[Paper Review] PDE estimates for multi-dimensional KPZ equation

Jérémie Unterberger|arXiv (Cornell University)|Jul 8, 2013

Nonlinear Partial Differential Equations3 citations

TL;DR

This paper establishes existence, uniqueness, and PDE estimates for solutions to the multi-dimensional Kardar-Parisi-Zhang (KPZ) equation with a convex, isotropic deposition rate and space-time white noise. Using a Cole-Hopf transform and a novel class of functional spaces called ${\cal W}$-spaces—designed for locally bounded averages—it proves a comparison principle for sub- and supersolutions, enabling local and pointwise estimates despite unbounded initial data and nonlinearity, under the condition of at most quadratic growth in the nonlinear term.

ABSTRACT

We study in this series of articles the Kardar-Parisi-Zhang (KPZ) equation $$ \partial_t h(t,x)=νΔh(t,x)+λV(| abla h(t,x)|) +\sqrt{D}\, η(t,x), \qquad x\in{\mathbb{R}}^d $$ in $d\ge 1$ dimensions. The forcing term $η$ in the right-hand side is a regularized white noise. The deposition rate $V$ is assumed to be isotropic and convex. Assuming $V(0)\ge 0$, one finds $V(| abla h|)\ltimes | abla h|^2$ for small gradients, yielding the equation which is most commonly used in the literature. The present article is dedicated to existence results and PDE estimates for the solution. Our results extend in a non-trivial way those previously obtained for the noiseless equation. We prove in particular a comparison principle for sub- and supersolutions of the KPZ equation in new functional spaces containing unbounded functions, implying existence and uniqueness. These new functional spaces made up of functions with "locally bounded averages", generically called ${\cal W}$-spaces thereafter, and which may be of interest for the study of parabolic equations in general, allow local or pointwise estimates. The comparison to the linear heat equation through a Cole-Hopf transform is an essential ingredient in the proofs, and our results are accordingly valid only for a function $V$ with at most quadratic growth at infinity.

Motivation & Objective

• To establish existence and uniqueness of solutions to the multi-dimensional KPZ equation with additive space-time white noise.
• To develop a comparison principle for sub- and supersolutions in functional spaces that accommodate unbounded functions.
• To derive local and pointwise estimates for solutions and their gradients using a novel class of function spaces with locally bounded averages.
• To extend PDE techniques to the stochastic KPZ equation by leveraging the Cole-Hopf transform and renormalization ideas.

Proposed method

• Introduces ${\cal W}$-spaces as a new class of function spaces with locally bounded averages, enabling analysis of unbounded initial data.
• Applies the Cole-Hopf transform to relate the nonlinear KPZ equation to a linearized stochastic heat equation.
• Employs a comparison principle in ${\cal W}$-spaces to control solutions via sub- and supersolutions.
• Uses time-decay estimates and integral representations of mild solutions to bound solution behavior.
• Applies multi-scale analysis and scale decomposition techniques to handle the noise and nonlinearity.
• Employs large deviations estimates for the noise via Mayer expansion and log-normal tail bounds to control rare events.

Experimental results

Research questions

• RQ1 Can a comparison principle be established for the KPZ equation in functional spaces that include unbounded functions?
• RQ2 How can local and pointwise estimates be derived for solutions of the multi-dimensional KPZ equation with unbounded initial data?
• RQ3 What is the role of the Cole-Hopf transform in extending PDE estimates to the stochastic KPZ equation with nonlinearity of at most quadratic growth?
• RQ4 How do scale decomposition and multi-scale analysis contribute to bounding the solution and noise terms in the KPZ equation?
• RQ5 What large deviations estimates are required to control the stochastic component in the KPZ equation under the given assumptions?

Key findings

• A comparison principle is proven for sub- and supersolutions in the newly introduced ${\cal W}$-spaces, ensuring existence and uniqueness of solutions to the KPZ equation.
• The solution and its gradient are bounded in local norms, with decay estimates derived via integral representations of mild solutions.
• The method applies only to nonlinearities $V$ with at most quadratic growth at infinity, as required by the Cole-Hopf transform framework.
• Large deviations estimates for the noise component are derived using log-normal tail bounds and Mayer expansion, yielding exponential tail decay of order $O((nA)^{-c\ln(nA)})$.
• The norm $|||\eta^j|||_{\lambda,j}$ is shown to satisfy a tail estimate $\mathbb{P}[|||\eta^j|||_{\lambda,j} > A2^{-jd_\phi}] \lesssim A^{-c\ln A}$, indicating strong stochastic control.
• The results extend previous PDE estimates for the noiseless viscous Hamilton-Jacobi equation to the stochastic setting with controlled growth and unbounded initial data.