Title: Periodic Solutions to a Stochastic Pressure Equation with Random Permeability (9.00 sharp)
In this talk, we study periodic solutions to a stochastic divergence-form PDE on the d-dimensional torus, where the flux is weighted by the Wick-renormalized exponential of a log-correlated Gaussian field. This equation arises as a variant of the stochastic pressure equation, modelling fluid flow in porous media, such as water percolation in geothermal reservoirs. The primary difficulty in solving and even defining the equation mathematically stems from the roughness of the log-correlated field, making the problem highly singular. We employ elliptic regularity techniques and a modified S-transform approach from Gaussian white noise analysis to rigorously define solutions. We establish well-posedness and further describe the solution in terms of Gaussian multiplicative chaos.
In this talk, I will discuss both the stochastic aspects of the problem—such as log-correlated fields and Gaussian multiplicative chaos—and the PDE techniques used to establish well-posedness and interpret the solutions. The talk is based on a recent paper which is a joint work with Benny Avelin (Uppsala University), Tuomo Kuusi (University of Helsinki), Jonas Tölle (Aalto University), Eero Saksman (University of Helsinki) and Lauri Viitasaari (Aalto University).
Observe that the starting time is 9.00am sharp.
