Seminario de Probabilidad

En este seminario se expone trabajos recientes en teoría de probabilidades.
17:00 hrs.
Christian Sadel. PUC
From Anderson models to GOE statistics
Sala 5, FAcultad de matemáticas. PUC
We first prove some SDE limit for product of random matrices. We then apply this to transfer matrices of block-Jacobi operators which we use to obtain limiting statistics for Anderson models on long strips under proper rescaling of the randomness. With the correct sequence of limits we obtain a random matrix ensemble and finally the Sine_1 kernel.

Finally we construct a sequence of graphs (antitrees) where some averaging effect of a random potential mimics the rescaling in the step before. This way we obtain a sequence of random matrices with randomness of fixed strength (disorder) only along the diagonal for which we have limiting GOE statistics (Sine_1 kernel).
16:30 hrshrs.
Claire Delplanck. Universidad de Chile
intertwinings and Stein's method for birth-death processes
Sala John Von Neumann, 7mo piso, CMM
In this talk, I will present intertwinings between Markov processes and gradients, which are functional relations relative to the space-derivative of a Markov semigroup. I will focus on the discrete case involving birth-death processes, and recall a first-order relation as well as introduce a new second-order relation for a discrete Laplacian. As the main application, new quantitative bounds on the Stein factors of discrete distributions are provided. Stein's factors are a key component of Stein's method, a collection of techniques to bound the distance between probability distribution.
Chiranjib Mukherjee. University of Muenster
Sala 5, facultad de matemáticas
We consider the smoothed multiplicative noise stochastic heat equation (SHE) in dimension d ≥ 3. If β > 0 is a parameter that denotes the disorder strength (i.e., inverse temperature), we show the existence of a critical β ∈ (0, ∞) so that, as ε → 0, the solution of the SHE exhibits “weak disorder” when β < β and “strong disorder” when β > β. Furthermore, we investigate the behavior of the “quenched” and “annealed” path measures arising from the solution in the weak and strong disorder phases. Part of this talk is based on a joint work with A. Shamov (Weizmann Institute) and O. Zeitouni (Weizmann Institute/ Courant Institute) and Yannic Broeker (Muenster).
Daniel Remenik. Universidad de Chile
The KPZ fixed point
Sala de seminarios, 5to piso, CMM Universidad de Chile
I will describe the construction and main properties of the KPZ fixed point, which is the scaling invariant Markov process conjectured to arise as the universal scaling limit of all models in the KPZ universality class, and which contains all the fluctuation behavior seen in the class. The construction follows from an exact solution of the totally asymmetric exclusion process (TASEP) for arbitrary initial condition. This is joint work with K. Matetski and J. Quastel.
Amitai Linker. U. Chile
The Contact Process on Evolving Scale-Free Networks
Sala John Von Neumann, 7mo piso, CMM
In this talk we present some results on the contact process running on large scale-free networks, where nodes update their connections at independent random times. We will show that depending on the parameters of the model we can observe either slow extinction for all infection rates, or fast extinction if the infection rate is small enough. This differs from previous results in the case of static scale-free networks where only the first behaviour is observed. We will also show that the analysis of the asymptotic form of the metastable density of the process and its dependency on the model parameters can be used to understand the optimal mechanisms used by the infection to survive. Joint work with Peter Mörters and Emmanuel Jacob.
Santiago Saglietti. Technion
On (a proof for) Kesten's theorem on supercritical Branching Brownian Motion with Absorption.
sala 2, Facultad de matemáticas, PUC
Consider a (continuous time) branching Markov process in which:
  • One starts with a single particle initially located at some x>0, whose position evolves according to a Brownian motion with negative drift  -c which is absorbed upon reaching the origin.
  • This particle waits for an independent random exponential time of parameter r>0 and then branches, dying on the spot and giving birth to a random number m\geq0 of new particles at its current position.
  • These m new particles now evolve independently, each following the same stochastic behavior of their parent (evolve and then branch, and so on…).

It is well-known that if r(E(m)-1)>c^2/2 then this process is supercritical: with positive probability the process does not die out, in the sense that there are particles strictly above the origin for all times.

In this talk we will show that, whenever the process does not die out, as t\to\infty one has that the number of particles at time t inside any given set B grows like its expectation and, furthermore, that its proportion over their total number behaves like the (minimal) quasi-stationary distribution associated to the Brownian motion with drift -c and absorption at 0. In particular, this proves a result stated by Kesten in [1] for which there was no proof available until now.

Joint work with Oren Louidor.

