Analysis for Probabilistic Models with phase transition

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Probabilistic models such as Domany-Kinzel model, Continuum perclation model, systems of interacting Brqwnian particles and vicious walkers, were studied during the term of this research. Result obtained are explained here for each model.1) Domany-Kinzel model is a stochastic model with two parameters and includes oriented site bond percolatin models and rule 90 model as special cases. For this model we obtain the relation between local survival probability and global survival probability. Applying this relation we studied properties of its stationary distributions and show the limit theorem for the model for non-attractive case.2) A Skorohod equation in infinite dimensional was studied. The solution of this equation represents a system of infinte Brownian balls, that is, a system of infinite Brownian motions with hard core interaction. The existence and uniqueness of the solutions was proved for the class of interactions with hard core and short range potential.3) Continuum percolation models which are composed by convex sets were studeid. We show that, the critical covered area fraction of the model does depend on shapes of sets and attains its minimum when the sets are common triangles.4) Vicious walkers model is a system of finite many independent random walks conditioned not to collide with each other. We show functional central limit theorems for the model and obtained a temporally inhomogeneous diffusion process which is associated with "Two matrix model".