Núcleo Modelos Estocásticos de Sistemas Complejos y Desordenados

Particle systems and propagation of chaos for some kinetic models

Event Date: Jun 11, 2018 in Núcleo Modelos Estocásticos de Sistemas Complejos y Desordenados, Seminars

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...

Read More

Two-time distribution for KPZ growth in one dimension

Event Date: May 29, 2018 in Núcleo Modelos Estocásticos de Sistemas Complejos y Desordenados, Seminars

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...

Read More

Invariant measures of discrete interacting particle systems: Algebraic aspects

Event Date: Jan 09, 2018 in Núcleo Modelos Estocásticos de Sistemas Complejos y Desordenados, Seminars

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...

Read More

Construction of geometric rough paths

Event Date: Nov 27, 2017 in Núcleo Modelos Estocásticos de Sistemas Complejos y Desordenados, Seminars

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...

Read More

The KPZ fixed point

Event Date: Oct 16, 2017 in Núcleo Modelos Estocásticos de Sistemas Complejos y Desordenados, Seminars

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.

Read More

A glimpse on excursion theory for the two-dimensional continuum Gaussian free field.

Event Date: Aug 14, 2017 in Núcleo Modelos Estocásticos de Sistemas Complejos y Desordenados, Seminars

Resumen:   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 cannot 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...

Read More