The aim of this short survey is to describe various models of diffusion processes where explicit computations are at hand. Most of them arise from geometric models encountered in various settings, and are associated with different families of orthogonal polynomials. The aim is mainly to collect in a common place the various formulae spread out in many different notes. We shall see that those difusion processes and their laws, that is diffusion semigroups and their generators, offer a rich interplay between probability, analysis, geometry and and algebra.

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