Collision of eigenvalues for matrix-valued processes

Abstract

We examine the probability that at least two eigenvalues of a Hermitian matrix-valued Gaussian process, collide. In particular, we determine sharp conditions under which such probability is zero. As an application, we show that the eigenvalues of a real symmetric matrix-valued fractional Brownian motion of Hurst parameter [Formula: see text], collide when [Formula: see text] and do not collide when [Formula: see text], while those of a complex Hermitian fractional Brownian motion collide when [Formula: see text] and do not collide when [Formula: see text]. Our approach is based on the relation between hitting probabilities for Gaussian processes with the capacity and Hausdorff dimension of measurable sets.