Scientific

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.

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.

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.