Synchronisation by noise for the stochastic quantisation equation in dimensions $2$ and $3$

TL;DR

This paper proves that noise induces uniform synchronization in the stochastic quantisation equation in 2D and 3D, using advanced probabilistic techniques to handle complex state spaces.

Contribution

It extends the framework of order-preserving Markov semigroups to H"older spaces of negative exponent for the first time.

Findings

Established uniform synchronization rates for the stochastic quantisation equation.

Applied the framework to negative exponent H"older spaces.

Demonstrated the effectiveness of coming down from infinity estimates.

Abstract

We prove uniform synchronisation by noise with rates for the stochastic quantisation equation in dimensions two and three. The proof relies on a combination of coming down from infinity estimates and the framework of order-preserving Markov semigroups derived in [Butkovsky, Scheutzow; 2019]. In particular, it is shown that this framework can be applied to the case of state spaces given in terms of H\"older spaces of negative exponent.