Seminario de Probabilidad

En este seminario se expone trabajos recientes en teoría de probabilidades.
2018-11-20
17:00hrs.
Atilla Yilmaz. Temple
Homogenization of a class of one-dimensional nonconvex viscous Hamilton-Jacobi equations with random potential
Sala 1
Abstract:
I will present joint work with Elena Kosygina and Ofer Zeitouni in which we prove the homogenization of a class of one-dimensional viscous Hamilton-Jacobi equations with random Hamiltonians that are nonconvex in the gradient variable. Due to the special form of the Hamiltonians, the solutions of these PDEs with linear initial conditions have representations involving exponential expectations of controlled Brownian motion in a random potential. The effective Hamiltonian is the asymptotic rate of growth of these exponential expectations as time goes to infinity and is explicit in terms of the tilted free energy of (uncontrolled) Brownian motion in a random potential. The proof involves large deviations, construction of correctors which lead to exponential martingales, and identification of asymptotically optimal policies.
 

http://nm.cmm.uchile.cl/seminarios/
2018-10-10
16:00hrs.
Sandro Gallo. Universidade Federal de Sao Carlos
?Soon
Facultad de matemáticas, sala por confirmar
Abstract:
The return time picture for stationary processes was originally addressed by Poincaré with his famous recurrence theorem. In the case of rare events (events of small measure), it is naturally associated to long time behavior, since the Kac Lemma states that the expected return time to a set is the inverse of the measure of the set. It is now well understood that the short time behavior, through the control of the probability of “as soon as possible”-returns to the set, plays a fundamental role in the Poincaré recurrence theory. The main objective of the talk will be to explain how this probability of “soon” returns shows up in the calculations and results of some statistical properties of recurrence times.
http://nm.cmm.uchile.cl/seminarios/
2018-09-10
16:00hrs.
Codina Cotar. University College of London
Disorder relevance for non-convex random gradient Gibbs measures in d<=2
Sala K301
Abstract:
t is a famous result of statistical mechanics that, at low enough temperature, the random field Ising model is disorder relevant for d<=2, i.e. the phase transition between uniqueness/non-uniqueness of Gibbs measures disappears,  and disorder irrelevant otherwise (Aizenman-Wehr 1990). Generally speaking, adding disorder to a model tends to destroy the non-uniqueness of Gibbs measures. 


In this talk we consider - in non-convex potential regime - a random gradient model with disorder in which the interface feels like a bulk term of random fields. We show that this model is disorder relevant with respect to the question of uniqueness of gradient Gibbs measures for a  class of non-convex potentials and a disorders.

No previous knowledge of gradient models will be assumed in the talk

2018-09-05
16:00hrs.
Marcelo Hilario. Ufmg
Random Walks in Dynamic Random Environment
sala 2
2018-08-29
16:00hrs.
Gerardo Barrera. Universidad de Alberta
termalización abrupta para perturbaciones aleatorias de sistemas dinámicos
sala 2
Abstract:

En este minicurso, estudiaremos el fenómeno de la convergencia abrupta (cut–off) a su medida invariante. Este fenómeno es notado por Diaconis, Aldous et al en los modelos Markovianos de barajamiendo de cartas. Nuestro modelo será una ecuación diferencial ordinaria (EDO) con un único punto fijo, el cual asumiremos es un atractor global. Agregamos una pequeña perturbación a esta ecuación y obtenemos un sistema dinámico aleatorio (SDA).

Si la perturbación es Gaussiana, bajo condiciones generales el SDA converge a una única distribución de equilibrio y dicha distribución de equilibrio es bien aproximada por una distribución Gaussiana con desviación estándar proporcional a la perturbación. Más aún, para cada perturbación fija, la convergencia de la distribución del SDA a su distribución de equilibrio es exponencialmente rápida. En este caso demostraremos que la convergencia es abrupta: en una ventana de tiempo pequeña comparada con el escala natural del proceso, la distancia al equilibrio cae desde su máximo valor posible a cerca de cero, y solo después de esa ventana de tiempo la convergencia es exponencialmente rápida. Esto es conocido en el contexto de Cadenas de Markov como el fenómeno de cut–off. Cuando el punto fijo de la EDO no es hiperbólico, demostraremos que no tenemos el fenómeno de cut–off. (Este es un trabajo conjunto con Milton Jara).

