Large-Scale Behavior of Some Stochastic PDES

Abstract

In this dissertation, we include four projects that study the large-scale behavior of some stochastic PDEs. The stochastic PDEs that we study are the Kardar-Parisi-Zhang (KPZ) equation and the stochastic heat equation (SHE).

The Kardar-Parisi-Zhang equation is a nonlinear stochastic PDE, known as the default model for random interface growth in physics. To study the Kardar-Parisi-Zhang equation, we use the approach of Hopf-Cole transform, which relates the KPZ equation to the stochastic heat equation. In spatial dimension one, the KPZ equation and the stochastic heat equation have received extensive studies, but the equations that are boundary driven or in higher dimensions remain more mysterious.

In the first part of this dissertation, we consider the stochastic heat equation and KPZ equation in spatial dimension two, which is a critical dimension, and the solutions can only be made sense as scaling limits. In the first project, we consider a nonlinear version of the stochastic heat equation and prove its limiting second-order Gaussian fluctuation. This result has been published in paper [1]. In the second project, we show a mesoscopic averaging phenomenon for the local averages of the two-dimensional KPZ equation. This result has been published in [2].

The second major part of this dissertation is a project that considers the KPZ equation in a one-dimensional half-space domain with a Neumann boundary condition. This half-space KPZ model exhibits a “depinning” phase transition as the boundary condition changes. We study the half-space KPZ equation starting from stationary Brownian initial data. We obtain its optimal fluctuation exponents in both the subcritical and critical regimes of the phase transition, and give an optimal upper bound for the fluctuation exponents in the supercritical regime. We also compute the average growth rate as a function of the boundary parameter. This result has been submitted for publication in paper [3].

In the last part of this dissertation, we also include a work that studies the time-dependent spatial averages of a long-range critically correlated stochastic heat equation in spatial dimension three or higher. These spatial averages have different limits under different space-time scales. This result has been included in preprint [4].