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Markovian Solutions to Discontinuous ODEs

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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

  1. 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  \( \displaystyle \sum _{j=1}^\infty j 2^{1-j} = 4\).

References

  1. Alberti, G., Bianchini, S., Crippa, G.: A uniqueness result for the continuity equation in two dimensions. J. Eur. Math. Soc. 16, 201–234 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ambrosio, L., Crippa, G.: Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields. In: Transport Equations and Multi-D Hyperbolic Conservation Laws. Lecture Notes of the Unione Matematica Italiana. vol. 5, pp. 3–57. Springer, Berlin (2008)

  3. Ancona, F., Bressan, A.: Patchy vector fields and asymptotic stabilization. Control Opt. Calc. Var. 4, 419–444 (1999)

    MathSciNet  MATH  Google Scholar 

  4. Binding, P.: The differential equation \(\dot{x}=f \circ x\). J. Differ. Equ. 31, 183–199 (1979)

    Article  MATH  Google Scholar 

  5. Bressan, A.: Unique solutions for a class of discontinuous differential equations. Proc. Am. Math. Soc. 104, 772–778 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bressan, A., Colombo, G.: Existence and continuous dependence for discontinuous O.D.E’s. Boll. Un. Mat. Italiana B 4, 295–311 (1990)

    MathSciNet  MATH  Google Scholar 

  7. Bressan, A., Mazzola, M., Nguyen, K.T.: Diffusion approximations of Markovian solutions to discontinuous ODEs (to appear)

  8. Bressan, A., Shen, W.: Uniqueness for discontinuous O.D.E. and conservation laws. Nonlinear Anal. 34, 637–652 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  9. Clarke, F.H., Ledyaev, Y.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory. Springer, New York (1998)

    MATH  Google Scholar 

  10. Colombini, F., Rauch, J.: Uniqueness in the Cauchy problem for transport in \(\mathbb{R}^2\) and \(\mathbb{R}^{1+2}\). J. Differ. Equ. 211, 162–167 (2005)

    Article  MATH  Google Scholar 

  11. Colombo, R.M., Marson, A.: A Hölder continuous ODE related to traffic flow. Proc. R. Soc. Edinb. Sect. A 133, 759–772 (2003)

    Article  MATH  Google Scholar 

  12. Dafermos, C.: Generalized characteristics and the structure of solutions of hyperbolic conservation laws. Indiana Univ. Math. J. 26, 1097–1119 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dafermos, C.: Generalized characteristics in hyperbolic systems of conservation laws. Arch. Ration. Mech. Anal. 107, 127–155 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dieci, L., Lopez, L.: A survey of numerical methods for IVPs of ODEs with discontinuous right-hand side. J. Comput. Appl. Math. 236, 3967–3991 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. DiPerna, R.J., Lions, P.L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  16. Weinan, E., Vanden-Eijnden, E.: A note on generalized flows. Physica D 183, 159–174 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Filippov, A.: Differential Equations with Discontinuous Right-Hand Sides. Kluwer Academic, Dordrecht (1988)

    Book  MATH  Google Scholar 

  18. Flandoli, F., Langa, J.: Markov attractors: a probabilistic approach to multivalued flows. Stoch. Dyn. 8, 59–75 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Marigo, A., Piccoli, B.: Regular syntheses and solutions to discontinuous ODEs. ESAIM Control Optim. Calc. Var. 7, 291–307 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Sentis, R.: Équations différentielles à second membre mesurable. Boll. Un. Mat. Italiana B. 15, 724–742 (1978)

    MathSciNet  MATH  Google Scholar 

  21. Piccoli, B., Sussmann, H.: Regular synthesis and sufficiency conditions for optimality. SIAM J. Control Optim. 39, 359–410 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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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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Authors and Affiliations

  1. Department of Mathematics, Penn State University, State College, USA

    Alberto Bressan

  2. IMJ-PRG, CNRS, Sorbonne Université, Paris, France

    Marco Mazzola

  3. Department of Mathematics, North Carolina State University, Raleigh, USA

    Khai T. Nguyen

Authors
  1. Alberto Bressan
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  2. Marco Mazzola
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  3. Khai T. Nguyen
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Correspondence to Khai T. Nguyen.

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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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