51³Ô¹Ï

Topic

Renormalized limits of the stochastic heat equation and the Kardar-Parisi-Zhang equation in high dimensions

Department of Mathematics and Statistics

Date & location

• Thursday, July 10, 2025
• 8:30 A.M.
• Virtual Defence

Examining Committee

Supervisory Committee

• Dr. Yu-Ting Chen, Department of Mathematics and Statistics, 51³Ô¹Ï (Supervisor)
• Dr. Anthony Quas, Department of Mathematics and Statistics, UVic (Member)
• Dr. Kristan Jensen, Department of Physics and Astronomy, UVic (Outside Member)

External Examiner

• Dr. Clément Cosco, Centre de recherche en mathématiques de la decision, Université Paris-Dauphine

Chair of Oral Examination

• Dr. Falk Herwig, Department of Physics and Astronomy, UVic

Abstract

This thesis investigates the mollified versions of the stochastic heat equation (SHE) and the Kardar–Parisi–Zhang (KPZ) equation in high dimensions (d ≥ 3), focusing on their limiting behaviors as the mollification is removed. These limiting behaviors are closely related to a key parameter, known as the coupling constant, which characterizes the strength of the driving noise.

34 The first part of this article analyzes the space-time fluctuations of the mollified SHE and mollified KPZ equation around their stationary states. These rescaled fluctuations, previously considered in [23, 24], provide a new viewpoint for describing the convergence of the above mollified equations. For every coupling constant that is strictly less than a specific critical threshold, called the L²-critical point, we establish Gaussian limits, possibly with additional random perturbations, for these rescaled fluctuations.

The second part of this article investigates the limiting higher moments for the mollified SHE in high dimensions. Motivated by a recent result [21] in two dimensions, a natural question is whether the higher moments also converge at the L²-critical point in high dimensions. Our main theorem gives a negative answer: in high dimensions, the limiting higher moments diverge for all coupling constants in a nontrivial interval containing the L²-critical point. As an application, we derive sharp estimates for quantities, which are believed to be closely related to probability distributions of the limiting partition function of the continuous directed polymer.

As a follow-up investigation, the third part of this article analyzes the above conjecture on the higher moments in three dimensions by considering a specific version of the limiting higher moment at the L²-critical point, referred to as the sub-limiting higher moment. This special limiting higher moment can be regarded as a three-dimensional analogue of the limiting higher moment in two dimensions at the corresponding L²-critical point, established in [21]. Our main result proves the spatial pointwise divergence of this limiting object. This divergence can be related to a well-known phenomenon in quantum physics known as the Efimov effect.