Friday June 8

Utrecht, Janskerkhof 15a, room 001

11:15-13:00

Speaker: Anton Thalmaier (Luxembourg)

**Title : Brownian motion, Ricci curvature, functional inequalities and**

** geometric flows III: Heat equations under geometric flows and**

** entropy formulas**

14:30-15:15

Speaker: Reka Szabó (Groningen)

**Title : Inhomogeneous percolation on ladder graphs**

15:30-16:15

Speaker: Federico Sau (Delft)

**Title : Exclusion process in symmetric dynamic environment: quenched**

** hydrodynamics**

Abstract Anton Thalmaier

We continue our description of Ricci curvature and Ricci flow in terms

of Brownian motion. After a general discussion of heat equations under

a geometric flow, we outline a probabilistic approach to entropy

formulas. In particular, we define variants of Perelman’s entropy

functionals, using the Wiener measure as reference measure, and

investigate their monotonicity along the flow.

(The slides of the first and second lecture are on the Mark Kac website)

Abstract Reka Szabó

We define an inhomogeneous percolation model on “ladder graphs” obtained

as direct products of an arbitrary graph G = (V,E) and the set of

integers (vertices are thought of as having a “vertical” component

indexed by an integer). We make two natural choices for the set of

edges, producing an unoriented graph and an oriented graph. These graphs

are endowed with percolation configurations in which independently,

edges inside a fixed infinite “column” are open with probability q, and

all other edges are open with probability p. We prove that the function

that maps q into the corresponding critical percolation threshold is

continuous in (0,1).

Abstract Federico Sau

For the simple exclusion process evolving in a symmetric dynamic random

environment, we derive the hydrodynamic limit from the quenched

invariance principle of the corresponding random walk. For instance, if

the limiting behavior of a test particle resembles that of Brownian

motion on a diffusive scale, the empirical density, in the limit and

suitably rescaled, evolves according to the heat equation.

In this talk we make this connection explicit for the simple exclusion

process and show how self-duality of the process enters the problem.

This allows us to extend the result to other conservative particle

systems (e.g. IRW, SIP) which share a similar property.

Work in progress with F. Collet, F. Redig and E. Saada. http://www.win.tue.nl/markkac/