Some results in weak KPZ universality

Stochastic partial differential equations (SPDEs) are a central object of study in the field of stochastic analysis. Their study involves a number of different tools coming from probability theory, functional analysis, harmonic analysis, statistical mechanics, and dynamical systems. Conversely SPDEs are an extremely useful paradigm to study scaling limit phenomena encountered throughout many other areas of mathematics and physics. The present thesis is concerned mainly with one particular SPDE called the Kardar-Parisi-Zhang (KPZ) equation, which appears universally as a fluctuation limit of height profiles of microscopic models such as interacting particle systems, directed polymers, and corner growth models. Such limit results are deemed instances of ``weak KPZ universality," a field born from the seminal paper of Bertini and Giacomin. We extend results on weak KPZ universality in a number of different directions. In one direction, we prove a version of Bertini-Giacomin's result in a half-space by adapting their methods to this setting, thus extending a result of Corwin and Shen and completing the final step towards the proof of a conjecture about fluctuation behavior of half-space KPZ. In another direction, we also prove a result for the free energy for directed polymers in an octant converging to the KPZ equation in a half-space with a nontrivial normalization at the boundary. In a third direction, we return to the whole-space regime and extend the Bertini-Giacomin result to the case of several different initial data coupled together, proving joint convergence of ASEP with its basic coupling to KPZ driven by the same realization of its noise. Finally we prove a ``nonlinear" version of the law of the iterated logarithm for the KPZ equation in a weak-noise but strong-nonlinearity regime. Beyond their intrinsic purpose, one application of all these extensions and generalizations is to take limits of known results and identities for discrete systems and pass them to the limit to obtain nontrivial information about the KPZ equation itself, which is a well-known methodology launched by I. Corwin and coauthors.