Related papers: Jointly invariant measures for the Kardar-Parisi-Z…
The Kardar-Parisi-Zhang (KPZ) equation on the real line is well-known to admit Brownian motion with a linear drift as a stationary distribution (modulo additive constants). We show that these solutions are attractive, a result known as a…
We construct a family of invariant measures from the perspective of a shock in the KPZ fixed point. These measures are parameterized by a positive number $\theta > 0$, and are supported on functions $f$ satisfying $\lim_{|x| \to \infty}…
We construct explicit jointly invariant measures for the periodic KPZ equation (and therefore also the stochastic Burgers' and stochastic heat equations) for general slope parameters and prove their uniqueness via a one force--one solution…
The Kardar-Parisi-Zhang (KPZ) universality class describes a large class of 2-dimensional models of random growth, which exhibit universal scaling exponents and limiting statistics. The last ten years has seen remarkable progress in this…
The Kardar-Parisi-Zhang (KPZ) fixed point is a Markov process that is conjectured to be at the core of the KPZ universality class. In this article we study two aspects the KPZ fixed point that share the same Brownian limiting behaviour: the…
The Kardar-Parisi-Zhang (KPZ) fixed point is a Markov process, recently introduced by Matetski, Quastel, Remenik (arXiv:1701.00018), that describes the limit fluctuations of the height function associated to the totally asymmetric simple…
Using Stein's method and a Gaussian integration by parts, we provide a direct proof of the known fact that drifted Brownian motions are invariant measures (modulo height) for the KPZ equation.
In [arXiv:2409.08465], Quastel and Gu use Stein's equation and integration by parts to give a direct proof that drifted Brownian motions are stationary (modulo height shifts) for the full-line KPZ equation. In this article, we consider the…
The ergodic theory of the open KPZ equation has seen significant progress in recent years, with explicit invariant measures described in a series of works by Corwin--Knizel, Barraquand--Le Doussal, and Bryc--Kuznetsov--Wang--Weso{\l}owski.…
We show that the increments of the KPZ fixed point started from arbitrary initial data are \emph{mutually} absolutely continuous with respect to Brownian motion with diffusion parameter $2$ on compacts, extending the one-sided Brownian…
The Kardar-Parisi-Zhang (KPZ) equation is a stochastic partial differential equation which is ill-posed because the nonlinearity is marginally defined with respect to the roughness of the forcing noise. However, its Cole-Hopf solution,…
We derive the KPZ equation as a continuum limit of height functions in asymmetric simple exclusion processes with drift that depends on the local particle configuration. To our knowledge, it is a first such result for a class of particle…
Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or regularity) and expanding the breadth of its…
This paper studies the large scale limits of multi-type invariant distributions and Busemann functions of planar stochastic growth models in the Kardar-Parisi-Zhang (KPZ) class. We identify a set of sufficient hypotheses for convergence of…
We construct explicit one-parameter families of stationary measures for the Kardar-Parisi-Zhang equation in half-space with Neumann boundary conditions at the origin, as well as for the log-gamma polymer model in a half-space. The…
The stationary measures of the Kardar-Parisi-Zhang equation on an interval have been computed recently. We present a rather direct derivation of this result by taking the weak asymmetry limit of the matrix product ansatz for the asymmetric…
We study exact stationary properties of the one-dimensional Kardar-Parisi-Zhang (KPZ) equation by using the replica approach. The stationary state for the KPZ equation is realized by setting the initial condition the two-sided Brownian…
We present a variational formulation for the Kardar-Parisi-Zhang (KPZ) equation that leads to a thermodynamic-like potential for the KPZ as well as for other related kinetic equations. For the KPZ case, with the knowledge of such a…
We present a variational formulation for the Kardar-Parisi-Zhang (KPZ) equation that leads to a thermodynamic-like potential for the KPZ as well as for other related kinetic equations. For the KPZ case, with the knowledge of such a…
The Kardar-Parisi-Zhang (KPZ) equation is a celebrated non-linear stochastic dynamical equation yielding non-equilibrium universal scaling. It exhibits notorious non-perturbative aspects. The KPZ fixed point is strong-coupling, all the more…