Por otro lado cuando la perturbación es una proceso de Lévy, nuevamente bajo condiciones generales el SDA converge a una única distribución de equilibrio y dicha distribución de equilibrio es bien aproximada por una distribución Q–descomponible. Como toy model estudiaremos el proceso de Ornstein–Uhlenbeck dirigido por un proceso de Lévy. Bajo condiciones de log–integrabilidad en la medida de Lévy, tendremos que el SDA posee una única distribución de equilibrio. Asumiendo la condición de Orey–Masuda tendremos regularidad en la distribuciones al tiempo 0 < t ≤ ∞ y esto nos permitirá probar el fenómeno de cut–off en la distancia de variación total. El tiempo de cut–off y la ventana de cut–off solo dependen de la parte deterministica del SDA. (Este es un trabajo conjunto con Juan Carlos Pardo).

Asumiendo que tenemos momentos en la medida de Lévy, el fenómeno continua siendo cierto. (Este es un trabajo en progreso con Juan Carlos Pardo y Michael Hoegele).

2018-08-27
16:00hrs.
Gerardo Barrera. Universidad de Alberta
Termalización abrupta para perturbaciones aleatorias de sistemas dinámicos
Sala 5
Abstract:

En este minicurso, estudiaremos el fenómeno de la convergencia abrupta (cut–off) a su medida invariante. Este fenómeno es notado por Diaconis, Aldous et al en los modelos Markovianos de barajamiendo de cartas. Nuestro modelo será una ecuación diferencial ordinaria (EDO) con un único punto fijo, el cual asumiremos es un atractor global. Agregamos una pequeña perturbación a esta ecuación y obtenemos un sistema dinámico aleatorio (SDA).

Si la perturbación es Gaussiana, bajo condiciones generales el SDA converge a una única distribución de equilibrio y dicha distribución de equilibrio es bien aproximada por una distribución Gaussiana con desviación estándar proporcional a la perturbación. Más aún, para cada perturbación fija, la convergencia de la distribución del SDA a su distribución de equilibrio es exponencialmente rápida. En este caso demostraremos que la convergencia es abrupta: en una ventana de tiempo pequeña comparada con el escala natural del proceso, la distancia al equilibrio cae desde su máximo valor posible a cerca de cero, y solo después de esa ventana de tiempo la convergencia es exponencialmente rápida. Esto es conocido en el contexto de Cadenas de Markov como el fenómeno de cut–off. Cuando el punto fijo de la EDO no es hiperbólico, demostraremos que no tenemos el fenómeno de cut–off. (Este es un trabajo conjunto con Milton Jara).

Por otro lado cuando la perturbación es una proceso de Lévy, nuevamente bajo condiciones generales el SDA converge a una única distribución de equilibrio y dicha distribución de equilibrio es bien aproximada por una distribución Q–descomponible. Como toy model estudiaremos el proceso de Ornstein–Uhlenbeck dirigido por un proceso de Lévy. Bajo condiciones de log–integrabilidad en la medida de Lévy, tendremos que el SDA posee una única distribución de equilibrio. Asumiendo la condición de Orey–Masuda tendremos regularidad en la distribuciones al tiempo 0 < t ≤ ∞ y esto nos permitirá probar el fenómeno de cut–off en la distancia de variación total. El tiempo de cut–off y la ventana de cut–off solo dependen de la parte deterministica del SDA. (Este es un trabajo conjunto con Juan Carlos Pardo).

Asumiendo que tenemos momentos en la medida de Lévy, el fenómeno continua siendo cierto. (Este es un trabajo en progreso con Juan Carlos Pardo y Michael Hoegele).

2018-08-22
16:00hrs.
Gerardo Barrera. Universidad de Alberta
Termalización abrupta para perturbaciones aleatorias de sistemas dinámicos
Sala 2
Abstract:

En este minicurso, estudiaremos el fenómeno de la convergencia abrupta (cut–off) a su medida invariante. Este fenómeno es notado por Diaconis, Aldous et al en los modelos Markovianos de barajamiendo de cartas. Nuestro modelo será una ecuación diferencial ordinaria (EDO) con un único punto fijo, el cual asumiremos es un atractor global. Agregamos una pequeña perturbación a esta ecuación y obtenemos un sistema dinámico aleatorio (SDA).

