We study a class of ordinary differential equations with a non-Lipschitz point singularity that admits non-unique solutions through this point. As a selection criterion, we introduce stochastic regularizations depending on a parameter $\nu $ : the regularized dynamics is globally defined for each $\nu> 0$ , and the original singular system is recovered in the limit of vanishing $\nu $ . We prove that this limit yields a unique statistical solution independent of regularization when the deterministic system possesses a chaotic attractor having a physical measure with the convergence to equilibrium property. In this case, solutions become spontaneously stochastic after passing through the singularity: they are selected randomly with an intrinsic probability distribution.

中文翻译：

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非李普希茨动力系统中的统计决定论

我们研究一类具有非利普希茨点奇点的常微分方程，它允许通过该点的非唯一解。作为选择标准，我们根据参数引入随机正则化 $\n$ ：正则化动态是为每个全局定义的 $\nu> 0$ ，在消失极限下恢复原奇异系统 $\n$ 。我们证明这个极限产生独特的统计解决方案当确定性系统具有混沌吸引子且具有收敛到平衡特性的物理量度时，独立于正则化。在这种情况下，解决方案在通过奇点后变得自发随机：它们是按照内在概率分布随机选择的。