Discretisation of regularity structures

We introduce a general framework allowing to apply the theory of regularity structures to discretisations of stochastic PDEs. The approach pursued in this article is that we do not focus on any one specific discretisation procedure. Instead, we assume that we are given a scale $\varepsilon > 0$ and a "black box" describing the behaviour of our discretised objects at scales below $\varepsilon$

Asymptotics of the critical time in Wiener sausage percolation with a small radius

We consider a continuum percolation model on $\R^d$, where $d\geq 4$.The occupied set is given by the union of independent Wiener sausages with radius $r$ running up to time $t$ and whoseinitial points are distributed according to a homogeneous Poisson point process.It was established in a previous work by Erhard, Mart\'{i}nez and Poisat~\cite{EMP13} that (1) if $r$ is small enough there is a non-trivial percolation transitionin $t$ occuring at a critical time $t\_c(r)$ and (2) in the supercritical regime the unbounded cluster is unique. In this paper we investigate the asymptotic behaviour of the critical time when the radius $r$ converges to $0$. The latter does not seem to be deducible from simple scaling arguments. We prove that for $d\geq 4$, there is a positive constant $c$ such that$c^{-1}\sqrt{\log(1/r)}\leq t\_c(r)\leq c\sqrt{\log(1/r)}$ when $d=4$ and $c^{-1}r^{(4-d)/2}\leq t\_c(r) \leq c\ r^{(4-d)/2}$when$d\geq 5$, as$r$converges to$0$. We derive along the way moment estimates on the capacity of Wiener sausages, which may be of independent interest

The parabolic Anderson model in a dynamic random environment: basic properties of the quenched Lyapunov exponent

In this paper we study the parabolic Anderson equation \partial u(x,t)/\partial t=\kappa\Delta u(x,t)+\xi(x,t)u(x,t), x\in\Z^d, t\geq 0, where the u-field and the \xi-field are \R-valued, \kappa \in [0,\infty) is the diffusion constant, and $\Delta$ is the discrete Laplacian. The initial condition u(x,0)=u_0(x), x\in\Z^d, is taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump at rate 2d\kappa, split into two at rate \xi\vee 0, and die at rate (-\xi)\vee 0. Our goal is to prove a number of basic properties of the solution u under assumptions on $\xi$ that are as weak as possible. Throughout the paper we assume that $\xi$ is stationary and ergodic under translations in space and time, is not constant and satisfies \E(|\xi(0,0)|)<\infty, where \E denotes expectation w.r.t. \xi. Under a mild assumption on the tails of the distribution of \xi, we show that the solution to the parabolic Anderson equation exists and is unique for all \kappa\in [0,\infty). Our main object of interest is the quenched Lyapunov exponent \lambda_0(\kappa)=\lim_{t\to\infty}\frac{1}{t}\log u(0,t). Under certain weak space-time mixing conditions on \xi, we show the following properties: (1)\lambda_0(\kappa) does not depend on the initial condition u_0; (2)\lambda_0(\kappa)<\infty for all \kappa\in [0,\infty); (3)\kappa \mapsto \lambda_0(\kappa) is continuous on [0,\infty) but not Lipschitz at 0. We further conjecture: (4)\lim_{\kappa\to\infty}[\lambda_p(\kappa)-\lambda_0(\kappa)]=0 for all p\in\N, where \lambda_p (\kappa)=\lim_{t\to\infty}\frac{1}{pt}\log\E([u(0,t)]^p) is the p-th annealed Lyapunov exponent. Finally, we prove that our weak space-time mixing conditions on \xi are satisfied for several classes of interacting particle systems.Comment: 50 pages. The comments of the referee are incorporated into the paper. A missing counting estimate was added in the proofs of Lemma 3.6 and Lemma 4.

Uniqueness and tube property for the Swiss cheese large deviations

We consider the simple random walk on the Euclidean lattice, in three dimensions and higher, conditioned to visit fewer sites than expected, when the deviation from the mean scales like the mean. The associated large deviation principle was first derived in 2001 by van den Berg, Bolthausen and den Hollander in the continuous setting, that is for the volume of a Wiener sausage, and later taken up by Phetpradap in the discrete setting. One of the key ideas in their work is to condition the range of the random walk to a certain skeleton, that is a sub-sequence of the random walk path taken along an appropriate mesoscopic scale. In this paper we prove that (i) the rate function obtained by van den Berg, Bolthausen and den Hollander has a unique minimizer over the set of probability measures modulo shifts, at least for deviations of the range well below the mean, and (ii) the empirical measure of the skeleton converges under the conditioned law, in a certain manner, to this minimizer. To this end we use an adaptation of the topology recently introduced by Mukherjee and Varadhan to compactify the space of probability measures