Si la perturbación es Gaussiana, bajo condiciones generales el SDA converge a una única distribución de equilibrio y dicha distribución de equilibrio es bien aproximada por una distribución Gaussiana con desviación estándar proporcional a la perturbación. Más aún, para cada perturbación fija, la convergencia de la distribución del SDA a su distribución de equilibrio es exponencialmente rápida. En este caso demostraremos que la convergencia es abrupta: en una ventana de tiempo pequeña comparada con el escala natural del proceso, la distancia al equilibrio cae desde su máximo valor posible a cerca de cero, y solo después de esa ventana de tiempo la convergencia es exponencialmente rápida. Esto es conocido en el contexto de Cadenas de Markov como el fenómeno de cut–off. Cuando el punto fijo de la EDO no es hiperbólico, demostraremos que no tenemos el fenómeno de cut–off. (Este es un trabajo conjunto con Milton Jara).

Por otro lado cuando la perturbación es una proceso de Lévy, nuevamente bajo condiciones generales el SDA converge a una única distribución de equilibrio y dicha distribución de equilibrio es bien aproximada por una distribución Q–descomponible. Como toy model estudiaremos el proceso de Ornstein–Uhlenbeck dirigido por un proceso de Lévy. Bajo condiciones de log–integrabilidad en la medida de Lévy, tendremos que el SDA posee una única distribución de equilibrio. Asumiendo la condición de Orey–Masuda tendremos regularidad en la distribuciones al tiempo 0 < t ≤ ∞ y esto nos permitirá probar el fenómeno de cut–off en la distancia de variación total. El tiempo de cut–off y la ventana de cut–off solo dependen de la parte deterministica del SDA. (Este es un trabajo conjunto con Juan Carlos Pardo).

Asumiendo que tenemos momentos en la medida de Lévy, el fenómeno continua siendo cierto. (Este es un trabajo en progreso con Juan Carlos Pardo y Michael Hoegele).

2018-06-19
16:00hrs.
Mauricio Duarte. Uab
Hard Ball Collisions
Facultas de Matemáticas (sala 5)
Abstract:
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 $n^2$.
2018-06-12
16:00 hrs.
Roberto Cortez.
Particle systems and propagation of chaos for some kinetic models
Salade seminarios John von Neuman, 7mo piso, CMM
Abstract:
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.
2018-05-29
16:45hrs.
Remy Sanchis. Universidade Federal de Minas Gerais
TBA
sala de seminarios John Von Neumann, CMM, 7mo piso
Abstract:
TBA
2018-05-29
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
Abstract:
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.
2018-05-15
16:00hrs.
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
Abstract:

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. 


http://nm.cmm.uchile.cl/seminarios/
2018-05-03
14:30hrs.
Rangel Baldasso. Bar Ilan University
Noise sensitivity for Voronoi Percolation
Sala de seminarios John Von Neumann, CMM, 7mo piso
Abstract:
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.

http://nm.cmm.uchile.cl
2018-04-02
16:30hrs.
Jean-Dominique Deuschel. T.u. Berlin
Harnack inequality for degenerate balanced random walks.
Sala 3, Facultad de Matemáticas, PUC
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.
 

http://nm.cmm.uchile.cl/seminarios/
2018-01-09
15:00hrs.
Luis Fredes. Bordeaux
Invariant mesures of discrete interacting particle systems: algebraic aspects
Sala John Von Neumann, 7mo piso, CMM
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.
http://nm.cmm.uchile.cl/seminarios/
2017-11-27
16:00hrs.
Nikolas Tapia. Universidad de Chile
Construction of geometric rough paths
Sala de seminarios del 5to piso, CMM
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 ?-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.
http://nm.cmm.uchile.cl/seminarios/
2017-11-20
17:00 hrs.
Christian Sadel. PUC
From Anderson models to GOE statistics
Sala 5, FAcultad de matemáticas. PUC
Abstract:
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). 
 

http://nm.cmm.uchile.cl/seminarios/
2017-11-13
16:30 hrshrs.
Claire Delplanck. Universidad de Chile
intertwinings and Stein's method for birth-death processes
Sala John Von Neumann, 7mo piso, CMM
Abstract:
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. 
  
http://nm.cmm.uchile.cl/seminarios/
2017-11-06
17:00hrs.
Chiranjib Mukherjee. University of Muenster
WEAK AND STRONG DISORDER FOR THE STOCHASTIC HEAT EQUATION IN d ? 3
Sala 5, facultad de matemáticas
Abstract:
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).
http://nm.cmm.uchile.cl/seminarios/
2017-10-16
16:30hrs.
Daniel Remenik. Universidad de Chile
The KPZ fixed point
Sala de seminarios, 5to piso, CMM Universidad de Chile
Abstract:
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.
http://nm.cmm.uchile.cl/