Equilibration for the dynamical $\Phi^4$ model

Hendrik Weber, University of Warwick. Part of the probability, stochastic modelling and financial mathematics seminar series.

In this talk I will discuss the long term behaviour of the stochastic PDE

\partial_t \\phi = \\Delta \\phi - \\phi^3 + \\xi,

where $\\xi $ denotes space-time white noise and the space variables $x $ takes values in the $d$ dimensional torus for either $d=2,3$. This equation was proposed in the eighties by Parisi and Wu to give a dynamical construction of the Euclidean $\\Phi^4$ quantum field theory which (at least formally) arises as the invariant measure of this SPDE.

Due to the irregularity of the driving white noise, the constructing solutions to the SPDE was an open problem for many years - the construction of short time solutions in the more difficult three dimensional case was accomplished by Hairer only a few years ago.

In this talk I will go back to Parisi and Wu’s original question and study the long term behaviour of solutions. In the two dimensional case $d=2$ I will show that solutions converge to equilibrium exponentially fast. I will also outline the proof of a similar statement in the three dimensional case.

This is based on joint work with Pavlos Tsatsoulis and Jean-Christophe Mourrat.

Hendrik Weber, University of Warwick