These are short notes from a series of lectures given at the University of Rennes in June 2013, at the University of Bonn in July 2013, at the XVIIth Brazilian School of Probability in Mambucaba in August 2013, and at ETH Zurich in September 2013. They give a concise overview of the theory of regularity structures as exposed in the article [Invent. Math. DOI: 10.1007/s00222-014-0505-4]. In order to allow to focus on the conceptual aspects of the theory, many proofs are omitted and statements are simplified. We focus on applying the theory to the problem of giving a solution theory to the stochastic quantisation equations for the Euclidean $\Phi^{4}_{3}$ quantum field theory.

These are the notes of my lectures at EBP, August 2013. I have added some proofs which being of more technical nature have been omitted in the talks. The notes and the lectures are based on a course on atomistic and continuous descriptions of matter which I gave a few years earlier in Sperlonga, Italy. In the present notes, I have tried to underline the more probabilistic aspects of the theory.

I am afraid I have not been able to reproduce in the written notes the very lively atmosphere of the talks. Many old friends of mine were attending the lectures and helped me a lot with questions, comments and criticism, it was a pleasure for me to speak at EBP and I hope also the audience enjoined all that. In particular, I want to renew my deepest thanks to Stefan Luckhaus who is undoubtedly the best help for a speaker to have in the audience.

These notes are divided into four chapters, like in the lectures I have mostly avoided proofs trying to give qualitatively the main ideas of the theory. The only exception is in the second lecture of these notes where I have given more details on the proof of phase transitions in the canonical Ising model. This partly for completeness and partly because I have been asked by several people for details after the talk.

I am not very good with bibliography so I just quoted some of the papers I am most familiar with, the reader will forgive me, I hope, for the many omissions.

I conclude these preface by renewing my warmest thanks to the people who attended EBP and in particular to Maria Eulalia Vares for inviting me at EBP, for the nice words when she introduced my lectures but especially for the very long friendship (in the past and hopefully in the future).

We consider the TASEP with site disorder and show that several representations of the flux are equivalent. We then describe heuristically the phase transition induced by site disorder and relate it to the problem of homogenization of conservation laws.

We discuss necessary and sufficient conditions for the convergence of disordered asymmetric zero-range process to the critical invariant measure.

We consider the Cahn–Hilliard equation in one space dimension, perturbed by the derivative of a space and time white noise of intensity $\varepsilon^{1/2}$, and we investigate the effect of the noise, as $\varepsilon\to0$, on the solutions when the initial condition is a front that separates the two stable phases. We prove that, given $\gamma<\frac{2}{3}$, with probability going to one as $\varepsilon\to0$, the solution remains close to a front for times of the order of $\varepsilon^{-\gamma}$, and we study the fluctuations of the front in this time scaling. They are given by a one dimensional continuous process, self similar of order $\frac{1}{4}$ and non-Markovian, related to a fractional Brownian motion and for which a couple of representations are given.

We investigate a kinetic model of a system in contact with several thermal reservoirs at different temperatures $T_\alpha$. Our system is a spatially uniform dilute gas whose internal dynamics is described by the non-linear Boltzmann equation with Maxwellian collisions. Similarly, the interaction with reservoir $\alpha$ is represented by a Markovian process that has the Maxwellian $M_{T_\alpha}$ as its stationary state. We prove existence and uniqueness of a non-equilibrium steady state (NESS) and show exponential convergence to this NESS in a metric on probability measures introduced into the study of Maxwellian collisions by Gabetta, Toscani and Wennberg (GTW). This shows that the GTW distance between the current velocity distribution to the steady-state velocity distribution is a Lyapunov functional for the system. We also derive expressions for the entropy production in the system plus the reservoirs which is always positive.

We consider a finite number of particles that move in $\mathbb{Z}$ as independent random walks. The particles are of two species that we call $a$ and $b$. The rightmost $a$-particle becomes a $b$-particle at constant rate, while the leftmost $b$-particle becomes $a$-particle at the same rate, independently. We prove that in the hydrodynamic limit the evolution is described by a nonlinear system of two PDE’s with free boundaries.

