Markovian Solutions to Discontinuous ODEs

Alberto Bressan, Marco Mazzola, Khai T. Nguyen

Research output: Contribution to journalArticlepeer-review

Abstract

Given a possibly discontinuous, bounded function f: R↦ R, we consider the set of generalized flows, obtained by assigning a probability measure on the set of Carathéodory solutions to the ODE 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 θk∈ [0 , 1] , describing the probability of moving up or down, at isolated points where two distinct trajectories can originate.

Original languageEnglish (US)
JournalJournal of Dynamics and Differential Equations
DOIs
StateAccepted/In press - 2021

All Science Journal Classification (ASJC) codes

  • Analysis

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