# Probability

## Monotonic IDLA forest and First Passage Percolation

We present a modification of the IDLA model on a rotated square lattice. The process results in a forest of trees covering the upper half plane. We show an equivalence between this model and first passage percolation. We prove that with probability 1 the trees resulting from the IDLA forest are finite.

## Interacting Particle Systems 11

Particles attempt to follow a simple dynamic (random walk, constant flow, etc) in some space (interval, line, cycle, arbitrary graph). Add a simple interaction between particles, and the behaviour can change completely. The resulting dynamical systems are far more complex than the ingredients suggest. These processes (interchange process, TASEP, sorting networks, etc) have diverse to many topics: growth processes, queuing theory, representation theory, algebraic combinatorics. I will discuss recent progress on and open problems arising from several models of interacting particle systems.

## Random Maps 11

The study of maps, that is of graphs embedded in surfaces, is a popular subject that has implications in many branches of mathematics, the most famous aspects being purely graph-theoretical, such as the four-color theorem. The study of random maps has met an increasing interest in the recent years. This is motivated in particular by problems in theoretical physics, in which random maps serve as discrete models of random continuum surfaces. The probabilistic interpretation of bijective counting methods for maps happen to be particularly fruitful, and relates random maps to other important combinatorial random structures like the continuum random tree and the Brownian snake. This course will survey these aspects and present recent developments in this area.

- Read more about Random Maps 11
- 3186 reads

## Density of states for alpha-stable processes

We will show the existence of the density of states for $alpha$-stable processes ie existence of the deterministic measure that is a limit (when M goes to infinity) of random measures based on sequence of the eigenvalues of the generator of the $alpha$-stable process in a ball B(0,M) with Poissonian obstacles. We will give also estimate of the limit measure near zero.

## Fluctuations study for type-dependent stochastic spin models

We study the fluctuations process for the type-dependent stochastic spin models proposed by Fernández et al.[2], which were used to model biological signaling networks. Using the results of Ethier & Kurtz [1], we analyse the asymmetric basic clock [3], a extension for the simplest cyclic-interaction module, that provides the basic functionality of generating oscillations. Particularly, we apply the central limit theorem for fluctuations process; the dynamics of this limit process is our aim. References. [1] S.N. Ethier, T.G. Kurtz, Markov Processes, Characterization and Convergence. Wiley, New York, 1986. [2] R. Fernandez, L.R. Fontes, E.J. Neves, Density-Profile Processes Describing Biological Signaling Networks: Almost Sure Convergence to Deterministic Trajectories. J Stat Phys (2009) 136: 875-901. [3] M.A. González Navarrete, Sistemas de partículas interagentes dependentes de tipo e aplicaçoes ao estudo de redes de sinalizaçao biológica. Master thesis, Instituto de Matemática e Estatística USP, 2011.

## Cesaro limit and ergodicity of locally eventually periodic measures

We study translation invariant deterministic dynamics (phi) on the lattice (cellular automata). In particular the evolution and limit of probability measures that give the set of locally eventually phi-periodic points full measure. We prove the convergence of the mean averages under phi of this measures. We characterize the ergodicity of the limit measures (solving a question posed by Blanchard and Tisseur) and we prove that in the limit phi is a mixture measure theoretical odometers.

## Interacting Particle Systems 10

Particles attempt to follow a simple dynamic (random walk, constant flow, etc) in some space (interval, line, cycle, arbitrary graph). Add a simple interaction between particles, and the behaviour can change completely. The resulting dynamical systems are far more complex than the ingredients suggest. These processes (interchange process, TASEP, sorting networks, etc) have diverse to many topics: growth processes, queuing theory, representation theory, algebraic combinatorics. I will discuss recent progress on and open problems arising from several models of interacting particle systems.

## A model of migration under constraint

We will present a random model of population, where individuals live on several islands, and will move from one to another when they run out of resources. Our main goal is to study how the population spreads on the different islands, when the number of initial individuals and available resources tend to infinity. Finding this limit relies on asymptotics for critical random walks and (not so classical) functionals of the Brownian excursion.

## Multi-dimensional Brownian Motion with Darning

The reason that we define multi-dimensional Brownian motion as a darning process is that, even for the simplest case which is R^2 being unioned with R^1, such a process cannot be defined in the usual sense, because 2-dimensional Brownian motion never hits a singleton. Constructions of darning processes are based on one-point extension theory which was first studied by M. Fukushima. Lots of very interesting examples, for instance, circular Brownian motion, Brownian motion with a ``knot", etc., can be constructed in this way, some of which will be provided in the talk. The rest of the talk will be focusing on the heat kernel estimates of multi-dimensional Brownian motion with darning.

## Stochastic Equations of Super-L\'{e}vy Process with General Branching Mechanism

The process of distribution functions of a one-dimensional super-L\'{e}vy process with general branching mechanism is characterized as the pathwise unique solution of a stochastic integral equation driven by time-space white noises and Poisson random measures. This generalizes a recent result of Xiong (2012), where the result for a super-Brownian motion with binary branching mechanism was obtained. To establish the main result, we prove a generalized It\^o's formula for backward stochastic integrals and study the pathwise uniqueness for a general backward doubly stochastic equation with jumps. Furthermore, we also present some results on the SPDE driven by a one-sided stable noise without negative jumps. This is a joint work with Hui He and Zenghu Li.