### A limit theorem for the position of a particle in the Lorentz model.

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

The parabolic Anderson model is the Cauchy problem for the heat equation with a random potential. We consider this model in a setting which is continuous in time and discrete in space, and focus on time-constant, independent and identically distributed potentials with polynomial tails at infinity. We are concerned with the long-term temporal dynamics of this system. Our main result is that the periods, in which the profile of the solutions remains nearly constant, are increasing linearly over time,...

In his 2003 paper, Varadhan proves the averaged large deviation principle for the mean velocity of a particle taking a nearest-neighbor random walk in a uniformly elliptic i.i.d. environment on ℤd with d≥1, and gives a variational formula for the corresponding rate function Ia. Under Sznitman’s transience condition (T), we show that Ia is strictly convex and analytic on a non-empty open set , and that the true velocity of the particle is an element (resp. in the boundary) of when the walk is non-nestling...

We prove the Einstein relation, relating the velocity under a small perturbation to the diffusivity in equilibrium, for certain biased random walks on Galton–Watson trees. This provides the first example where the Einstein relation is proved for motion in random media with arbitrarily slow traps.

A linear Boltzmann equation is interpreted as the forward equation for the probability density of a Markov process $\left(K\right(t),i(t),Y(t\left)\right)$ on $({\mathbb{T}}^{2}\times \{1,2\}\times {\mathbb{R}}^{2})$, where ${\mathbb{T}}^{2}$ is the two-dimensional torus. Here $\left(K\right(t),i(t\left)\right)$ is an autonomous reversible jump process, with waiting times between two jumps with finite expectation value but infinite variance. $Y\left(t\right)$ is an additive functional of $K$, defined as ${\int}_{0}^{t}v\left(K\left(s\right)\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}s$, where $\left|v\right|\sim 1$ for small $k$. We prove that the rescaled process ${(NlnN)}^{-1/2}Y\left(Nt\right)$ converges in distribution to a two-dimensional Brownian motion. As a consequence, the appropriately...

We consider an energy conserving linear dynamics that we perturb by a Glauber dynamics with random site dependent intensity. We prove hydrodynamic limits for this non-reversible system in random media. The diffusion coefficient turns out to depend on the random field only by its statistics. The diffusion coefficient defined through the Green–Kubo formula is also studied and its convergence to some homogenized diffusion coefficient is proved.

We develop a potential theoretic approach to the problem of metastability for reversible diffusion processes with generators of the form $-\u03f5\Delta +\nabla F(\xb7)\nabla $ on ${\mathbb{R}}^{d}$ or subsets of ${\mathbb{R}}^{d}$, where $F$ is a smooth function with finitely many local minima. In analogy to previous work on discrete Markov chains, we show that metastable exit times from the attractive domains of the minima of $F$ can be related, up to multiplicative errors that tend to one as $\u03f5\downarrow 0$, to the capacities of suitably constructed sets. We show that these capacities...

We continue the analysis of the problem of metastability for reversible diffusion processes, initiated in [BEGK3], with a precise analysis of the low-lying spectrum of the generator. Recall that we are considering processes with generators of the form $-\u03f5\Delta +\nabla F(\xb7)\nabla $ on ${\mathbb{R}}^{d}$ or subsets of ${\mathbb{R}}^{d}$, where $F$ is a smooth function with finitely many local minima. Here we consider only the generic situation where the depths of all local minima are different. We show that in general the exponentially small part of the spectrum...

We consider the parabolic Anderson model, the Cauchy problem for the heat equation with random potential in ℤd. We use i.i.d. potentials ξ:ℤd→ℝ in the third universality class, namely the class of almost bounded potentials, in the classification of van der Hofstad, König and Mörters [Commun. Math. Phys.267 (2006) 307–353]. This class consists of potentials whose logarithmic moment generating function is regularly varying with parameter γ=1, but do not belong to the class of so-called double-exponentially...

We consider a bounded step size random walk in an ergodic random environment with some ellipticity, on an integer lattice of arbitrary dimension. We prove a level 3 large deviation principle, under almost every environment, with rate function related to a relative entropy.