TY - JOUR

T1 - Markovian Solutions to Discontinuous ODEs

AU - Bressan, Alberto

AU - Mazzola, Marco

AU - Nguyen, Khai T.

N1 - Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

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U2 - 10.1007/s10884-021-09974-4

DO - 10.1007/s10884-021-09974-4

M3 - Article

AN - SCOPUS:85102488785

JO - Journal of Dynamics and Differential Equations

JF - Journal of Dynamics and Differential Equations

SN - 1040-7294

ER -