非Lipschitz动力系统中的统计确定性
Statistical determinism in non-Lipschitz dynamical systems
AI总结：
本文研究具有非Lipschitz点奇点的常微分方程，通过引入随机正则化作为选择准则，证明了在确定性系统具有混沌性质时，极限过程可产生独立于正则化的唯一统计解，使解在通过奇点后自发随机。
摘要：
我们研究一类具有非Lipschitz点奇点的常微分方程，该奇点允许通过此点的解不唯一。作为一种选择准则，我们引入了依赖于参数 $ν$ 的随机正则化：对于每个 $ν>0$，正则化动力学是全局定义的，且在 $ν$ 趋于零的极限下恢复了原始的奇异系统。我们证明，当确定性系统具有某些混沌性质时，该极限产生一个独立于正则化的唯一统计解。在这种情况下，解在通过奇点后变得自发随机：它们以一个内在的概率分布被随机选择。
英文摘要：
We study a class of ordinary differential equations with a non-Lipschitz point singularity, which admit non-unique solutions through this point. As a selection criterion, we introduce stochastic regularizations depending on the parameter $ν$: the regularized dynamics is globally defined for each $ν> 0$, and the original singular system is recovered in the limit of vanishing $ν$. We prove that this limit yields a unique statistical solution independent of regularization, when the deterministic system possesses certain chaotic properties. In this case, solutions become spontaneously stochastic after passing through the singularity: they are selected randomly with an intrinsic probability distribution.
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