[Paper Review] Hairer-Quastel universality in non-stationarity via energy solution theory

Kevin Yang|arXiv (Cornell University)|Oct 30, 2020

Stochastic processes and financial applications27 references4 citations

TL;DR

This paper establishes a novel energy solution theory for the KPZ and stochastic Burgers equations with general continuous initial data, extending prior work limited to stationary or H"older-regular data. It proves universality for SPDEs with nonlinearities beyond smooth functions, provides explicit convergence rates to the invariant measure in Wasserstein and relative entropy, and establishes a log-Sobolev inequality, offering the first intrinsic solution theory for continuous initial conditions in singular SPDEs.

ABSTRACT

The paper addresses probabilistic aspects of the KPZ equation and stochastic Burgers equation by providing a solution theory that builds on the energy solution theory Goncalves-Jara '14, Gubinelli-Jara '13, Gubinelli-Perkowski '18, Gubinelli-Perkowski '20. The perspective we adopt is to study the stochastic Burgers equation by writing its solution as a probabilistic solution Gubinelli-Perkowski '17 plus a term that can be studied with deterministic PDE considerations. One motivation is universality of KPZ and stochastic Burgers equations for a certain class of stochastic PDE growth models, first studied in Hairer-Quastel '18. For this, we prove universality for SPDEs with general nonlinearities, thereby extending Hairer-Quastel '18, Hairer-Xu '19, and for many non-stationary initial data, thereby extending Gubinelli-Perkowski '16. Our perspective lets us also prove explicit rates of convergence to white noise invariant measure of stochastic Burgers for non-stationary initial data, in particular extending the spectral gap result of Gubinelli-Perkowski '20 beyond stationary initial data, though for non-stationary data our convergence will be measured in Wasserstein distance and relative entropy, not via the spectral gap as in Gubinelli-Perkowski '20. Actually, we extend the spectral gap in Gubinelli-Perkowski '20 to a log-Sobolev inequality. Our methods can also analyze fractional stochastic Burgers equations; we discuss this briefly. Lastly, we note that our perspective on the KPZ and stochastic Burgers equations provides a first intrinsic notion of solutions for general continuous initial data, in contrast to Holder regular data needed for regularity structures, paracontrolled distributions, and Holder-regular Brownian bridge data for energy solutions.

Motivation & Objective

• To extend energy solution theory beyond stationary or H"older-regular initial data to general continuous initial conditions for the KPZ and stochastic Burgers equations.
• To prove universality of the KPZ equation for a broad class of nonlinearities, including non-smooth ones like |x|, beyond the scope of regularity structures.
• To establish explicit convergence rates to the white noise invariant measure for non-stationary initial data, using Wasserstein distance and relative entropy.
• To extend the spectral gap result of [23] to a log-Sobolev inequality for non-stationary initial data.
• To provide a first intrinsic solution theory for singular SPDEs with arbitrary continuous initial data, avoiding the need for H"older regularity or Brownian bridge assumptions.

Proposed method

• Formalize the solution to the stochastic Burgers equation as a probabilistic solution plus a deterministic correction term, enabling analysis via PDE techniques.
• Use a scaling limit of SPDEs with general nonlinearity F(ε^{1/2}∇g^ε) to recover the KPZ equation in the ε→0 limit.
• Apply energy solution theory to handle the singular nature of the SPDEs, leveraging martingale and a priori estimates.
• Employ relative entropy and Wasserstein distance to quantify convergence to the invariant measure for non-stationary initial data.
• Utilize the contraction principle for relative entropy under continuous maps to control path-space measures and establish tightness.
• Extend the spectral gap framework to a log-Sobolev inequality for non-stationary initial data, using path-space measure decomposition and entropy bounds.

Experimental results

Research questions

• RQ1 Can energy solution theory be extended to handle arbitrary continuous initial data for the KPZ and stochastic Burgers equations?
• RQ2 Does universality of the KPZ equation hold for general nonlinearities, including non-smooth ones like |x|, beyond the smooth case?
• RQ3 What explicit rates of convergence to the white noise invariant measure can be established for non-stationary initial data?
• RQ4 Can the spectral gap result for stationary data be extended to a log-Sobolev inequality in the non-stationary setting?
• RQ5 Is there a first intrinsic solution theory for singular SPDEs with continuous initial data, independent of H"older regularity or Brownian bridge assumptions?

Key findings

• The paper establishes a solution theory for the KPZ and stochastic Burgers equations with arbitrary continuous initial data, overcoming the H"older regularity requirement of regularity structures and paracontrolled distributions.
• Universality is proven for SPDEs with general nonlinearities, including non-smooth ones such as F(x) = |x|, extending prior results limited to smooth F.
• Explicit convergence rates to the white noise invariant measure are obtained in Wasserstein distance and relative entropy for non-stationary initial data, extending the spectral gap result of [23] beyond stationary cases.
• A log-Sobolev inequality is established for the stochastic Burgers equation with non-stationary initial data, strengthening the spectral gap result.
• The method enables analysis of fractional stochastic Burgers equations, suggesting broader applicability beyond the standard case.
• The framework provides the first intrinsic solution theory for singular SPDEs with continuous initial data, avoiding the need for Brownian bridge or H"older-regular data.