Renormalized solutions for stochastic transport equations and the regularization by bilinear multiplicative noise

Bilinear Coagulation Equations

We consider coagulation equations of Smoluchowski or Flory type where the total merge rate has a bilinear form $π(y)\cdot Aπ(x)$ for a vector of conserved quantities $π$, generalising the multiplicative kernel. For these kernels, a gelation transition occurs at a finite time $t_\mathrm{g}\in (0,\infty)$, which can be given exactly in terms of an eigenvalue problem in finite dimensions. We prove a hydrodynamic limit for a stochastic coagulant, including a corresponding phase transition for the largest particle, and exploit a coupling to random graphs to extend analysis of the limiting process b…

The Edwards-Wilkinson limit of the random heat equation in dimensions three and higher

We consider the heat equation with a multiplicative Gaussian potential in dimensions $d\geq 3$. We show that the renormalized solution converges to the solution of a deterministic diffusion equation with an effective diffusivity. We also prove that the renormalized large scale random fluctuations are described by the Edwards-Wilkinson model, that is, the stochastic heat equation (SHE) with additive white noise, with an effective variance.

Bilinear control and growth of Sobolev norms for the nonlinear Schrödinger equation

We consider the nonlinear Schrödinger equation (NLS) on a torus of arbitrary dimension. The equation is studied in presence of an external potential field whose time-dependent amplitude is taken as control. Assuming that the potential satisfies a saturation property, we show that the NLS equation is approximately controllable between any pair of eigenstates in arbitrarily small time. The proof is obtained by developing a multiplicative version of a geometric control approach introduced by Agrachev and Sarychev. We give an application of this result to the study of the large time behavior of th…

通过正则性结构得到随机多孔介质方程的先验界

A priori bounds for stochastic porous media equations via regularity structures

该研究在基于L¹的正则性类中证明了带乘性噪声的奇异随机多孔介质方程解的先验界，通过动力学形式和重归一化能量不等式，平衡扩散退化性与噪声阻尼效应，实现解的模型化。

具有对数相关高斯系数的重整化随机压力方程

Renormalized stochastic pressure equation with log-correlated Gaussian coefficients

本文研究含对数相关高斯系数与 Wick 重整化项的随机压力方程，借助椭圆正则性和改造的 S 变换定义解，并证明周期解的存在唯一性及其高斯乘性混沌表示。

乘性噪声驱动的随机Schrödinger-Korteweg de Vries系统

Stochastic Schrödinger-Korteweg de Vries systems driven by multiplicative noises

本文研究乘性噪声驱动的随机S-KdV系统在$H_x^1\times H_x^1$中的适定性，通过建立Bourgain范数估计和引入物理与频率空间截断的近似方程，证明了系统的整体适定性。

Optimization based model order reduction for stochastic systems

In this paper, we bring together the worlds of model order reduction for stochastic linear systems and $\mathcal H_2$-optimal model order reduction for deterministic systems. In particular, we supplement and complete the theory of error bounds for model order reduction of stochastic differential equations. With these error bounds, we establish a link between the output error for stochastic systems (with additive and multiplicative noise) and modified versions of the $\mathcal H_2$-norm for both linear and bilinear deterministic systems. When deriving the respective optimality conditions for mi…

Localization of the Gaussian multiplicative chaos in the Wiener space and the stochastic heat equation in strong disorder

We consider a {\it{Gaussian multiplicative chaos}} (GMC) measure on the classical Wiener space driven by a smoothened (Gaussian) space-time white noise. For $d\geq 3$ it was shown in \cite{MSZ16} that for small noise intensity, the total mass of the GMC converges to a strictly positive random variable, while larger disorder strength (i.e., low temperature) forces the total mass to lose uniform integrability, eventually producing a vanishing limit. Inspired by strong localization phenomena for log-correlated Gaussian fields and Gaussian multiplicative chaos in the finite dimensional Euclidean s…

方差爆破之外色散偏微分方程的概率适定性 I：Benjamin-Bona-Mahony 方程

Probabilistic well-posedness of dispersive PDEs beyond variance blowup I: Benjamin-Bona-Mahony equation

该研究拓展非线性色散PDE概率适定性理论至方差爆破外，以BBM方程为对象，通过两种重整化方法证明解的依分布收敛性，且结果适用于任意低正则高斯初值，还推广到分数阶噪声驱动情形。

乘性莱维噪声驱动的随机二维流体型演化方程的长期动力学

Long-time dynamics of stochastic 2D hydrodynamic-type evolution equations driven by multiplicative Lévy noise

本文研究了由乘性莱维噪声驱动的抽象非线性随机流体型方程解的长期动力学，建立了全局解的存在性，并证明了弱拉回均随机吸引子的存在唯一性，探讨了不变测度的存在性和双极限行为，以及在附加条件下拉回测度吸引子的唯一性和渐近自洽稳定性。

Model reduction and uncertainty quantification of multiscale diffusions with parameter uncertainties using nonlinear expectations

In this paper we study model reduction of linear and bilinear quadratic stochastic control problems with parameter uncertainties. Specifically, we consider slow-fast systems with unknown diffusion coefficient and study the convergence of the slow process in the limit of infinite scale separation. The aim of our work is two-fold: Firstly, we want to propose a general framework for averaging and homogenisation of multiscale systems with parametric uncertainties in the drift or in the diffusion coefficient. Secondly, we want to use this framework to quantify the uncertainty in the reduced system …

维纳空间中的次临界高斯乘性混沌：构造、矩与体积衰减

Subcritical Gaussian Multiplicative Chaos in the Wiener Space: Construction, Moments and Volume Decay

本文在维纳空间上构造了弱无序区域中的次临界高斯乘性混沌，证明其支撑于γ-厚路径、相对于维纳测度奇异，并确定了测度的分形性质与总质量的负矩和L^p矩。