Introduction to Regularity Structures

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. We give a concise overview of the theory of regularity structures as exposed in Hairer (2014). 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.Comment: 33 page

Ergodicity of Stochastic Differential Equations Driven by Fractional Brownian Motion

We study the ergodic properties of finite-dimensional systems of SDEs driven by non-degenerate additive fractional Brownian motion with arbitrary Hurst parameter $H\in(0,1)$. A general framework is constructed to make precise the notions of ``invariant measure'' and ``stationary state'' for such a system. We then prove under rather weak dissipativity conditions that such an SDE possesses a unique stationary solution and that the convergence rate of an arbitrary solution towards the stationary one is (at least) algebraic. A lower bound on the exponent is also given.Comment: 49 pages, 8 figure

Renormalisation of parabolic stochastic PDEs

We give a survey of recent result regarding scaling limits of systems from statistical mechanics, as well as the universality of the behaviour of such systems in so-called cross-over regimes. It transpires that some of these universal objects are described by singular stochastic PDEs. We then give a survey of the recently developed theory of regularity structures which allows to build these objects and to describe some of their properties. We place particular emphasis on the renormalisation procedure required to give meaning to these equations. These are expanded notes of the 20th Takagi lectures held at Tokyo University on November 4, 2017

Periodic homogenization with an interface

We consider a diffusion process with coefficients that are periodic outside of an 'interface region' of finite thickness. The question investigated in the articles [1,2] is the limiting long time / large scale behaviour of such a process under diffusive rescaling. It is clear that outside of the interface, the limiting process must behave like Brownian motion, with diffusion matrices given by the standard theory of homogenization. The interesting behaviour therefore occurs on the interface. Our main result is that the limiting process is a semimartingale whose bounded variation part is proportional to the local time spent on the interface. We also exhibit an explicit way of identifying its parameters in terms of the coefficients of the original diffusion. Our method of proof relies on the framework provided by Freidlin and Wentzell for diffusion processes on a graph in order to identify the generator of the limiting process.Comment: ISAAC 09 conference proceeding

A spatial version of the It\^{o}-Stratonovich correction

We consider a class of stochastic PDEs of Burgers type in spatial dimension 1, driven by space-time white noise. Even though it is well known that these equations are well posed, it turns out that if one performs a spatial discretization of the nonlinearity in the "wrong" way, then the sequence of approximate equations does converge to a limit, but this limit exhibits an additional correction term. This correction term is proportional to the local quadratic cross-variation (in space) of the gradient of the conserved quantity with the solution itself. This can be understood as a consequence of the fact that for any fixed time, the law of the solution is locally equivalent to Wiener measure, where space plays the role of time. In this sense, the correction term is similar to the usual It\^{o}-Stratonovich correction term that arises when one considers different temporal discretizations of stochastic ODEs.Comment: Published in at http://dx.doi.org/10.1214/11-AOP662 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org