Moments of Brownian motions on Lie groups (Q2573765)

It is known that a stochastic process \(X_t\) on \(\mathbb R\) is (up to a modification) a Brownian motion if and only if the processes \(R_k(X_t,t)\) are martingales for \(k=1,2,3,4\). Here \(R_k(x,t)=t^{k/2}H_k(x/\sqrt{t/2})\) where \(H_k\) are the Hermite polynomials; see \textit{J. Wesołowski} [Bull. Pol. Acad. Sci., Math. 38, No. 1--12, 49--53 (1990; Zbl 0762.60035)]. The author obtains a generalization of this result for processes on \(GL(n,\mathbb R)\). The generalization extends to the case of an arbitrary locally compact group admitting a faithful finite-dimensional representation, or even a representation with a discrete kernel. The technique is based on the detailed study of moment matrices of the Brownian motion on \(GL(n,\mathbb R)\).