Open Access
2023 Hairer-Quastel universality in non-stationarity via energy solution theory
Kevin Yang
Author Affiliations +
Electron. J. Probab. 28: 1-26 (2023). DOI: 10.1214/23-EJP908


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 [15, 19, 20, 23]. The perspective we adopt is to study the stochastic Burgers equation by writing its solution as a probabilistic solution [22] 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 [28]. For this, we prove universality for SPDEs with general nonlinearities, thereby extending [28, 29], and for many non-stationary initial data, thereby extending [21].

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 [23] 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 [23]. Actually, we extend the spectral gap in [23] to a log-Sobolev inequality. Our methods can also analyze fractional stochastic Burgers equations [23]; we discuss this briefly.

Lastly, we note our perspective on the KPZ and stochastic Burgers equations gives a first intrinsic notion of solutions for general continuous initial data, in contrast to Hölder regular data needed for regularity structures [26], paracontrolled distributions [14], and Hölder-regular Brownian bridge data for energy solutions [19, 20].

Funding Statement

The author thanks the Northern California Chapter of the ARCS Foundation, under whose funding this research was conducted.


The author would like to thank Amir Dembo for helpful discussion and advice. The author thanks the referees for the immensely helpful feedback.


Download Citation

Kevin Yang. "Hairer-Quastel universality in non-stationarity via energy solution theory." Electron. J. Probab. 28 1 - 26, 2023.


Received: 27 November 2021; Accepted: 17 January 2023; Published: 2023
First available in Project Euclid: 26 January 2023

Digital Object Identifier: 10.1214/23-EJP908

Primary: 60H17

Keywords: KPZ , Stochastic pde , Universality

Vol.28 • 2023
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