arXiv subjects

Martin Hairer

Publications and source records attributed to Martin Hairer.

At least 73 records · Page 4Linked to original sources

arXiv2018

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.

math.PR↗

arXiv2018

An analytic BPHZ theorem for regularity structures

We prove a general theorem on the stochastic convergence of appropriately renormalized models arising from nonlinear stochastic PDEs. The theory of regularity structures gives a fairly automated framework for studying these problems but previous works had to expend significant effort to obtain these stochastic estimates in an ad-hoc manner. In contrast, the main result of this article operates as a black box which automatically produces these estimates for nearly all of the equations that fit within the scope of the theory of regularity structures. Our approach leverages multi-scale analysis strongly reminiscent to that used in constructive field theory, but with several significant twists. These come in particular from the presence of "positive renormalizations" caused by the recentering procedure proper to the theory of regularity structure, from the difference in the action of the group of possible renormalization operations, as well as from the fact that we allow for non-Gaussian driving fields. One rather surprising fact is that although the "canonical lift" is of course typically not continuous on any Hölder-type space containing the noise (which is why renormalization is required in the first place), we show that the "BPHZ lift" where the renormalization constants are computed using the formula given in arXiv:1610.08468, is continuous in law when restricted to a class of stationary random fields with sufficiently many moments.

math.PR↗

arXiv2018

Tightness of the Ising-Kac model on the two-dimensional torus

We consider the sequence of Gibbs measures of Ising models with Kac interaction defined on a periodic two-dimensional discrete torus near criticality. Using the convergence of the Glauber dynamic proven by H. Weber and J.C. Mourrat and a method by H. Weber and P. Tsatsoulis, we show tightness for the sequence of Gibbs measures of the Ising-Kac model near criticality and characterise the law of the limit as the $Φ^4_2$ measure on the torus. Our result is very similar to the one obtained by M. Cassandro, R. Marra and E. Presutti on $\mathbb{Z}^2$, but our strategy takes advantage of the dynamic, instead of correlation inequalities. In particular, our result covers the whole critical regime and does not require the large temperature / large mass / small coupling assumption present in earlier results.

math.PR↗

arXiv2017

A solution theory for quasilinear singular SPDEs

We give a construction allowing to construct local renormalised solutions to general quasilinear stochastic PDEs within the theory of regularity structures, thus greatly generalising the recent results of [BDH16,FG16,OW16]. Loosely speaking, our construction covers quasilinear variants of all classes of equations for which the general construction of [Hai14,BHZ16,CH16] applies, including in particular one-dimensional systems with KPZ-type nonlinearities driven by space-time white noise. In a less singular and more specific case, we furthermore show that the counterterms introduced by the renormalisation procedure are given by local functionals of the solution. The main feature of our construction is that it allows to exploit a number of existing results developed for the semilinear case, so that the number of additional arguments it requires is relatively small.

math.AP↗

arXiv2017

Singular SPDEs in domains with boundaries

We study spaces of modelled distributions with singular behaviour near the boundary of a domain that, in the context of the theory of regularity structures, allow one to give robust solution theories for singular stochastic PDEs with boundary conditions. The calculus of modelled distributions established in Hairer (Invent. Math. 198, 2014) is extended to this setting. We formulate and solve fixed point problems in these spaces with a class of kernels that is sufficiently large to cover in particular the Dirichlet and Neumann heat kernels. These results are then used to provide solution theories for the KPZ equation with Dirichlet and Neumann boundary conditions and for the 2D generalised parabolic Anderson model with Dirichlet boundary conditions. In the case of the KPZ equation with Neumann boundary conditions, we show that, depending on the class of mollifiers one considers, a "boundary renormalisation" takes place. In other words, there are situations in which a certain boundary condition is applied to an approximation to the KPZ equation, but the limiting process is the Hopf-Cole solution to the KPZ equation with a different boundary condition.

math.PR↗

arXiv2017

The reconstruction theorem in Besov spaces

The theory of regularity structures sets up an abstract framework of modelled distributions generalising the usual Hölder functions and allowing one to give a meaning to several ill-posed stochastic PDEs. A key result in that theory is the so-called reconstruction theorem: it defines a continuous linear operator that maps spaces of "modelled distributions" into the usual space of distributions. In the present paper, we extend the scope of this theorem to analogues to the whole class of Besov spaces $\mathcal{B}^γ_{p,q}$ with non-integer regularity indices. We then show that these spaces behave very much like their classical counterparts by obtaining the corresponding embedding theorems and Schauder-type estimates.

math.FA↗

arXiv2017

Discretisations of rough stochastic PDEs

We develop a general framework for spatial discretisations of parabolic stochastic PDEs whose solutions are provided in the framework of the theory of regularity structures and which are functions in time. As an application, we show that the dynamical $Φ^4_3$ model on the dyadic grid converges after renormalisation to its continuous counterpart. This result in particular implies that, as expected, the $Φ^4_3$ measure with a sufficiently small coupling constant is invariant for this equation and that the lifetime of its solutions is almost surely infinite for almost every initial condition.

math.PR↗

arXiv2017

A simple construction of the continuum parabolic Anderson model on $\mathbf{R}^2$

We propose a simple construction of the solution to the continuum parabolic Anderson model on $\mathbf{R}^2$ which does not rely on any elaborate arguments and makes extensive use of the linearity of the equation. A logarithmic renormalisation is required to counterbalance the divergent product appearing in the equation. Furthermore, we use time-dependent weights in our spaces of distributions in order to construct the solution on the unbounded space $\mathbf{R}^2$.

