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Multivariate Bessel and Jacobi processes describe Calogero-Moser-Sutherlandparticle models. They depend on a parameter $k$ and are related totime-dependent classical random matrix models like Dysom Brownian motions,where $k$ has the interpretation of an inverse temperature. There are severalstochastic limit theorems for $k\to\infty$ were the limits depend on thesolutions of associated ODEs where these ODEs admit particular simple solutionswhich are connected with the zeros of the classical orthogonal polynomials. Inthis paper we show that these solutions attract all solutions. Moreover wepresent a connection between the solutions of these ODEs with associatedinverse heat equations. These inverse heat equations are used to compute theexpectations of some determinantal formulas for the Bessel and Jacobiprocesses.

On the differential equations of frozen Calogero-Moser-Sutherland particle models