统计力学和量子场论中出现的矩阵模型与具有对数库仑交互作用的随机粒子系统有密切的联系，而这些模型中的外场位势函数一般不是二次函数，此随机粒子系统在数学和物理的各个领域和分支都有非常重要的作用，但其随机动力学模型，即广义Dyson Brown运动，在概率论和随机分析中没有被深入研究。由于广义Dyson布朗运动与矩阵值扩散之间有对应，本课题主要研究广义Dyson Brown运动对应的矩阵值扩散过程的收敛， 通过引入酉矩阵值布朗运动，得到带有一般位势的酉矩阵值过程的收敛性，希望给出酉矩阵空间上的布朗运动收敛到自由可乘布朗运动的新证明；在已经证明了广义Dyson 布朗运动经验测度过程的大数定律和中心极限定理的前提下，当外场位势满足适当条件时给出广义Dyson布朗运动经验测度过程的大偏差原理的动态结果。

The particle system with logarithmic Coulomb interaction are closely connected with the Matrix model of statistical mechanics and in quantum field theory, but the external potential are usually not quadratic functions. This particle systems has been used in various areas in mathematics and physics, but the dynamics model, i.e. the generalized Dyson Brownian motion has not been intensive studied in probability theory and stochastic analysis. Since the generalized Dyson Brownian motion can be realized as the eigenvalues processes of some matrix valued diffusion process, this proposed project mainly studies the asymptotic behaviour of these valued diffusion process.By introduce the Unitary valued Brown motion, we can have the convergence of the Unitary valued diffusion process with general potential, then we want to give another method to prove the Unitary valued Brownian motion converge in distribution towards a free multiplicative Brownian motion; As we proved the Law of Large Numbers and the Central Limit Theorem for the empirical measure processes associated with the Generalized Dyson Brownian motion, we want to prove the large deviations principle for the Generalized Dyson Brownian motion with reasonable condition on potential.