Analysis for Probabilistic Models with phase transition

在本研究期间，我们研究了Domany-Kinzel模型、连续介质渗流模型、相互作用的Brqwnian粒子系统和恶性步行者等概率模型。1）Domany-Kinzel模型是一个双参数随机模型，其中包括定向粘结模型和规则90模型。对于该模型，我们得到了局部生存概率和全局生存概率之间的关系。应用这一关系，我们研究了它的平稳分布的性质，并证明了非吸引情形下模型的极限定理。2）研究了无穷维Skorohod方程。该方程的解表示一个无限布朗球系统，即一个具有硬核相互作用的无限布朗运动系统。证明了一类具有硬核和短程势相互作用的连续渗流模型解的存在唯一性。3）研究了由凸集构成的连续渗流模型。我们证明了模型的临界覆盖面积分数依赖于集合的形状，并且当集合是公共三角形时，临界覆盖面积分数达到最小值。4）Vicious Walkers模型是一个有限多个相互独立且不碰撞的随机游动系统。我们证明了该模型的泛函中心极限定理，并得到了一个与“双矩阵模型”相联系的时间非齐次扩散过程。

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".