Assigning probabilities to non-Lipschitz mechanical systems

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: $\ddot{r} = r^\alpha$, $r(t=0) =0$, and $\dot{r}\mid_{r(t=0)} =0$, where the dots indicate derivatives to time and $\alpha \in ]0,1[$. This example has a physical interpretation as a mass in a uniform gravitational field on a dome of particular shape; the case with $\alpha=1/2$ is known as Norton's dome. Our approach is based on (1) finite difference equations, which are deterministic, (2) a uniform prior on the phase space, and (3) non-standard analysis, which involves infinitesimals and which is conceptually close to numerical methods from physical praxis. This allows us to assign probabilities to the solutions of the initial value problem in the original, indeterministic model.