Seminario de Probabilidad

En este seminario se expone trabajos recientes en teoría de probabilidades.
Mauricio Duarte. Uab
Hard Ball Collisions
Facultas de Matemáticas (sala 5)
We will explore the behavior of systems of a large number of hard balls under elastic collisions. We prove by example that the  number of elastic collisions of $n$ balls of equal mass and equal size in $d$-dimensional space can be greater than $n^3/27$ for $n\geq 3$ and $d\geq 2$. The previously known lower bound was of order $n^2$.
16:00 hrs.
Roberto Cortez.
Particle Systems and Propagation of Chaos for Some Kinetic Models
Salade seminarios John von Neuman, 7mo piso, CMM
In this talk we will make a quick historical review of some equations arising in the classical kinetic theory of gases and related models. We will start with the Boltzmann equation, which describes the evolution of the distribution of positions and velocities of infinitely many small particles of a gas in 3-dimensional space, subjected to elastic binary collisions. We consider a finite $N$-particle system and introduce the important concept of propagation of chaos: the convergence, as $N\to\infty$ and for each time $t\geq 0$, of the distribution of the particles towards the solution of the equation. We present some recent quantitative propagation of chaos results for the spatially homogeneous Boltzmann equation and Kac's model. Lastly, we will introduce a relatively new class of one-dimensional kinetic equations modelling wealth redistribution in a population performing binary trades. When trades preserve wealth only on average, these models can exhibit an equilibrium distribution with heavy tails, as is seen in real-world economies. We focus on the corresponding finite $N$-particle system and study how the heaviness of the tails of its distribution relates to that of the limit kinetic equation. Unless wealth is preserved exactly in each trade, we find important qualitative differences between both cases.
15:30 hrs.hrs.
Bao Nguyen. Kth
Two-Time Distribution for Kpz Growth in One Dimension
Sala de seminarios John Von Neumann, CMM, 7mo piso
Consider the height fluctuations H(x,t) at spatial point x and time t of one-dimensional growth models in the Kardar-Parisi-Zhang (KPZ) class. The spatial point process at a single time is known to converge at large time to the Airy processes (depending on the initial data). The multi-time process however is less well understood. In this talk, I will discuss the result by Johansson on the two-time problem, namely the joint distribution of (H(x,t),H(x,at)) with a>0, in the case of droplet initial data. I also show how to adapt his approach to the flat initial case. This is based on joint work with Kurt Johansson.
Remy Sanchis. Universidade Federal de Minas Gerais
sala de seminarios John Von Neumann, CMM, 7mo piso
Javiera Barreea. Uai
Sharp Bounds for The Reliability of a K-Out-Of-N Sys- Tem Under Dependent Failures Using Cutoff Phenomenon Techniques
Facultad de matemáticas PUC, Sala 5

In this work we consider the reliability of a network where link failures are correlated. We define the reliability as the probability of the network to be working at a given time instant. Our main contribution is a collection of results giving a detailed analysis of a non-trivial scaling regime for the probability of the network being working at a certain time, as the time and size of network scales. Here we consider that the network fails when there are no links working, or more generally when less than k-out-of-n edges are working (with k close to n) like in [2] and [5]. Our results allow to study the common-cause failure models describe in [3] on networks in a realistic, relevant, yet practical, fashion: it allows to capture correlated components in the network; it allows to estimate and give error bounds for the failure probabilities of the system; and at same time only needs to specify a reduced family of parameters. Moreover, our results for the k-out-of-n failure model allow to give new scaling regimes for the probabilistic behavior of the last-ordinals in the theory of extreme values for dependent tuples. The techniques are similar to those used to estimate the asymptotic convergence profile for ergodic Markov chains [1] or [4]. 

  1. [1]  J. Barrera and B. Ycart Bounds for Left and Right Window Cutoffs. ALEA, Lat. Am. J. Probab. Math. Stat., 11 (2): 445–458, 2014.

  2. [2]  I. Bayramoglu and M. Ozkut. The Reliability of Coherent Systems Subjected to Marshall?Olkin Type Shocks. EEE Transactions on Reliability,, 64 (1): 435–443, 2015.

  3. [3]  U. Cherubini, F. Durante, and S. Mulinacci. Marshall – Olkin Distributions-Advances in Theory and Applications: Bologna, Italy, October 2013, volume 141. Springer, 2015. 

  4. [4]  B. Lachaud and B. Ycart Convergence Times for Parallel Markov Chains. Positive systems, 169–176, 2006.

  5. [5]  T. Yuge, M. Maruyama, and S. Yanagi Reliability of a k-out-of-n Systemwith Common-Cause Failures Using Multivariate Exponential Distribution. Procedia Computer Sci- ence, 96: 968?976, 2016.
Rangel Baldasso. Bar Ilan University
Noise Sensitivity for Voronoi Percolation
Sala de seminarios John Von Neumann, CMM, 7mo piso
Noise sensitivity is a concept that measures if the outcome of Boolean function can be predicted when one is given its value for a perturbation of the configuration. A sequence of functions is noise sensitive when this is asymptotically not possible. A non-trivial example of a sequence that is noise sensitive is the crossing functions in critical two-dimensional Bernoulli bond percolation. In this setting, noise sensitivity can be understood via the study of randomized algorithms. Together with a discretization argument, these techniques can be extended to the continuum setting. In this talk, we prove noise sensitivity for critical Voronoi percolation in dimension two, and derive some consequences of it.
Based on a joint work with D. Ahlberg.
Jean-Dominique Deuschel. T.u. Berlin
Harnack Inequality for Degenerate Balanced Random Walks.
Sala 3, Facultad de Matemáticas, PUC
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.
Luis Fredes. Bordeaux
Invariant Mesures of Discrete Interacting Particle Systems: Algebraic Aspects
Sala John Von Neumann, 7mo piso, CMM
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.
Nikolas Tapia. Universidad de Chile
Construction of Geometric Rough Paths
Sala de seminarios del 5to piso, CMM
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 ?-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.
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
Weak and Strong Disorder for The Stochastic Heat Equation in D ? 3
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.