Singular HJB equations with applications to KPZ on the real line

奇异随机偏微分方程的链式法则对称性：从多指标到装饰树

Chain rule symmetry for singular SPDEs from multi-indices to decorated trees

本文回顾广义与几何KPZ方程链式法则的最新结果，通过B级数展开引入正则结构，用多指标与装饰树分别刻画，后者需operadic与同调工具，并总结证明要点。

具有奇异相互作用的布朗粒子双无穷系统与KPZ方程

Bi-infinite systems of singularly interacting Brownian particles and the KPZ equation

研究实线上具奇异非对称相互作用的布朗粒子双无穷系统，通过碰撞链局部化等方法，建立适定性理论与确定平稳分布，证明在弱非对称下粒子数函数平衡涨落收敛到乘性随机热方程，将其置于KPZ普适类。

The Gaussian Structure of the Singular Stochastic Burgers Equation

We consider the stochastically forced Burgers equation with an emphasis on spatially rough driving noise. We show that the law of the process at a fixed time $t$, conditioned on no explosions, is absolutely continuous with respect to the stochastic heat equation obtained by removing the nonlinearity from the equation. This establishes a form of ellipticity in this infinite dimensional setting. The results follow from a recasting of the Girsanov Theorem to handle less spatially regular solutions while only proving absolute continuity at a fixed time and not on path-space. The results are proven…

Some recent progress in singular stochastic PDEs

Stochastic PDEs are ubiquitous in mathematical modeling. Yet, many such equations are too singular to admit classical treatment. In this article we review some recent progress in defining, approximating and studying the properties of a few examples of such equations. We focus mainly on the dynamical Phi4 equation, KPZ equation and Parabolic Anderson Model, as well as touch on a few other equations which arise mainly in physics.

HJB equations for certain singularly controlled diffusions

Given a closed, bounded convex set $\mathcal{W}\subset{\mathbb {R}}^d$ with nonempty interior, we consider a control problem in which the state process $W$ and the control process $U$ satisfy \[W_t= w_0+\int_0^t\vartheta(W_s) ds+\int_0^tσ(W_s) dZ_s+GU_t\in \mathcal{W},\qquad t\ge0,\] where $Z$ is a standard, multi-dimensional Brownian motion, $\vartheta,σ\in C^{0,1}(\mathcal{W})$, $G$ is a fixed matrix, and $w_0\in\mathcal{W}$. The process $U$ is locally of bounded variation and has increments in a given closed convex cone $\mathcal{U}\subset{\mathbb{R}}^p$. Given $g\in C(\mathcal{W})$, $κ\in{…

A (2+1)-dimensional Anisotropic KPZ growth model with a smooth phase

Stochastic growth processes in dimension $(2+1)$ were conjectured by D. Wolf, on the basis of renormalization-group arguments, to fall into two distinct universality classes, according to whether the Hessian $H_ρ$ of the speed of growth $v(ρ)$ as a function of the average slope $ρ$ satisfies $\det H_ρ>0$ ("isotropic KPZ class") or $\det H_ρ\le 0$ ("anisotropic KPZ (AKPZ)" class). The former is characterized by strictly positive growth and roughness exponents, while in the AKPZ class fluctuations are logarithmic in time and space. It is natural to ask (a) if one can exhibit interesting growth m…

开放KPZ方程的不变测度：解析视角

Invariant measures for the open KPZ equation: an analytic perspective

本文通过随机分析方法，证明了开放KPZ方程的不变测度公式，利用高斯测度、测度变换和正则结构理论，建立了边界层内的中心极限定理。

拟线性奇异SPDE的正则性结构

Regularity structures for quasilinear singular SPDEs

该研究证明拟线性广义KPZ方程正则性结构表述的适定性，给出全次临界区域重整化方程显式形式，在BPHZ模型收敛假设下得到正则化重整化方程解的收敛性，涵盖时空白噪声情形。

Martingale-driven approximations of singular stochastic PDEs

We define multiple stochastic integrals with respect to càdlàg martingales and prove moment bounds and chaos expansions, which allow to work with them in a way similar to Wiener stochastic integrals. In combination with the discretization framework of Erhard and Hairer (2017), our results give a tool for proving convergence of interacting particle systems to stochastic PDEs using regularity structures. As examples, we prove convergence of martingale-driven discretizations of the $3$-dimensional stochastic quantization equation and the KPZ equation.

各向异性KPZ方程的弱耦合极限

Weak coupling limit of the Anisotropic KPZ equation

本文研究二维各向异性KPZ方程在弱耦合极限下的行为，证明当耦合常数按对数标度衰减时，方程收敛到显式有效粘性的线性随机热方程，首次给出无法通过Cole-Hopf变换线性化的临界奇异SPDE的完整重整化结果。

2D Anisotropic KPZ at stationarity: scaling, tightness and non triviality

In this work we focus on the two-dimensional anisotropic KPZ (aKPZ) equation, which is formally given by \begin{equation*}\partial_t h =\fracν{2}Δh + λ((\partial_1 h)^2 - (\partial_2 h)^2) + ν^\frac{1}{2}ξ,\end{equation*} where $ξ$ denotes a noise which is white in both space and time, and $λ$ and $ν$ are positive constants. Due to the wild oscillations of the noise and the quadratic nonlinearity, the previous equation is classically ill-posed. It is not possible to linearise it via the Cole-Hopf transformation and the pathwise techniques for singular SPDEs (the theory of Regularity Structures…

非平移不变奇异 SPDE 的典范解

Canonical solutions to non-translation invariant singular SPDEs

本文研究含一般一致椭圆算子的非平移不变奇异 SPDE，构造有限维典范解族，求解 g-PAM、ϕ^4_2、ϕ^4_3 与 KPZ，并证明重整化函数的局部性及解映射连续性。

奇异 SPDEs 的链式法则对称性

Chain rule symmetry for singular SPDEs

本文利用 operad 理论和同调代数思想，刻画了亚临界机制下几何随机热方程的链式法则对称性，并给出了拟线性 KPZ 方程的正确重整化。

临界维数下随机 Burgers 方程的超扩散中心极限定理

Superdiffusive Central Limit Theorem for the Stochastic Burgers Equation at the critical dimension

本文证明二维临界随机 Burgers 方程具有 $(\log t)^{2/3}$ 对数超扩散性，精确给出前因子，并建立超扩散中心极限定理与显式 Gaussian 重整化群不动点。