论文标题
将概率分配给非lipschitz机械系统
Assigning probabilities to non-Lipschitz mechanical systems
论文作者
论文摘要
我们提出了一种将概率分配给具有Lipschitz奇异性的最初价值问题解决方案的方法。为了说明该方法，我们关注以下玩具示例：$ \ frac {d^2r（t）} {dt^2} = r^a $，$ r（t = 0）= 0 $，和$ \ frac {dr（t）} {dt} {dt} {dt} {dt} \ mid_ {r（t = 0）}这个示例具有物理解释为在特定形状的无摩擦，刚性圆顶上的均匀重力场中的质量。 $ a = 1/2 $的表壳称为诺顿的圆顶。我们的方法基于（1）有限差方程，这些方程是确定性的； （2）Alpha理论的基本技术，这是一个简化的非标准分析框架，使我们能够研究无穷小的扰动； （3）规范相空间上的统一先验。我们确定性的高铁网格模型使我们能够将概率分配给原始不确定模型中初始值问题的解决方案。
We present a method for assigning probabilities to the solutions of initial value problems that have a Lipschitz singularity. To illustrate the method, we focus on the following toy example: $\frac{d^2r(t)}{dt^2} = r^a$, $r(t=0) =0$, and $\frac{dr(t)}{dt}\mid_{r(t=0)} =0$, with $a \in ]0,1[$. This example has a physical interpretation as a mass in a uniform gravitational field on a frictionless, rigid dome of a particular shape; the case with $a=1/2$ is known as Norton's dome. Our approach is based on (1) finite difference equations, which are deterministic; (2) elementary techniques from alpha-theory, a simplified framework for non-standard analysis that allows us to study infinitesimal perturbations; and (3) a uniform prior on the canonical phase space. Our deterministic, hyperfinite grid model allows us to assign probabilities to the solutions of the initial value problem in the original, indeterministic model.