Let ${S}$ be a countable set provided with a partial order and a minimal element. Consider a Markov chain on $S\cup\{0\}$ absorbed at $0$ with a quasi-stationary distribution. We use Holley inequality to obtain sufficient conditions under which the following hold. The trajectory of the chain starting from the minimal state is stochastically dominated by the trajectory of the chain starting from any probability on ${S}$, when both are conditioned to nonabsorption until a certain time. Moreover, the Yaglom limit corresponding to this deterministic initial condition is the unique minimal quasi-stationary distribution in the sense of stochastic order. As an application, we provide new proofs to classical results in the field.

We consider discrete-time stationary processes with long-range dependencies, $X_{n}\in\{\pm1\}$, ${n\in\mathbb{Z}}$, specified by a regular attractive $g$-function, similar to those considered by Bramson and Kalikow [Israel J. Math. 84 (1993) 153–160]. We give an explicit set of conditions that imply the existence of at least two distinct processes specified by the same $g$-function, and consider a few examples that emphasize the role played by the smoothness of the majority rule at the origin.

This paper address the problem of identifying pairs of interacting sites from a finite sample of independent realizations of the Ising model. We consider Ising models in a infinite countable set of sites under Dobrushin uniqueness condition. The observed sample contains only the values assigned by the Ising model to a finite set of sites. Our main result is an upperbound for the probability of misidentification of the pairs of interacting sites in this finite set.

Large families of noisy interacting units (cells, individuals, components in a circuit, …) exhibiting synchronization often exhibit oscillatory behaviors too. This is a well established empirical observation that has attracted a remarkable amount of attention, notably in life sciences, because of the central role played by internally generated rhythms. A certain number of elementary models that seem to capture the essence, or at least some essential features, of the phenomenon have been set forth, but the mathematical analysis is in any case very challenging and often out of reach. We focus on phase models, proposed and repeatedly considered by Y. Kuramoto and coauthors, and on the mathematical results that can be established. In spite of the fact that noise plays a crucial role, and in fact these models in abstract terms are just a special class of diffusions in high dimensional spaces, the core of the analysis is at the level of the PDE that provides an accurate description of the limit of a very large number of units in interaction. We will stress how the fundamental difficulty in dealing with these models is in their non-equilibrium character and the results we present for phase models are crucially related to the fact that, with a very special choice of the parameters, they reduce to an equilibrium statistical mechanics model.

In the last decade, the theory of large deviations has become a main tool in statistical mechanics especially in the study of non-equilibrium. In a rational reconstruction of the story, one must recognize the ideal connection and debt of some recent work, to discussions taking place at the beginning of the twentieth century. The famous equation $S=k\ln W$ usually attributed to Boltzmann, actually written in this final form by Planck on his route to the quantum hypothesis, was interpreted by Einstein as a large deviation formula. This interpretation, on which he based his theory of thermodynamic equilibrium fluctuations, has been a source of inspiration in recent developments of non-equilibrium statistical mechanics. In this paper, we briefly illustrate this aspect.

The pressure of a gas of particles with a uniformly repulsive pair interaction in a finite container is shown to satisfy (exactly as a formal object) a “viscous” Hamilton–Jacobi (H–J) equation whose solution in power series is recursively given by the variation of constants formula. We investigate the solution of the H–J and of its Legendre transform equation by the Cauchy-majorant method and provide a lower bound to the radius of convergence on the virial series of the fluid which goes beyond the threshold established by Lagrange’s inversion formula. Such results obtained in (On the virial series for a gas of particles with uniformly repulsive pairwise interactions (2014) Preprint) are reviewed and regarded as the first step towards the solution of a problem posed by Kac, Uhlenbeck and Hemmer (J. Math. Phys. 4 (1963) 216–228), questioning on the relation of the approach to the mean field theory with Ursell–Mayer expansion.

We obtain macroscopic adiabatic thermodynamic transformations by space–time scalings of a microscopic Hamiltonian dynamics subject to random collisions with the environment. The microscopic dynamics is given by a chain of oscillators subject to a varying tension (external force) and to collisions with external independent particles of “infinite mass”. The effect of each collision is to change the sign of the velocity without changing the modulus. This way the energy is conserved by the resulting dynamics. After a diffusive space–time scaling and coarse-graining, the profiles of volume and energy converge to the solution of a deterministic diffusive system of equations with boundary conditions given by the applied tension. This defines an irreversible thermodynamic transformation from an initial equilibrium to a new equilibrium given by the final tension applied. Quasi-static reversible adiabatic transformations are then obtained by a further time scaling. Then we prove that the relations between the limit work, internal energy and thermodynamic entropy agree with the first and second principle of thermodynamics.