Convergence of the two-dimensional dynamic Ising-Kac model to Φ 4 2

The Ising-Kac model is a variant of the ferromagnetic Ising model in which each spin variable interacts with all spins in a neighborhood of radius γ−1 for γ ≪ 1 around its base point. We study the Glauber dynamics for this model on a discrete two-dimensional torus inline image for a system size N ≫ γ−1 and for an inverse temperature close to the critical value of the mean field model. We show that the suitably rescaled coarse-grained spin field converges in distribution to the solution of a nonlinear stochastic partial differential equation. This equation is the dynamic version of the inline image quantum field theory, which is formally given by a reaction-diffusion equation driven by an additive space-time white noise. It is well-known that in two spatial dimensions such equations are distribution valued and a Wick renormalization has to be performed in order to define the nonlinear term. Formally, this renormalization corresponds to adding an infinite mass term to the equation. We show that this need for renormalization for the limiting equation is reflected in the discrete system by a shift of the critical temperature away from its mean field value. © 2016 Wiley Periodicals, Inc

The dynamic phi^4_3 model comes down from infinity

We prove an a priori bound for the dynamic $\Phi^4_3$ model on the torus wich is independent of the initial condition. In particular, this bound rules out the possibility of finite time blow-up of the solution. It also gives a uniform control over solutions at large times, and thus allows to construct invariant measures via the Krylov-Bogoliubov method. It thereby provides a new dynamic construction of the Euclidean $\Phi^4_3$ field theory on finite volume. Our method is based on the local-in-time solution theory developed recently by Gubinelli, Imkeller, Perkowski and Catellier, Chouk. The argument relies entirely on deterministic PDE arguments (such as embeddings of Besov spaces and interpolation), which are combined to derive energy inequalities

Random data wave equations

Nowadays we have many methods allowing to exploit the regularising properties of the linear part of a nonlinear dispersive equation (such as the KdV equation, the nonlinear wave or the nonlinear Schroedinger equations) in order to prove well-posedness in low regularity Sobolev spaces. By well-posedness in low regularity Sobolev spaces we mean that less regularity than the one imposed by the energy methods is required (the energy methods do not exploit the dispersive properties of the linear part of the equation). In many cases these methods to prove well-posedness in low regularity Sobolev spaces lead to optimal results in terms of the regularity of the initial data. By optimal we mean that if one requires slightly less regularity then the corresponding Cauchy problem becomes ill-posed in the Hadamard sense. We call the Sobolev spaces in which these ill-posedness results hold spaces of supercritical regularity. More recently, methods to prove probabilistic well-posedness in Sobolev spaces of supercritical regularity were developed. More precisely, by probabilistic well-posedness we mean that one endows the corresponding Sobolev space of supercritical regularity with a non degenerate probability measure and then one shows that almost surely with respect to this measure one can define a (unique) global flow. However, in most of the cases when the methods to prove probabilistic well-posedness apply, there is no information about the measure transported by the flow. Very recently, a method to prove that the transported measure is absolutely continuous with respect to the initial measure was developed. In such a situation, we have a measure which is quasi-invariant under the corresponding flow. The aim of these lectures is to present all of the above described developments in the context of the nonlinear wave equation.Comment: Lecture notes based on a course given at a CIME summer school in August 201

A quantitative version of the Kipnis-Varadhan theorem and Monte-Carlo approximation of homogenized coefficients

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