Jointly invariant measures for the KPZ equation with periodic noise
We present an explicit coupling of Brownian bridges plus affine shifts that are jointly invariant for the Kardar-Parisi-Zhang equation with periodic noise. These are described by Pitman-like transforms of independent Brownian bridges. We obtain these invariant measures by working with a semi-discrete model known as the O’Connell-Yor polymer in a periodic environment. In that setting, the relevant Markov process is described by a system of coupled SDEs. We show how to transform this Markov process to an auxiliary Markov process with a more tractable invariant measure. We discuss connections of this method to works of Ferrari and Martin in the mid 2000s in the context of multi-species particle systems. Furthermore, we present an application of this work to give an explicit formula for the covariance function of a limiting Gaussian process obtained from the coupled stochastic heat equation. Based on forthcoming joint work with Ivan Corwin and Yu Gu.
Date: 27 May 2024, 14:00 (Monday, 6th week, Trinity 2024)
Venue: Venue to be announced
Speaker: Evan Sorensen
Organising department: Department of Statistics
Organisers: Matthias Winkel (Department of Statistics, University of Oxford), Julien Berestycki (University of Oxford), Christina Goldschmidt (Department of Statistics, University of Oxford), James Martin (Department of Statistics, University of Oxford)
Part of: Probability seminar
Booking required?: Not required
Audience: Public
Editors: James Martin, Julien Berestycki