Jürgen Gärtner

Large deviations from the mckean-vlasov limit for weakly interacting diffusions

A system of N diffusions on R

Parabolic problems for the Anderson model. I. Intermittency and related topics

SUP:D

The Parabolic Anderson Model

ESUP:in which the interaction is expressed in terms of the empirical measure is considered. The limiting behavior as N →∞ is described by a McKean_Vlasov equation. The purpose of this paper is to show that the large deviations from the McKean-Vlasov limit can be described by a generalization of the theory of Freidlin and Wentzell and to obtain a characterization of the action functional. In order to obtain this action functional we first obtain results on projective limits of large deviation systems, large deviations on dual vector spaces and a Sanov type theorem for vectors of empirical measures

Almost sure asymptotics for the continuous parabolic Anderson model

The objective of this paper is a mathematically rigorous investigation of intermittency and related questions intensively studied in different areas of physics, in particular in hydrodynamics. On a qualitative level, intermittent random fields are distinguished by the appearance of sparsely distributed sharp “peaks” which give the main contribution to the formation of the statistical moments. The paper deals with the Cauchy problem (∂/∂t)u(t,x)=Hu(t, x), u(0,x)=t0(x) ≥ 0, (t, x) ∈ ℝ+ × ℤd, for the Anderson HamiltonianH = κΔ + ξ(·), ξ(x),x ∈ ℤd where is a (generally unbounded) spatially homogeneous random potential. This first part is devoted to some basic problems. Using percolation arguments, a complete answer to the question of existence and uniqueness for the Cauchy problem in the class of all nonnegative solutions is given in the case of i.i.d. random variables. Necessary and sufficient conditions for intermittency of the fieldsu(t,·) ast→∞ are found in spectral terms ofH. Rough asymptotic formulas for the statistical moments and the almost sure behavior ofu(t,x) ast→∞ are also derived.

Survival Probability of a Random Walk Among a Poisson System of Moving Traps

This is a survey on the intermittent behavior of the parabolic Anderson model, which is the Cauchy problem for the heat equation with random potential on the lattice ℤd. We first introduce the model and give heuristic explanations of the long-time behavior of the solution, both in the annealed and the quenched setting for time-independent potentials. We thereby consider examples of potentials studied in the literature. In the particularly important case of an i.i.d. potential with double-exponential tails we formulate the asymptotic results in detail. Furthermore, we explain that, under mild regularity assumptions, there are only four different universality classes of asymptotic behaviors. Finally, we study the moment Lyapunov exponents for space-time homogeneous catalytic potentials generated by a Poisson field of random walks.

Multilevel large deviations and interacting diffusions

Abstract. We consider the parabolic Anderson problem ∂tu = κΔu + ξ(x)u on ℝ+×ℝd with initial condition u(0,x) = 1. Here κ > 0 is a diffusion constant and ξ is a random homogeneous potential. We concentrate on the two important cases of a Gaussian potential and a shot noise Poisson potential. Under some mild regularity assumptions, we derive the second-order term of the almost sure asymptotics of u(t, 0) as t→∞.

Quenched Lyapunov Exponent for the Parabolic Anderson Model in a Dynamic Random Environment

We review some old and prove some new results on the survival probability of a random walk among a Poisson system of moving traps on \({\mathbb{Z}}^{d}\), which can also be interpreted as the solution of a parabolic Anderson model with a random time-dependent potential. We show that the annealed survival probability decays asymptotically as e\({}^{-{\lambda }_{1}\sqrt{t}}\) for d = 1, as e\({}^{-{\lambda }_{2}t/\log t}\) for d = 2, and as e\({}^{-{\lambda }_{d}t}\) for d ≥ 3, where λ1 and λ2 can be identified explicitly. In addition, we show that the quenched survival probability decays asymptotically as e\({}^{-\tilde{{\lambda }}_{d}t}\), with \(\tilde{{\lambda }}_{d} > 0\) for all d ≥ 1. A key ingredient in bounding the annealed survival probability is what is known in the physics literature as the Pascal principle, which asserts that the annealed survival probability is maximized if the random walk stays at a fixed position. A corollary of independent interest is that the expected cardinality of the range of a continuous time symmetric random walk increases under perturbation by a deterministic path.

Stable limit laws for the parabolic Anderson model between quenched and annealed behaviour

SummaryLet (ξN) be a sequence of random variables with values in a topological space which satisfy the large deviation principle. For eachM and eachN, let ΞM, N denote the empirical measure associated withM independent copies of ξN. As a main result, we show that (ΞM, N) also satisfies the large deviation principle asM,N→∞. We derive several representations of the associated rate function. These results are then applied to empirical measure processes ΞM, N(t) =M−1 Σi=1N δξiN(t) 0≦t≦T, where (ξ1N,..., ξMN(t)) is a system of weakly interacting diffusions with noise intensity 1/N. This is a continuation of our previous work on the McKean-Vlasov limit and related hierarchical models ([4], [5]).

Analytic aspects of multilevel large deviations

We continue our study of the parabolic Anderson equation ∂u ∕ ∂t = κΔu + γξu for the space–time field \(u: \,{\mathbb{Z}}^{d} \times [0,\infty ) \rightarrow \mathbb{R}\), where κ ∈ [0, ∞) is the diffusion constant, Δ is the discrete Laplacian, γ ∈ (0, ∞) is the coupling constant, and \(\xi : \,{\mathbb{Z}}^{d} \times [0,\infty ) \rightarrow \mathbb{R}\) is a space–time random environment that drives the equation. The solution of this equation describes the evolution of a “reactant” u under the influence of a “catalyst” ξ, both living on \({\mathbb{Z}}^{d}\).

Multilevel Models of Interacting Diffusions and Large Deviations

We consider the solution to the parabolic Anderson model with homogeneous initial condition in large time-dependent boxes. We derive stable limit theorems, ranging over all possible scaling parameters, for the rescaled sum over the solution depending on the growth rate of the boxes. Furthermore, we give sufficient conditions for a strong law of large numbers. Resume. Nous considerons la solution du modele parabolique d’Anderson avec condition initiale homogene sur de grandes boites dependantes du temps. Nous derivons des theoremes limites stables, pour toutes les valeurs possibles des parametres d’echelle, pour la somme de la solution changee d’echelle en fonction du taux de croissance des boites. De plus, nous donnons des conditions suffisantes pour une loi des grands nombres. MSC: Primary 60K37; 82C44; secondary 60H25; 60F05