37 problems

Let the KPZ equation be the stochastic growth equation whose solution is denoted by $\mathcal{H}_\beta(T,X)$, and let the KPZ universality class consist of models sharing its chara…

One motivation for studying martingale problems for singular SPDEs is that they provide a tool for deriving equations as scaling limits. Weak KPZ universality conjecture. A wide ra…

Let $\mathsf{u},\mathsf{v}\in\mathbb{R}$ be the boundary slope parameters for the open KPZ equation, and consider the probabilistic description of its invariant measure obtained by…

The KPZ equation and its companion, the stochastic Burgers equation, describe universal fluctuation laws for several random growth interfaces near a stationary state. Weak KPZ univ…

Let $B_1$ be the first component of the two-layer process defined by the measure in. For parameters $u,v$, let $\Lambda_1,\Lambda_2$ be distributed according to the two-layer measu…

Let $H_{u,v}(t,x)$ denote the Hopf--Cole solution to the open KPZ equation with Neumann boundary parameters $u$ and $v$, and set … Let $h_{u,v,w,r}^{(N)}(t,x)$ be the diffusively s…

Consider the open ASEP and its height-function process under the weak asymmetry and height-function scaling assumptions used in the paper, with boundary parameters satisfying the s…

Let the open ASEP height function process be defined with ASEP parameters, and let Liggett's condition denote the parameter restriction used in existing convergence results for the…

The KPZ equation describes a random height function, and its large-time behavior is studied through the KPZ fixed point and the KPZ sheet. KPZ fixed-point conjecture. The KPZ fixed…

KPZ scaling-theory conjecture. The one-parameter family $(P_\lambda)_{\lambda\in\mathbb{R}}$ completely describes the spatially translation-invariant stationary measures for the KP…

Only-ergodic-stationary-distributions conjecture. The only ergodic stationary distributions of the KPZ equation are Brownian motions with linear drift.

Only-ergodic-stationary-distributions conjecture. The only ergodic stationary distributions of the KPZ equation are Brownian motions with linear drift.

Fix $u,v\in\R$. Let $h_0,\tilde h_0\in\operatorname{C}([0,1])$ be arbitrary random functions on the same probability space, and let $\xi$ be white noise on $\mathbb{R}\times[0,1]$.…

Let $u,v e 0$ be boundary parameters, and let $mathcal{h}{u,v}$ denote the stationary measure of the open KPZ equation. For $u,v0$, this measure is represented as the law of $W+X$,…

Let $H$ denote the stationary height function and let the path-integral representation associated with it be defined by the formulas referenced as … , with boundary parameters…

Consider the open KPZ equation with stationary measures $h_{u,v}(\cdot)$ indexed by each pair of boundary parameters $u,v$. A one-force one-solution principle means that, when two…

Let $\Delta_h=h_{{\rm opt},B}(+\infty,t)-h_{{\rm opt},B}(-\infty,t)$ be the height difference for the optimal profile with Brownian initial condition, and let $\kappa_0$,…

Let $h_\varepsilon$ denote the KPZ equation solution with initial data, and suppose the initial data are varied from the narrow-wedge data considered in Theorem; for symmetrically…

The half-line KPZ stationary measures described above are considered in the large-scale limit as $x$ tends to infinity; this limit is expected to define stationary measures of the…

Let $\mathfrak{h}_t$ denote the KPZ height fluctuation at time $t$, under initial data more general than the narrow wedge, and consider its large peaks and valleys. KPZ scaling con…

Let $h(t,x)$ denote the solution of the KPZ equation with narrow wedge initial data, and consider its fluctuations under horizontal scaling by $t^{2/3}$, vertical scaling by…

Fix $\delta\in(0,2/5)$. Define … and let $\mathbf{I}_x=\mathbf{I}_x(s,\delta)$ be the interval … For $x\in\mathbf{I}_x$, set $\bar{\aleph}:=\aleph(x,\bar{\gamma})$. Uniform asympto…

The KPZ equation is considered with deterministic initial data, and its lower tail refers to the regime in which the solution or partition function has unusually low values. The pa…

For any $\delta>0$, let $\mathcal{C}_{\mathfrak{s};\mathrm{loc}}^{-\delta}(\mathbf{R}_{>0}\times\mathbf{T}^{2})$ and $\mathcal{C}^{-\delta}(\mathbf{T}^{2})$ denote the correspondin…

Consider the $(2+1)$-dimensional KPZ equation with diffusivity $u$ and noise strength $D$ fixed, and let $lambda$ be its nonlinearity parameter. A nontrivial scaling limit is a li…