[1] Kesten, H. (1978). Branching Brownian motion with absorption. Stochastic Processes and their Applications, 7(1), 9-47.
16:00 hrshrs.
Avelio Sepúlveda. Eth Zurich
A glimpse on excursion theory for the two-dimensional continuum Gaussian free field.
Sala de seminarios John Von Neumann, 7mo piso, CMM
Based on joint work with Juhan Aru, Titus Lupu and Wendelin Werner. Two-dimensional continuum Gaussian free field (GFF) has been one of the main objects of conformal invariant probability theory in the last ten years. The GFF is the two-dimensional analogue of Brownian motion when the time set is replaced by a 2-dimensional domain. Although one can not make sense of the GFF as a proper function, it can be seen as a “generalized function” (i.e. a Schwartz distribution). The main objective of this talk is to go through recent development in the understanding of the analogue, in the GFF context, of Ito’s excursion theory for Brownian motion. As a corollary, we will see how this theory can be used to define the Lévy transform of the GFF.
Julian Tugaut. Université Jean Monnet
Exit time of a self-stabilizing diffusion
sala John Von Neumann, 7mo piso, CMM, U. Cjile
In this talk, we briefly present some Freidlin and Wentzell results then we give a Kramers’type law satisfied by the McKean-Vlasov diffusion when the confining potential is uniformly strictly convex. We briefly present two previous proofs of this result before giving a third proof which is simpler, more intuitive and less technical.
Ciprian Tudor. Lille
A link between the zeta function and stochastic calculus
Seminarios John Von Neumann CMM, ubicada en Beauchef 851, Torre Norte, Piso 7 (ingreso ascensores torre poniente)
 The study of the zeros of the Riemann zeta function constitutes one of the most challenging problems in mathematics. A large literature in devoted to the study of the behavior of the zeta zeros. We will  discuss  how  tools from stochastic analysis, and in particular from Malliavin calculus (multiple integrals,  Wiener chaos, Stein method etc) can be used in the study of some aspects of the behavior of  the zeta function.
Milica Tomacevic. Tosca Team, Inria Sophia-Antipolis Mediterranee
A new probabilistic interpretation of Keller-Segel model for chemotaxis, application to 1-d.
Sala John Von Neumann, 7mo piso, CMM, U. Chile
The Keller Segel (KS) model for chemotaxis is a two-dimensional system of parabolic or elliptic PDEs. Motivated by the study of the fully parabolic model using probabilistic methods, we give rise to a non-linear SDE of McKean-Vlasov type with a highly non standard and singular interaction kernel.

In this talk I will briefly introduce the KS model, point out some of the PDE analysis results related to the model and then, in detail, analyze our probabilistic interpretation in the case d=1.
This is a joint work with Denis Talay (TOSCA team, INRIA Sophia-Antipolis Mediterranee).
Christophe Profeta. Universite D'evry Val D'essonne
Stable Langevin model with diffusive-reflective boundary conditions. 
Sala 5, Facultad de matemáticas, Campus San Joaquín. PUC.
We consider a one-dimensional stable Langevin process confined in the upper half-plane and submitted to a diffusive-reflective boundary condition whenever the particle position hits 0. We show that different regimes appear according to the value of the chosen parameters. We then use this study to construct the law of a (free) stable Langevin process conditioned to stay positive, thus extending earlier works on the integrated Brownian motion. Such construction finally enables us to improve some recent persistence probability estimates. This is a joint work with Jean-François Jabir.
16:30 hrs.hrs.
Karl Liechty . Depaul
Propagation of critical behavior for unitary invariant plus GUE random matrices
Sala John Von Neumann, 7mo piso, CMM.
 It is a well known and celebrated fact that the eigenvalues of random Hermitian matrices from a unitary invariant ensemble form a determinantal point process with correlation kernel given in terms of a system of orthogonal polynomials on the real line. It is a much more recent result that the eigenvalues of the sum of such a random matrix with a matrix from the Gaussian unitary ensemble (GUE) also forms a determinantal point process, with the kernel given in terms of the Weierstrass transform of the original kernel. I'll talk about the case in which the limiting distribution of eigenvalues is critical in the sense that there is a non-generic scaling limit for the correlation kernel, and discuss the effect of a Gaussian perturbation on the limiting critical kernel. This is joint work with Tom Claeys, Arno Kuijlaars, and Dong Wang.
Bernardo Nunes Borges de Lima . Ufmg
Embedding binary sequences into Bernoulli site percolation on Z3
Sala Jacques L. Lions, CMM, Universidad de Chile 
We investigate the problem of embedding infinite binary sequences into Bernoulli site percolation on Zd with parameter p. In 1995, I. Benjamini and H. Kesten proved that, for d≥10 and p=1/2, all sequences can be embedded, almost surely. They conjectured that the same should hold for d≥3. We consider d≥3 and p∈(pc(d),1−pc(d)), where pc(d)<1/2 is the critical threshold for site percolation on Zd. We show that there exists an integer M=M(p), such that, a.s., every binary sequence, for which every run of consecutive {0s} or {1s} contains at least M digits, can be embedded. Joint work with M. Hilário (UFMG), P. Nolin (ETH) and V. Sidoravicius (IMPA)
Christophe Profeta. Université D’Evry Val D’Essonn
Sala 2 - Facultad de Matemáticas a las 16:30 Hrs.
Ioannis Papageorgiou. UBA
The Log-Sobolev Inequality for Unbounded Spin Systems on The Lattice.
Sala 2 - facultad de Matemáticas a las 16:30 Hrs.
Aurelia Deshayes. UBA
Scaling Limit of Subcritical Contact Process
Sala 2 - Facultad de Matemáticas a las 16:30 Hrs.
Chi Nguyen. UBA
Sala 2 - Faculad de Matemáticas a las 16:30 Hrs.
Santiago Saglietti. P. Universidad Católica de Chile
The Kesten-Stigum Theorem in L^2 for Abbmd
CMM Sala por confirmar a las 16:30 Hrs.
Mauricio Duarte. UBA
Gravitation Versus Brownian Motion
Sala 2, Facultad de Matemáticas a las 16:30 Hrs.
Julián Martinez. UBA
Hydrodynamic Limit for Branching Brownian Particles With Spatial Selection and The Kpp Equation.
Sala 2 de la Facultad de Matemáticas a las 16:30 Hrs.