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 - 2023/3
Y1 - 2023/3
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.
UR - http://www.scopus.com/inward/record.url?scp=85102488785&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85102488785&partnerID=8YFLogxK
U2 - 10.1007/s10884-021-09974-4
DO - 10.1007/s10884-021-09974-4
M3 - Article
AN - SCOPUS:85102488785
SN - 1040-7294
VL - 35
SP - 135
EP - 162
JO - Journal of Dynamics and Differential Equations
JF - Journal of Dynamics and Differential Equations
IS - 1
ER -