具有强剪切的随机流——第一部分：强完备性与集合吸引子
Stochastic Flows with Strong Shear - Part I: Strong Completeness and Set Attractors
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中文总结 AI 辅助
本文研究一类带强剪切项的二维随机微分方程诱导的随机流，给出强完备性与集合吸引子存在的充要条件，并证明角速度导数增长过快时紧集直径可指数增长，甚至导致全局流不存在。
中文摘要 AI 辅助
我们研究一类二维随机微分方程所诱导的随机流的强完备性以及集合吸引子的存在性，该方程类似于平面Ornstein-Uhlenbeck过程，并附加一个依赖于半径的旋转漂移项。我们的主要结果给出了随机流强完备以及集合吸引子存在的充分条件和必要条件。特别地，我们证明：如果角速度$\rho(r)$关于半径$r$的导数满足$|\rho'(r)| \geq K_2\\, r^3$（对于大的$r$和某个普适常数$K_2$），则紧致集合的直径可以以正概率指数级增长。此外，我们证明当$|\rho'(r)|\geq r^{3+\varepsilon}$（$\varepsilon>0$）时，以概率1，例外初始条件在有限时间内发散至无穷，从而排除了全局随机流的存在。
英文摘要
We study strong completeness and the existence of a set attractor for the stochastic flows induced by a class of two-dimensional stochastic differential equations resembling a planar Ornstein-Uhlenbeck process with an additional radius-dependent rotational drift term. Our main results give both sufficient and necessary conditions for the stochastic flows to be strongly complete and for the existence of set attractors. In particular, we demonstrate that if the derivative of the angular velocity $ρ(r)$ with respect to the radius $r$ satisfies $|ρ'(r)| \geq K_2\, r^3$, for large $r$ and some universal constant $K_2$, the diameter of a compact set can grow exponentially fast with positive probability. Furthermore, we show that for $|ρ'(r)|\geq r^{3+\varepsilon}$, $\varepsilon>0$, with probability one, exceptional initial conditions diverge to infinity in finite time, ruling out the existence of a global stochastic flow.
