WEAK AND STRONG DISORDER FOR THE STOCHASTIC HEAT EQUATION IN d ≥ 3

We consider the smoothed multiplicative noise stochastic heat equation (SHE) in dimen-

sion 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).

Sala 5, Facultad de Matemáticas, Universidad Católica, 17:00 hrs.

nov / 2017
06

The KPZ fixed point

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.

Sala de seminarios 5to piso CMM

oct / 2017
16

The Contact Process on Evolving Scale-Free Networks

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.

sep / 2017
04

On (a proof for) Kesten's theorem on supercritical Branching Brownian Motion with Absorption.

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.

ago / 2017
21

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

*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.

ago / 2017
14

Pontificia Universidad Católica de Chile (PUC-Chile)

Av. Vicuña Mackenna 4860, Macul,

Santiago – Chile

(+56 2) 2354 5779

Facultad de Ciencias Físicas y Matemáticas (FCFM)

Universidad de Chile

Beauchef 851, Edificio Norte, Piso 7,

Santiago – Chile