Skip to main content
SpringerLink
Log in

Diffusion Approximations of Markovian Solutions to Discontinuous ODEs

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

Abstract

In a companion paper, the authors have characterized all deterministic semigroups, and all Markov semigroups, whose trajectories are Carathéodory solutions to a given ODE \(\dot{x} = f(x)\), where f is a possibly discontinuous, regulated function. The present paper establishes two approximation results. Namely, every deterministic semigroup can be obtained as the pointwise limit of the flows generated by a sequence of ODEs \(\dot{x}=f_n(x)\) with smooth right hand sides. Moreover, every Markov semigroup can be obtained as limit of a sequence of diffusion processes with smooth drifts and with diffusion coefficients approaching zero.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Aizenman, M.: On vector fields as generators of flows: a counterexample to Nelson’s conjecture. Ann. Math. 107, 287–296 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  2. 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 

  3. Ambrosio, L.: Transport equation and Cauchy problem for BV vector fields. Invent. Math. 158, 227–260 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Aubin, J.P., Cellina, A.: Differential Inclusions, Set-Valued Maps and Viability Theory. Springer, Berlin (1984)

    Book  MATH  Google Scholar 

  5. Bianchini, S., Bonicatto, P.: A uniqueness result for the decomposition of vector fields in \({{\mathbb{R}}}^d\). Invent. Math. 220, 255–393 (2020)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  7. Bressan, A., Mazzola, M., Nguyen, K.T.: Markovian solutions to discontinuous ODEs. J. Dyn. Differ. Equ. (2021)

  8. Cellina, A.: Approximation of set valued functions and fixed point theorems. Ann. Mat. Pura Appl. 82, 17–24 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cellina, A.: Multivalued differential equations and ordinary differential equations. SIAM J. Appl. Math. 18, 533–538 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  10. Crippa, G., Gusev, N., Spirito, S., Wiedemann, E.: Non-uniqueness and prescribed energy for the continuity equation. Commun. Math. Sci. 13, 1937–1947 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ciampa, G., Crippa, G., Spirito, S.: Smooth approximation is not a selection principle for the transport equation with rough vector field. Calc. Var. Partial Differ. Equ. 59(1), 13 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  12. Biles, D.C., Federson, M., Pouso, R.L.: A survey of recent results for the generalizations of ordinary differential equations. Abstr. Appl. Anal. 260409 (2014)

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Dovetta, S., Marconi, E., Spinolo, L.V.: Initial-boundary value problems for merely bounded nearly incompressible vector fields in one space dimension. arXiv:2105.11157

  15. Filippov, A.: Differential equations with discontinuous right-hand side. Am. Math. Soc. Transl. 199–231 (1964)

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Gihman, I.I., Skorohod, A.V.: The Theory of Stochastic Processes, Vols. I, II, and III, Springer-Verlag, Berlin, 1974, 1975, and 1979

  18. Gusev, N.A.: On the one-dimensional continuity equation with a nearly incompressible vector field. Commun. Pure Appl. Anal. 18, 559–568 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  19. Karlin, S., Taylor, H.M.: A Second Course in Stochastic Processes. Academic Press, New York-London (1981)

    MATH  Google Scholar 

  20. Modena, S., Székelyhidi, L.: Non-uniqueness for the transport equation with Sobolev vector fields. Ann. PDE 4(2), 18 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  21. Oksendal, B.: Stochastic Differential Equations, An Introduction with Applications, 5th edn. Springer (2003)

  22. Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations. Prentice-Hall, Englewood Cliffs (1967)

    MATH  Google Scholar 

  23. Stroock, D.W., Varadhan, S.R.S.: Multidimensional Diffusion Processes. Springer, Berlin (1979)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This research by K. T. Nguyen was partially supported by a Grant from the Simons Foundation/SFARI (521811, NTK).

Author information

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
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marco Mazzola
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Khai T. Nguyen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Khai T. Nguyen.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bressan, A., Mazzola, M. & Nguyen, K.T. Diffusion Approximations of Markovian Solutions to Discontinuous ODEs. J Dyn Diff Equat (2023). https://doi.org/10.1007/s10884-023-10250-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10884-023-10250-w

Search

Navigation