Stochastic analysis for random matrices

Abstract

Let X^N(0) be a symmetric (resp. Hermitian) matrix with eigenvalues $(\lambda _N^1 (0), \cdots ,\lambda _N^N (0))$. Let, for t ? 0, $\lambda _N (t) = (\lambda _N^1 (t), \cdots ,\lambda _N^N (t))$ denote the (real) eigenvalues of $X^N (t) = X^N (0) + H^{N,\beta } (t)$ for t ? 0. We shall prove that (?N(t))t?0 is a semi-martingale with respect to the filtration F_t = ?(B_i,j(s), B_ij(s), 1 ? i, j ? N, s ? t) whose evolution is described by a stochastic differential system. This result was first stated by Dyson [85], and (?_N(t))t?0, when X ^N(0) = 0, has since then been called Dyson’s Brownian motion. To begin with, let us describe the stochastic differential system that governs the evolution of (?_N(t))t?0 and show that it is well defined.