Abstract
We consider the probability distribution on a classical group G which naturally generalizes the normal distribution (the “heat kernel”), defined in terms of Brownian motions on G. As Brownian motion can be defined in terms of the Laplacian on G, much of this work involves the computation of the Laplacian. These results are then used to study the behavior of the normal distribution on U(n( as \(n \mapsto \infty \). In addition, we show how the results on computing the Laplacian on the classical groups can be used to compute expectations with respect to Haar measure on those groups.
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Rains, E.M. Combinatorial Properties of Brownian Motion on the Compact Classical Groups. Journal of Theoretical Probability 10, 659–679 (1997). https://doi.org/10.1023/A:1022601711176
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DOI: https://doi.org/10.1023/A:1022601711176