Renormalized stochastic pressure equation with log-correlated Gaussian coefficients

Benny Avelin, Tuomo Kuusi, Patrik Nummi*, Eero Saksman, Jonas M. Tölle, Lauri Viitasaari

*Corresponding author for this work

Research output: Contribution to journalArticleScientific


We study periodic solutions to the following divergence-form stochastic partial differential equation with Wick-renormalized gradient on the d-dimensional flat torus Td,

where X is the log-correlated Gaussian field, F is a random vector and ⋄ 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⋄(−βX) is modeling the random permeability of the rock. The porosity field is given by a log-correlated Gaussian random field βX, where β<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.
Original languageEnglish
Number of pages35
Publication statusSubmitted - 28 May 2024
MoE publication typeB1 Non-refereed journal articles


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