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Integration formulas for Brownian motion on classical compact Lie groups. (English. French summary) Zbl 1388.58022
The author considers the Brownian motion \((G_t)\) on the classical compact Lie group \(G\in\{O(N), U(N),Sp(N)\}\).
He first obtains combinatorial formulas yielding the moments \(\mathbb{E}\big[G_t^{\otimes n}\big]\) (and also \(\mathbb{E}\big[G_t^{\otimes n}\otimes \overline{G_t}^{\otimes n}\big]\) for \(G=U(N)\)). The expressions got in this way are well suited to handle \(N\to\infty\) and \(t\to\infty\) asymptotics. In particular, letting \(t\to\infty\), the author then derives the analogous formulas when expectation is replaced by the Haar measure.
Moreover, using the preceding, the author gives a new proof, based on stochastic calculus, for the so-called first fundamental theorem of invariants (describing the subspaces of tensor spaces made of the invariants under the \(G\)-action) and Schur-Weyl duality (describing the commutant of the tensorial \(G\)-action).

MSC:
58J65 Diffusion processes and stochastic analysis on manifolds
60J65 Brownian motion
15A72 Vector and tensor algebra, theory of invariants
14L35 Classical groups (algebro-geometric aspects)
46L53 Noncommutative probability and statistics
60H30 Applications of stochastic analysis (to PDEs, etc.)
60G15 Gaussian processes
05E10 Combinatorial aspects of representation theory
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