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I will present an overview of the results and methods on the estimation of Poisson, Newton and transition(heat) kernels for Dyson Brownian Motions and the stochastic diffusion processes of Dunkl and Opdam-Cherednik. From an analytical perspective, this research falls within the analysis related to Dunkl and Opdam-Cherednik operators on $R^n$ equipped with a root system. Most of these results were obtained in collaboration with Patrice Sawyer (Sudbury).