Gradient-type estimates for the dynamic $\varphi^4_2$-model

TL;DR

This paper establishes gradient bounds and exponential contraction for the Markov semigroup of the dynamic ^4_2-model on a torus, using pathwise estimates and stopping time techniques, leading to a Poincare9 inequality for large mass.

Contribution

It introduces a novel method combining pathwise estimates and stopping times to derive gradient bounds and spectral gap inequalities for the ^4_2-model.

Findings

Proves gradient bounds for the Markov semigroup of the ^4_2-model.

Establishes exponential contraction of the semigroup for large mass.

Derives a Poincare9 inequality with an almost optimal spectral gap.

Abstract

We prove gradient bounds for the Markov semigroup of the dynamic $\varphi^4_2$-model on a torus of fixed size $L>0$. For sufficiently large mass $m>0$ these estimates imply exponential contraction of the Markov semigroup. Our method is based on pathwise estimates of the linearized equation. To compensate the lack of exponential integrability of the stochastic drivers we use a stopping time argument and the strong Markov property in the spirit of Cass--Litterer--Lyons. Following the classical approach of Bakry-\'Emery, as a corollary we prove a Poincar\'e/spectral gap inequality for the $\varphi^4_2$-measure of sufficiently large mass $m>0$ with almost optimal carr\'e du champ.