Matrix-valued stochastic processes

Abstract

In this thesis, we study the high-dimensional limits of the sequence of eigenvalue empirical measure processes of a class of matrix-valued stochastic processes as well as the fluctuations around its limits. We first obtain the almost sure relative compactness of the sequence of eigenvalue empirical measure processes as well as a deterministic equation for the high-dimensional limit measure by It\^{o} calculus and martingale theory. The results are extended to random particle systems that generalize the stochastic differential equations (SDEs) of the eigenvalue processes. For the eigenvalue processes and the particle systems, we obtain the central limit theorems (CLTs) to characterize the fluctuations of the empirical measure processes around the limiting measure-valued processes. We also introduce a new set of conditions for the existence and uniqueness of a strong solution for the SDEs of the eigenvalue processes and the particle system. In addition, we develop comparison principles for the random particle systems and the eigenvalue processes.

Moreover, we use fractional calculus to study the high-dimensional limit of the eigenvalue empirical measure processes of three types of real symmetric/complex Hermitian matrix-valued stochastic processes (Wigner-type matrix-valued processes, Wishart-type matrix-valued processes, and matrix-valued processes with locally dependent entries), whose entries are generated from the solution of Stratonovich SDE driven by fractional Brownian motion with Hurst parameter $H \in (1/2,1)$.

In addition, by developing a multi-dimensional Green's formula, we derive a system of stochastic partial differential equations (SPDEs) satisfied by the eigenvalue processes of the real symmetric matrix Brownian sheet. We also establish the It\^{o}'s formula for the eigenvalue processes.

Furthermore, we study the multiple collision of eigenvalue processes of real symmetric or complex Hermitian matrix-valued Gaussian fields. We characterize the conditions under which the probability that at least $k$ eigenvalues collide is strictly positive and we obtain the Hausdorff dimension of the set of collision times.