math.PR↗

arXiv2017

The strong Feller property for singular stochastic PDEs

We show that the Markov semigroups generated by a large class of singular stochastic PDEs satisfy the strong Feller property. These include for example the KPZ equation and the dynamical $Φ^4_3$ model. As a corollary, we prove that the Brownian bridge measure is the unique invariant measure for the KPZ equation with periodic boundary conditions.

math.PR↗

arXiv2017

Large scale behaviour of 3D continuous phase coexistence models

We study a class of three dimensional continuous phase coexistence models, and show that, under different symmetry assumptions on the potential, the large-scale behaviour of such models near a bifurcation point is described by the dynamical $Φ^p_3$ models for $p \in \{2,3,4\}$. This result is specific to space dimension $3$ and does not hold in dimension $2$.

math-ph↗

arXiv2016

A central limit theorem for the KPZ equation

We consider the KPZ equation in one space dimension driven by a stationary centred space-time random field, which is sufficiently integrable and mixing, but not necessarily Gaussian. We show that, in the weakly asymmetric regime, the solution to this equation considered at a suitable large scale and in a suitable reference frame converges to the Hopf-Cole solution to the KPZ equation driven by space-time Gaussian white noise. While the limiting process depends only on the integrated variance of the driving field, the diverging constants appearing in the definition of the reference frame also depend on higher order moments.

math.PR↗

arXiv2016

Weak error estimates for trajectories of SPDEs for Spectral Galerkin discretization

We consider stochastic semi-linear evolution equations which are driven by additive, spatially correlated, Wiener noise, and in particular consider problems of heat equation (analytic semigroup) and damped-driven wave equations (bounded semigroup) type. We discretize these equations by means of a spectral Galerkin projection, and we study the approximation of the probability distribution of the trajectories: test functions are regular, but depend on the values of the process on the interval $[0,T]$. We introduce a new approach in the context of quantative weak error analysis for discretization of SPDEs. The weak error is formulated using a deterministic function (Itô map) of the stochastic convolution found when the nonlinear term is dropped. The regularity properties of the Itô map are exploited, and in particular second-order Taylor expansions employed, to transfer the error from spectral approximation of the stochastic convolution into the weak error of interest. We prove that the weak rate of convergence is twice the strong rate of convergence in two situations. First, we assume that the covariance operator commutes with the generator of the semigroup: the first order term in the weak error expansion cancels out thanks to an independence property. Second, we remove the commuting assumption, and extend the previous result, thanks to the analysis of a new error term depending on a commutator.

math.PR↗

arXiv2016

A fractional kinetic process describing the intermediate time behaviour of cellular flows

This paper studies the intermediate time behaviour of a small random perturbation of a periodic cellular flow. Our main result shows that on time scales shorter than the diffusive time scale, the limiting behaviour of trajectories that start close enough to cell boundaries is a fractional kinetic process: A Brownian motion time changed by the local time of an independent Brownian motion. Our proof uses the Freidlin-Wentzell framework, and the key step is to establish an analogous averaging principle on shorter time scales. As a consequence of our main theorem, we obtain a homogenization result for the associated advection-diffusion equation. We show that on intermediate time scales the effective equation is a fractional time PDE that arises in modelling anomalous diffusion.

math.PR↗

arXiv2016

The motion of a random string

We review a series of forthcoming results leading to the construction of a natural evolution on the space of loops with values in a Riemannian manifold. In particular, this clarifies the algebraic structure of the renormalisation procedures appearing in the context of the theory of regularity structures.

math.PR↗

arXiv2016

Optimal rate of convergence for stochastic Burgers-type equations

Recently, a solution theory for one-dimensional stochastic PDEs of Burgers type driven by space-time white noise was developed. In particular, it was shown that natural numerical approximations of these equations converge and that their convergence rate in the uniform topology is arbitrarily close to $\frac{1}{6}$. In the present article we improve this result in the case of additive noise by proving that the optimal rate of convergence is arbitrarily close to $\frac{1}{2}$.

math.PR↗

arXiv2015

A class of growth models rescaling to KPZ

We consider a large class of $1+1$-dimensional continuous interface growth models and we show that, in both the weakly asymmetric and the intermediate disorder regimes, these models converge to Hopf-Cole solutions to the KPZ equation.

math-ph↗

arXiv2015

Regularity structures and the dynamical $Φ^4_3$ model

We give a concise overview of the theory of regularity structures as first exposed in [Hai14]. In order to allow to focus on the conceptual aspects of the theory, many proofs are omitted and statements are simplified. In order to provide both motivation and focus, we concentrate on the study of solutions to the stochastic quantisation equations for the Euclidean $Φ^4_3$ quantum field theory which can be obtained with the help of this theory. In particular, we sketch the proofs of how one can show that this model arises quite naturally as an idealised limiting object for several classes of smooth models.

math.PR↗

arXiv2015

A Wong-Zakai theorem for stochastic PDEs

We prove a version of the Wong-Zakai theorem for one-dimensional parabolic nonlinear stochastic PDEs driven by space-time white noise. As a corollary, we obtain a detailed local description of solutions. Dedicated to the memory of Kiyosi Itô on the occasion of the 100th anniversary of his birth.

math.PR↗