Les exposés ont lieu dans la salle de séminaire du CMI sauf mardi matin.

Lundi 23

9h30 Jan Nagel - 11h Matthias Meiners

14h Alessandra Faggionato

Mardi 24

exposés salle 101

9h Michele Salvi - 10h30 Pierre Mathieu

14h Andrey Piatnitski- 16h Jean-Christophe Mourrat

Mercredi 25

9h Jean-Chistophe Mourrat- 10h45 Andrey Piatnitski

Jeudi 26

9h Viviane Baladi - 10h45 Alejandro Ramirez

14h30 Alejandro Ramirez

Vendredi 27

9h Viviane Baladi - 11h Nina Gantert

Abstracts

**Viviane BALADI**

Réponse linéaire, un panorama de résultats et de contre-exemples

Lorsque chaque élément d'une famille f_t de dynamiques possède

une mesure invariante "naturelle" m_t (par exemple Sinai-Ruelle-Bowen)

il est naturel de chercher à calculer la dérivée de m_t

par rapport à t. C'est la question de la réponse linéaire.

Il y a une vingtaine d'années, Ruelle a établi une

formule pour la réponse linéaire dans le cas hyperbolique

lisse (Anosov). Cette formule reflète le fait que dans

ce cas m_t est le point fixe simple d'un opérateur (de transfert)

ayant un trou spectral. Cependant, des contre-exemples

à la réponse linéaire (obtenus avec Smania puis

Benedicks et Schnellmann, en dimension un, dans des familles

non structurellement stables) ont montré que la présence

d'un trou spectral ne suffit pas. Je présenterai

un panorama d'un quart de siècle de résultats rigoureux

sur la réponse linéaire en systèmes dynamiques, y compris

conjectures et questions ouvertes.

**Alessandra FAGGIONATO**

Gallavotti-Cohen type symmetries and linear response

In this talk we describe the Gallavotti-Cohen type symmetry for large deviation functionals

of Markov processes and show that, in the linear regime, it allows an alternative derivation of the Green-Kubo formula

and the Onsager reciprocity relations (as obtained by Lebowitz and Spohn).

**Nina GANTERT **

Einstein relation and linear response for one-dimensional Mott

variable-range hopping

We consider one-dimensional Mott variable-range hopping. This is a

random walk in random environment which can be interpreted as a

"long-range" random conductance model. We introduce a bias and prove the

linear response as well as the Einstein relation, under an assumption on

the exponential moments of the distances between neighbour points.

Based on joint work with Alessandra Faggionato and Michele Salvi.

Pierre MATHIEU

Diffusions on a torus: Nyquist relations and continuity of steady states.

**Matthias MEINERS**

Title: Biased random walk on a one-dimensional percolation cluster

Biased random walk on the infinite cluster of supercritical Bernoulli percolation on the integer lattice in dimension $d$

is a model for transport in an inhomogeneous medium.

Even though the model is easy to formulate, many natural questions remain unanswered.

In 2009, Axelson-Fisk and H\"aggstr\"om proposed a model of a biased random walk on a one-dimensional percolation cluster.

This model has been designed to be much simpler, but to show a similarly rich qualitative behavior.

In my talk, I will discuss some natural questions for the Axelson-Fisk-H\"aggstr\"om model

including the regularity of the speed of the walk as a function of its bias

or the speed of the walk at the critical bias.

The talk is based on joint work with Nina Gantert (Technische Universität München),

Jan-Erik Lübbers (Technische Universität Darmstadt) and Sebastian Müller (Aix-Marseille Université).

**JC MOURRAT**

Quantitative stochastic homogenization and the Einstein relation

Divergence-form operators with random coefficients homogenize over large scales. Recently, major progress has been made to make this convergence quantitative. I will outline some of the main results in this direction. I will then explain how this quantitative information can be used to prove more precise versions of the Einstein relation.

**Jan NAGEL**

The speed of biased random walk among random conductances

We consider a random walk on the d-dimensional lattice in the random conductance model. When we introduce a bias to the right, the process satisfies a law of large numbers with a nonzero effective speed. We are interested in properties of the speed as a function of the bias, in particular the differentiability and the monotonicity. We show that uniform ellipticity is not enough to guarantee monotonicity and prove strict monotonicity in the low-disorder regime. The talk is based on a joint work with Noam Berger and Nina Gantert.

**Alejandro RAMIREZ**

RANDOM WALK AT LOW DISORDER: NEW EXAMPLES OF

BALLISTIC RANDOM WALKS AND ASYMPTOTIC

EXPANSIONS

We consider a random walk whose jump probabilities are i.i.d. perturbations

of those of some random walk on the hypercubic lattice for dimensions larger than 2.

(i)

Asymptotic expansion of the environmental invariant measure for random walks at low disorder.

We will consider a random walk in the lattice in an i.i.d. environment which is a perturbation

of some random walk with homogeneous jump probabilities. We will show that the relevant invariant

measure of the environmental process admits an asymptotic expansion.

, where the terms of order 0 and 1 are explicit.

To prove this, we

first show how to represent an invariant measure as a Cesaro average

up to a geometric time, and express this in terms of Green functions.

Then, we expand the Green functions to obtain the expansion of the

invariant measure.

(ii)

Asymptotic expansion of the velocity for random walks at low

disorder

. Here we will show how to recover through the asymptotic

expansion for the invariant measure already obtained, an asymptotic

expansion for the velocity, which is a theorem proved by Sabot in 2004.

(iii)

Ballistic random walks which are perturbations of simple symmetric random walks.

We present a result proved by Sznitman in

2003, which gives explicit conditions under which a perturbed simple

symmetric random walk is ballistic, and explain the formal connection

between this result and the asymptotic expansion of Sabot for the velocity.

As a corollary, this result provides new examples of random walks which do not satisfy Kalikow condition.

(iv)

New examples of ballistic random walks in dimension

d= 3.

We show how to extend Sznitman’s theorem in pushing the average drift condition down

This provides new examples

of ballistic random walks which do not satisfy Kalikow condition.

This talk is based on joint works with David Campos in part(i)

, withClement Laurent, Christophe Sabot and Santiago Saglietti in part(iii)

and with Santiago Saglietti in part (iv)

**Andrey PIATNITSKI**

FDT for reversible diffusions in a random environment.

**Michele SALVI**

Regularity of biased random walks in random environment

We consider one-dimensional nearest-neighbor random walks in random environment under the influence of a bias. We analyze the regularity (differentiability, monotonicity, …) of the asymptotic velocity and of the diffusivity of the walk as functions of the intensity of the bias. Through a series of examples we exhibit some surprising properties of these quantities. In the case of the random conductance model we give a short and direct proof of the Einstein relation, extending the known set of hypothesis under which it is verified.

This is a joint work with Alessandra Faggionato.