Hairer-Quastel universality in non-stationarity via energy solution theory

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 [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].