Lectures
Algumas propriedades sobre cadeias de Markov em tempo contínuo
Invited speaker: Leandro Correia Araújo (UFBA)
Abstract: Vamos construir duas cadeias de Markov em tempo contínuo através de uma cadeia em tempo discreto e veremos conceitos relacionados, como a Forma de Dirichlet, a função de Green, alguns resultados da teoria do potencial e, por fim, a Fórmula de Feynman-Kac.
Date: 19/11 at 3 p.m. on Google Meet (https://meet.google.com/ynr-orjd-dqz)
Large time behaviour of the parabolic Anderson model
Invited speaker: Willem van Zuijlen (Weierstraß institute Berlin)
Abstract: We consider the parabolic Anderson model with a white noise potential in two dimensions. This model is also called the stochastic heat equation with a multiplicative noise. We study the large time asymptotics of the total mass of the solution. Due to the irregularity of the white noise, in two dimensions the equation is a priori not well-posed. Using paracontrolled calculus or regularity structures one can make sense of the equation by a renormalisation, which can be thought of as ``subtracting infinity of the potential''. To obtain the asymptotics of the total mass we use a spectral decomposition, an alternative Feynman-Kac type representation and heat-kernel estimates which come from joint works with Khalil Chouk, Wolfgang König and Nicolas Perkowski.
Date: 21/10 at 9 a.m. on Google Meet (meet.google.com/ksu-ycsy-odo)
Node immunization in networks: a scalable searching algorithm based on random rooted forests.
Invited speaker: Luca Avena (Leiden University)
Abstract: We are interested in the so-called multiple-node immunization for complex networks under attack of a virus. The latter is a hot topic in network science and it consists in identifying and removing a set of nodes of a given size in a graph to maximally impede the virus spread. Several approaches have been proposed in the literature based on numerical and theoretical insights on how classical models for virus spread (so-called compartmental models) evolve on graphs.
Based on the stability analysis of these compartmental models, the maximal eigenvalue of the adjacency matrix of the graph has been proposed as a measure for how much resilient the network is. Thus one of the most common approach for immunization consists in identifying the set of nodes of a given cardinality, for which the reduced network, obtained by removing these nodes, has maximal largest eigenvalue.This optimization problem turns out to be a computationally hard problem in the well known NP class and the available exact or proxy algorithms offer good solutions and performances only for small data sets.
We propose a novel randomized flexible method to identify these sets of nodes based on random walk kernels and random rooted forests. We explain the theoretical results behind this method, originally derived by Avena and Gaudilliere, and present experimental results where we test method and performances on classical synthetic and real-world benchmarks.
Joint work with Michael Emmerich, Alex Gaudilliere and Irina Gurewitsch.
Date: 14/10 at 9 a.m. on Google Meet (meet.google.com/zwc-ecum-zvk )
METASTABILITY FOR THE DILUTE CURIE–WEISS MODEL WITH GLAUBER DYNAMICS
Invited speaker: Elena Pulvirenti (Delft University)
Abstract: We analyse the metastable behaviour of the dilute Curie–Weiss model subject to a Glauber dynamics. The model is a random version of a mean-field Ising model, where the coupling coefficients are replaced by i.i.d. random coefficients, e.g. Bernoulli random variables with fixed parameter p. This model can be also viewed as an Ising model on the Erdos–Renyi random graph with edge probability p. The system is a Markov chain where spins flip according to a Metropolis dynamics at inverse temperature \beta. We compute the average time the system takes to reach the stable phase when it starts from a certain probability distribution on the metastable state (called the last-exit biased distribution), in the regime where the system size goes to infinity, the inverse temperature is larger than 1 and the magnetic field is positive and small enough. We obtain asymptotic bounds on the probability of the event that the mean metastable hitting time is approximated by that of the Curie–Weiss model. The proof uses the potential theoretic approach to metastability and concentration of measure inequalities.
Date: 7/10 at 9 a.m. on Google Meet (meet.google.com/txs-vcrn-mwn )
2D anisotropic KPZ at stationarity
Invited speaker: Dirk Erhard (UFBA)
Abstract: The KPZ equation is the stochastic partial differential equation in d space dimensions formally given by \partial_t h=\Delta h +\langle h,Q h\rangle +\xi, where \xi is the so called space time white noise, i.e., a gaussian process with short range correlations, and Q is a d dimensional matrix. This equation was introduced in the physics literature in the late eighties to model stochastic growth phenomena, is moreover connected to (d+1) dimensional directed polymers in a random potential and is supposed to arise as a scaling limit of a large class of interacting particle systems. In this talk I will try to explain where this equation comes from, why it is interesting, and how its behaviour depends on the spatial dimension. I will mostly focus on the case of dimension 2, and I will comment on a recent result which contradicts a folklore belief from the physics literature.
This is based on joint works with Giuseppe Cannizzaro, Philipp Schönbauer and Fabio Toninelli
Date: 7/09 at 9 a.m. on Google Meet (meet.google.com/btd-gihc-kvd )
Processo de Exclusão com saltos longos e elos lentos na reta
Invited speaker: Pedro Cardoso (IST-Lisboa)
Abstract: We consider an exclusion process with long jumps in the real line and give some ideas to derive a dynamical phase transition when there exist slow bonds at the origin. We deal with the diffusive case (finite variance) and the super-diffusive case (infinite variance).
Date: 7/09 at 9 a.m. on Google Meet (meet.google.com/uwt-vnri-xdy)
Metaestabilidade em um sistema de partículas .
Invited speaker: Ricardo Misturini (UFRGS)
Abstract: Dizemos que um processo possui comportamento metaestável quando ele fica por um tempo longo em um aparente estado de equilíbrio antes de uma transição repentina para outro estado estável. Apresentaremos esse fenômeno em um sistema de partículas conhecido como modelo ABC. Nesse modelo, partículas de três tipos (A, B e C), interagem em um círculo discreto (uma partícula por sítio) como uma cadeia de Markov a tempo contínuo através de transposições entre partículas vizinhas. As taxas com que essas transições ocorrem dependem de um parâmetro denominado temperatura do sistema. Veremos que quando a temperatura tende a zero, as configurações metaestáveis são aquelas em que as partículas segregam-se formando três regiões puras. Entenderemos como é a dinâmica na escala de tempo em que se pode observar transições entre essas configurações metaestáveis. Quando o círculo discreto é reescalado para o contínuo, veremos que essa forma segregada evolui como um movimento browniano no círculo.
Date: 08/07 at 9 a.m. on Google Meet
Spread of an infection on the zero range process .
Invited speaker: Rangel Baldasso (Leiden University)
Abstract: We consider the spread of an infection on top of a moving population. The environment evolves as a zero range process on the integer lattice starting in equilibrium. At time zero, the set of infected particles is composed by those which are on the negative axis, while particles at the right of the origin are considered healthy. A healthy particle immediately becomes infected if it shares a site with an infected particle. We prove that the front of the infection wave travels to the right with positive and finite velocity.
Date: 01/07 at 9 a.m. on Google Meet
Singular stochastic PDEs — meanings of infinities and their weak universalities.
Invited speaker: Weijun Xu (Oxford)
Date: 17/06 at 9 a.m. on Google Meet
Introdução à Teoria Espectral.
Invited speaker: Joedson Santana (UFBA)
Date: 10/06 at 9 a.m. on Google Meet