Collision of eigenvalues for matrix-valued processes

2018/02/15 by Jaramillo, Arturo, Nualart, David
#15B52 #60G22 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1802.05410

Abstract

We examine the probability that at least two eigenvalues of an 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 H, collide when H<1/2 and don't collide when H>(1)/(2), while those of a complex Hermitian fractional Brownian motion collide when H(1)/(3). Our approach is based on the relation between hitting probabilities for Gaussian processes with the capacity and Hausdorff dimension of measurable sets.

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