Colloquium de l'IDP

The dynamic $Phi^4$ model - Scaling limits and small noise behaviour
Hendrik Weber (Warwick)
Thursday 30 June 2016 14:00 -  Orléans -  Salle de Séminaire

Résumé :
In this talk I will discuss some recent progress on the dynamic $\Phi^4$ model, which is formally given by a non-linear stochastic PDE driven by space-time white noise. Due to the irregularity of the noise for spatial dimension $d \geq 2$ solutions are distribution valued and a renormalisation procedure has to be performed to interpret the non-linear term. I will discuss two situations in which such a renormalised solution appears naturally. First, I will discuss how it can be derived as a scaling limit of Ising-type models with a long range interaction. In this case the "infinite normalisation constant” has a natural interpretation as a shift of the inverse temperature. In the second part I will discuss the behaviour of solutions for small noise strength. I will argue that despite the infinite normalisation constant these renormalised PDE are the natural perturbation of the deterministic Allen-Cahn equation. I will illustrate this on the level of large deviations and then show that transitions between stable states of the deterministic dynamics are governed by an Eyring-Kramer type formula. Based on joint work with N. Berglund (Orléans), G. Di Gesù (Paris), M. Hairer (Warwick), J.C. Mourrat (Lyon).

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