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 Carathéodory 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.
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Notes
Indeed, differentiating the identity \(\displaystyle {1\over 1-x} = \sum \nolimits _{j=0}^\infty x^j\) one obtains \(\displaystyle {1\over (1-x)^2} = \sum \nolimits _{j=1}^\infty j x^{j-1}\). Setting \(x=1/2\) we get \( \displaystyle \sum _{j=1}^\infty j 2^{1-j} = 4\).
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Acknowledgements
This research by K. T. Nguyen was partially supported by a grant from the Simons Foundation/SFARI (521811, NTK).
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Bressan, A., Mazzola, M. & Nguyen, K.T. Markovian Solutions to Discontinuous ODEs. J Dyn Diff Equat 35, 135–162 (2023). https://doi.org/10.1007/s10884-021-09974-4
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DOI: https://doi.org/10.1007/s10884-021-09974-4