# Mathematics

## 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.

## Interacting Particle Systems 9

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 9

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.

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## Interacting Particle Systems 8

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.

## Limit theorems for conditioned non-generic Galton-Watson trees

We are interested in a particular type of subcritical Galton-Watson trees, which are called non-generic trees in the physics community. In contrast with the critical or supercritical case, it is known that condensation appears in large conditioned non-generic trees, meaning that with high probability there exists a unique vertex with macroscopic degree comparable to the total size of the tree. We investigate this phenomenon by studying scaling limits of such trees. In particular, we show that the height of such trees grows logarithmically in their size.

## Interacting Particle Systems 7

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 7

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.

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## Local relaxation for FA-1f out of equilibrium

We consider the Fredrickson and Andersen one spin facilitated model (FA1f)on Z^d. Each site with rate one refreshes its occupation variable to a filled or to an empty state with probability p or q=1-p respectively, provided that at least one of its nearest neighbours is empty. We study the non-equilibrium dynamics started from an initial distribution $\nu$ different from the stationary product p-Bernoulli measure $\mu$, which has enough zeros. We then prove local convergence to equilibrium when the vacancy density q is above a proper threshold. The convergence is exponential (d=1) or stretched exponential (d>1). Joint work with N. Cancrini, F. Martinelli, C. Roberto and C. Toninelli.

## Random Maps 6

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 6
- 4397 reads