Seminarios

Abstract:

Abstract:

Abstract:

We consider an i.i.d. balanced environment  $\omega(x,e)=\omega(x,-e)$,genuinely d dimensional on the lattice and show that there exist a
positive constant $C$ and a random radius $R(\omega)$ with streched
exponential tail such that every non negative
$\omega$ harmonic function $u$ on the ball  $B_{2r}$ of radius
$2r>R(\omega)$,
we have $\max_{B_r} u <= C \min_{B_r} u$. Our proof relies on a quantitative quenched invariance principle for the corresponding random walk in  balanced random environment and a careful analysis of the directed percolation cluster. This result extends Martins Barlow’s Harnack’s inequality for i.i.d. bond percolation to the directed case.

This is joint work with N.Berger  M. Cohen and X. Guo.

Sala 3, Facultad de Matemáticas, PUC

Abstract:

We consider a continuous time particle system on a graph L being either Z,  Z_n, a segment {1,…, n}, or Z^d, with state space Ek={0,…,k-1} for some k belonging to {infinity, 2, 3, …}. We also assume that the Markovian evolution is driven by some translation invariant local dynamics with bounded width dependence, encoded by a rate matrix T. These are standard settings, satisfied by many studied particle systems. We provide some sufficient and/or necessary conditions on the matrix T, so that this Markov process admits some simple invariant distribution, as a product measure, as the distribution of a Markov process indexed by Z or {1,…, n} (if L=Z or {1,…,n}), or as a Gibbs measure (if L=Z_n). These results are mainly obtained with some manipulations of finite words, with alphabet Ek, representing subconfigurations of the systems. For the case L=Z, we give a procedure to find the set of invariant i.i.d and Markov measures.

Abstract:

This talk is based on a joint work in progress with L. Zambotti (UPMC). First, I will give a brief introduction to the theory of rough paths focusing on the case of Hölder regularity between 1/3 and 1/2. After this, I will address the basic problem of construction of a geometric rough path over a given ɑ-Hölder path in a finite-dimensional vector space. Although this problem was already solved by Lyons and Victoir in 2007, their method relies on the axiom of choice and thus is not explicit; in exchange the proof is simpler. In an upcoming paper, we provide an explicit construction clarifying the connection between rough paths theory and free (nilpotent) Lie algebras. In particular, we use an explicit form of the Baker–Campbell–Hausdorff formula due to Loday in order to provide explicit expressions and bounds to achieve such a construction.

Sala de seminarios del 5to piso, CMM, 16:00 hrs.