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In this paper we study the convergence of dynamical discrete web (DyDW) to the dynamical Brownian web (DyBW) in the path space topology. We show that almost surely the DyBW has RCLL paths taking values in an appropriate metric space and as a sequence of RCLL paths, the scaled dynamical discrete web converges to the DyBW. This proves weak convergence of the DyDW process to the DyBW process.

We introduce a new metric for collections of aged paths and a robust set of criteria for compactness for a set of collection of aged paths in the topology corresponding to this metric. We show that the distribution of stable webs ($1< \alpha \leq 2$) made up of collections of stable paths is tight in this topology. We then show the weak convergence of appropriately normalized systems of coalescing random walks in the domain of attraction of stable laws for $1 < \alpha \leq 2$ under this metric to the corresponding stable web. We obtain some path results in the brownian case., Comment: 25 pages

Motivated by tumor growth models we establish conditions for the $R-$positivity of Markov processes and positive matrices. We then apply them to obtain the asymptotic behavior of the tumors sizes in the supercritical regime.

We provide a process on the space of coalescing cadlag stable paths and show convergence in the appropriate topology for coalescing stable random walks on the integer lattice.

The study of rumor models from a percolation theory point of view has gained a few adepts in the last few years. The persistence of a rumor, which may consistently spread out throughout a population can be associated to the existence of a giant component containing the origin of a graph. That is one of the main interest in percolation theory. In this paper we present a quick review of recent results on rumor models of this type., Comment: 30 pages. arXiv admin note: text overlap with arXiv:1305.6802 by other authors

We study a rumour model from a percolation theory and branching process point of view. The existence of a giant component is related to the event where the rumour, which started from the root of a tree, spreads out through an infinite number of its vertices. We present lower and upper bounds for the probability of that event, according to the distribution of the random variables that defines the radius of influence of each individual. We work with Galton-Watson branching trees (homogeneous and non-homogeneous) and spherically symmetric trees which includes homogeneous and $k-$periodic trees., Comment: The project has been deeply edited and now have a new co-author

We introduce a new self-interacting random walk on the integers in a dynamic random environment and show that it converges to a pure diffusion in the scaling limit. We also find a lower bound on the diffusion coefficient in some special cases. With minor changes the same argument can be used to prove the scaling limit of the corresponding walk in Z^d., Comment: 10 pages

We describe the hydrodynamic behavior of the $k$-step exclusion process. Since the flux appearing in the hydrodynamic equation for this particle system is neither convex nor concave, the set of possible solutions include in addition to entropic shocks and continuous solutions those with contact discontinuities. We finish with a limit theorem for the tagged particle., Comment: 18 pages