Coupling in SPDEs and spectral analysis of heavy-tailed random operators

This thesis contains the work of the author in stochastic analysis and random operators (random Schrödinger operators and random matrices). It is divided into three parts. In the first part, the author considers infinite dimensional stochastic PDEs defined on an abstract Hilbert space, and the main contribution is a solution theory for such SPDEs when the coefficient in front of the white noise is only Hölder continuous with respect to the solution. The proof assumes non-degeneracy of the noise, and uses a generalized coupling technique that is the mixture of a pathwise contraction argument and a probabilistic change of measure argument. Both stochastic heat and wave equations are considered, as well as long-time behavior and small-mass limit. Two related works on solving McKean-Vlasov SDEs with non-Lipschitz coefficients, and a support theorem on parabolic SPDEs with Hölder diffusion coefficients are also included. In the second part, the author investigates 1-d random Schrödinger operators (random potentials induced on the diagonal of a finite-volume 1-d Laplacian matrix) where the coefficients of the potential scale down with respect to the volume size. The results are: (1) a scaling limit at the spectral edge at a particular rate of downward scaling; and (2) a large deviation result in the light-tailed regime, and the distribution of the top eigenvalues in the heavy-tailed regime, under various tail assumptions on the random potential and the rate of downward scaling. In the third part, the author considers Wigner matrices, randomly weighted regular graphs and sample covariance matrices where the random variables decay at a particular rate (for dense matrices, they are regularly varying with index 4) so that the largest eigenvalue lies in the crossover regime from sticking to the edge and having Tracy-Widom fluctuation, to deviating from the edge and having a Poisson distribution at a larger magnitude. It is proved that in this crossover regime, the finitely many largest eigenvalues follow a deformed Poisson distribution (at the same magnitude of the spectrum), where the deformation function is characterized by the Stieltjes transform of the semicircle law or the Marchenko-Pastur law, signifying a BBP-type phenomenon. The eigenvectors to these outlying eigenvalues are also investigated.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2024

Author dc:creator

• Han, Yi

Advisor dc:contributor.advisor

• Norris, James

• Coupling
• Heavy-tailed
• random Schrödinger operators
• Stochastic wave equation

Licence

• https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/aecbbccf-3632-4391-b11a-7374c542dff8/download
• https://creativecommons.org/licenses/by/4.0/

Language dc:language

eng

Author Identifier

0000-0003-4909-7133

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/375746