Asymptotics for the almost sure Lyapunov Exponent for the solution of the parabolic Anderson problem

随机微分方程的有限时间爆炸：Khasminskii Lyapunov方法及其与Osgood准则一致性的综述

Finite Time Explosion of Stochastic Differential Equations: A survey into Khasminskii's Lyapunov Method and its Consistency with the Osgood Criterion

本文综述Khasminskii Lyapunov方法如何分类随机微分方程解的三类有限时间爆炸行为，并结合Feller检验、示例及向常数Poisson强度跳过程的扩展说明其适用性。

Almost sure stability of controlled degenerate diffusions

We develop a direct Lyapunov method for the almost sure open-loop stabilizability and asymptotic stabilizability of controlled degenerate diffusion processes. The infinitesimal decrease condition for a Lyapunov function is a new form of Hamilton-Jacobi-Bellman partial differential inequality of $2nd$ order. We give local and global versions of the First and Second Lyapunov Theorems assuming the existence of a lower semicontinuous Lyapunov function satisfying such inequality in the viscosity sense. An explicit formula for a stabilizing feedback is provided for affine systems with smooth Lyapuno…

Anderson 随机量子化方程

Anderson stochastic quantization equation

本文研究紧曲面上带乘性空间白噪声和独立时空白噪声驱动的抛物型散焦随机量子化方程，通过重整化构造随机 Gibbs 测度，并证明初值服从该测度时动力学的几乎必然全局适定性与 Gibbs 测度不变性。

Nonlinear Anderson localized states at arbitrary disorder

It is classical, following Furstenberg's theorem on positive Lyapunov exponent for products of random SL$(2, \mathbb R)$ matrices, that the one dimensional random Schrödinger operator has Anderson localization at arbitrary disorder. This paper proves a nonlinear analogue, thereby establishing a KAM-type persistence result for a non-integrable system.

Almost-sure exponential mixing of passive scalars by the stochastic Navier-Stokes equations

We deduce almost-sure exponentially fast mixing of passive scalars advected by solutions of the stochastically-forced 2D Navier-Stokes equations and 3D hyper-viscous Navier-Stokes equations in $\mathbb T^d$ subjected to non-denegenerate $H^σ$-regular noise for any $σ$ sufficiently large. That is, for all $s > 0$ there is a deterministic exponential decay rate such that all mean-zero $H^s$ passive scalars decay in $H^{-s}$ at this same rate with probability one. This is equivalent to what is known as \emph{quenched correlation decay} for the Lagrangian flow in the dynamical systems literature. …