  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.
