Markovian Solutions to Discontinuous ODEs

TL;DR

This paper characterizes Markovian solutions to discontinuous ODEs by describing how probability measures on solutions can be constructed using measures, Poisson variables, and probabilities at points of discontinuity.

Contribution

It provides a complete characterization of all Markovian generalized flows for discontinuous ODEs in terms of measures, Poisson variables, and transition probabilities.

Findings

Characterization of Markovian solutions via measures and random variables.

Explicit description of transition mechanisms at discontinuities.

Framework applicable to a broad class of discontinuous ODEs.

Abstract

Given a possibly discontinuous, bounded function $f:\mathbb{R}\mapsto\mathbb{R}$, we consider the set of generalized flows, obtained by assigning a probability measure on the set of Carath\'eodory solutions to the ODE ~$\dot x = f(x)$. The paper provides a complete characterization of all such flows which have a Markov property in time. This is achieved in terms of (i) a positive, atomless measure supported on the set $f^{-1}(0)$ where $f$ vanishes, (ii) a countable number of Poisson random variables, determining the waiting times at points in $f^{-1}(0)$, and (iii) a countable set of numbers $\theta_k\in [0,1]$, describing the probability of moving up or down, at isolated points where two distinct trajectories can originate.