Dirk Erhard

Bio: Dirk Erhard is an academic researcher from Federal University of Bahia. The author has contributed to research in topics: Lyapunov exponent & Random walk. The author has an hindex of 8, co-authored 33 publications receiving 193 citations. Previous affiliations of Dirk Erhard include Leiden University & University of Warwick.

The parabolic Anderson model in a dynamic random environment: space-time ergodicity for the quenched Lyapunov exponent

Abstract: We continue our study of the parabolic Anderson equation @u(x;t)=@t = u(x;t) + (x;t)u(x;t), x2 Z d , t 0, where 2 [0;1) is the diusion constant, is the discrete Laplacian, and plays the role of a dynamic random environment that drives the equation. The initial condition u(x; 0) = u0(x), x2 Z d , is taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a eld of particles performing independent simple random walks with binary branching: particles jump at rate 2d , split into two at rate _ 0, and die at rate ( )_ 0. We assume that is stationary and ergodic under translations in space and time, is not constant and satises E(j (0; 0)j) E( (0; 0)) for 2 (0;1). In the present paper we show that lim !1 0( ) = E( (0; 0)) under an additional @ ;

Discretisation of regularity structures

Abstract: Nous introduisons un cadre general permettant d’appliquer la theorie des structures de regularite a des discretisations d’EDP stochastiques. L’approche suivie dans cet article est que, au lieu de nous focaliser sur un type d’approximation specifique, nous supposons donnee une echelle $\varepsilon>0$ et une “boite noire” decrivant le comportement des objets discretises aux echelles plus petites.

2D Anisotropic KPZ at stationarity: scaling, tightness and non triviality

Abstract: In this work we focus on the two-dimensional anisotropic KPZ (aKPZ) equation, which is formally given by \begin{equation*}\partial _{t}h=\frac{ u }{2}\Delta h+\lambda \bigl((\partial _{1}h)^{2}-(\partial _{2}h)^{2}\bigr)+ u ^{\frac{1}{2}}\xi ,\end{equation*} where $\xi $ denotes a noise which is white in both space and time, and $\lambda $ and $ u $ are positive constants. Due to the wild oscillations of the noise and the quadratic nonlinearity, the previous equation is classically ill posed. It is not possible to linearise it via the Cole–Hopf transformation and the pathwise techniques for singular SPDEs (the theory of regularity structures by M. Hairer or the paracontrolled distributions approach of M. Gubinelli, P. Imkeller, N. Perkowski) are not applicable. In the present work we consider a regularised version of aKPZ which preserves its invariant measure. We prove the existence of subsequential limits once the regularisation is removed, provided $\lambda $ and $ u $ are suitably renormalised. Moreover, we show that, in the regime in which $ u $ is constant and the coupling constant $\lambda $ converges to $0$ as the inverse of the square root logarithm, any limit differs from the solution to the linear equation obtained by simply dropping the nonlinearity in aKPZ.

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

Abstract: 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.

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

Abstract: 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 whose initial 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 transition in $t$ 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.

Convergence of probability measures

Abstract: The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Methods Of Modern Mathematical Physics

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Data-Driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations

Abstract: We consider stochastic programs where the distribution of the uncertain parameters is only observable through a finite training dataset. Using the Wasserstein metric, we construct a ball in the space of (multivariate and non-discrete) probability distributions centered at the uniform distribution on the training samples, and we seek decisions that perform best in view of the worst-case distribution within this Wasserstein ball. The state-of-the-art methods for solving the resulting distributionally robust optimization problems rely on global optimization techniques, which quickly become computationally excruciating. In this paper we demonstrate that, under mild assumptions, the distributionally robust optimization problems over Wasserstein balls can in fact be reformulated as finite convex programs—in many interesting cases even as tractable linear programs. Leveraging recent measure concentration results, we also show that their solutions enjoy powerful finite-sample performance guarantees. Our theoretical results are exemplified in mean-risk portfolio optimization as well as uncertainty quantification.

An Introduction To The Theory Of Point Processes

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