Hermite and Laguerre polynomials and matrix-valued stochastic
Stephan Lawi, .
Abstract
We extend to matrix-valued stochastic processes, some well-known
relations between real-valued diffusions and classical orthogonal
polynomials, along with some recent results about Lévy processes
and martingale polynomials. In particular, joint semigroup densities
of the eigenvalue processes of the generalized matrix-valued
Ornstein-Uhlenbeck and squared Ornstein-Uhlenbeck processes are
respectively expressed by means of the Hermite and Laguerre
polynomials of matrix arguments. These polynomials also define
martingales for the Brownian matrix and the generalized Gamma
process. As an application, we derive a chaotic representation
property for the eigenvalue process of the Brownian matrix.
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