. S-transform at X and projections into a subspace

Key Result

Lemma 2.5

Let $X$ be a log-correlated Gaussian field on $(0,T)$, and $\beta \in (0,1)$ be a parameter. Let $0 < \varepsilon < \varepsilon'$, $x,y \in (0,T)$. Then the convolution approximation $X_\varepsilon$ satisfies:

Theorems & Definitions (62)

  • Definition 2.1: Log-correlated Gaussian field
  • Remark 2.2
  • Definition 2.3: Wick exponential
  • Remark 2.4
  • Lemma 2.5: Approximation lemma
  • proof : Proof of \ref{['lem:mollification-of-log-correlated-field']}
  • Theorem 2.6
  • Definition 3.1
  • Remark 3.2
  • Theorem 3.3
  • ...and 52 more