Skip to main content
SpringerLink
Log in

Front Propagation and Quasi-Stationary Distributions: Two Faces of the Same Coin

  • Conference paper
  • First Online:
Sojourns in Probability Theory and Statistical Physics - III

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 300))

Abstract

We analyze the connection between front propagation and quasi-stationary distributions in translation invariant one-dimensional Markov processes. We describe the link between them through the microscopic models known as Branching Brownian Motion with selection and Fleming–Viot.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Aronson, D.G., Weinberger, H.F.: Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30(1), 33–76 (1978)

    Article  MathSciNet  Google Scholar 

  2. Asselah, A., Ferrari, P.A., Groisman, P., Jonckheere, M.: Fleming-Viot selects the minimal quasi-stationary distribution: The Galton-Watson case. Ann. Inst. Henri Poincaré Probab. Stat. 52(2), 647–668 (2016)

    Article  MathSciNet  Google Scholar 

  3. Asselah, A., Thai, M.-N.: A note on the rightmost particle in a Fleming–Viot process. Preprint arXiv:1212.4168 (2012)

  4. Bérard, J., Gouéré, J.B.: Brunet-Derrida behavior of branching-selection particle systems on the line. Commun. Math. Phys. 298(2), 323–342 (2010)

    Article  MathSciNet  Google Scholar 

  5. Berestycki, J., Berestycki, N., Schweinsberg, J.: The genealogy of branching Brownian motion with absorption. Ann. Probab. 41(2), 527–618 (2013)

    Article  MathSciNet  Google Scholar 

  6. Bieniek, M., Burdzy, K., Finch, S.: Non-extinction of a Fleming-Viot particle model. Probab. Theory Relat. Fields 153(1–2), 293–332 (2012)

    Article  MathSciNet  Google Scholar 

  7. Bramson, M., Calderoni, P., De Masi, A., Ferrari, P., Lebowitz, J., Schonmann, R.H.: Microscopic selection principle for a diffusion-reaction equation. J. Statist. Phys. 45(5–6), 905–920 (1986)

    Article  MathSciNet  Google Scholar 

  8. Brunet, É., Derrida, B.: Shift in the velocity of a front due to a cutoff. Phys. Rev. E 56(3 part A), 2597–2604 (1997)

    Article  MathSciNet  Google Scholar 

  9. Brunet, É., Derrida, B.: Effect of microscopic noise on front propagation. J. Statist. Phys. 103(1–2), 269–282 (2001)

    Article  MathSciNet  Google Scholar 

  10. Brunet, É., Derrida, B., Mueller, A.H., Munier, S.: Noisy traveling waves: effect of selection on genealogies. Europhys. Lett. 76(1), 1–7 (2006)

    Article  MathSciNet  Google Scholar 

  11. Brunet, É., Derrida, B., Mueller, A.H., Munier, S.: Effect of selection on ancestry: an exactly soluble case and its phenomenological generalization. Phys. Rev. E 76(4), 041104 (2007)

    Article  MathSciNet  Google Scholar 

  12. Burdzy, K., Holyst, R., Ingerman, D., March, P.: Configurational transition in a Fleming–Viot-type model and probabilistic interpretation of Laplacian eigenfunctions. J. Phys. A, Math. Gen. 29(11), 2633–2642 (1996)

    Article  Google Scholar 

  13. Durrett, R., Remenik, D.: Brunet-Derrida particle systems, free boundary problems and Wiener-Hopf equations. Ann. Probab. 39(6), 2043–2078 (2011)

    Article  MathSciNet  Google Scholar 

  14. Ferrari, P.A., Marić, N.: Quasi stationary distributions and Fleming-Viot processes in countable spaces. Electron. J. Probab. 12(24), 684–702 (2007)

    Article  MathSciNet  Google Scholar 

  15. Fisher, R.A.: The wave of advance of advantageous genes. Ann. Eugenics 7, 353–369 (1937)

    MATH  Google Scholar 

  16. Fleming, W.H., Viot, M.: Some measure-valued Markov processes in population genetics theory. Indiana Univ. Math. J. 28(5), 817–843 (1979)

    Article  MathSciNet  Google Scholar 

  17. Grigorescu, I., Kang, M.: Immortal particle for a catalytic branching process. Probab. Theory Relat. Fields 153(1–2), 333–361 (2012)

    Article  MathSciNet  Google Scholar 

  18. Groisman, P., Jonckheere, M.: Simulation of quasi-stationary distributions on countable spaces. Markov Process. Relat. Fields 19(3), 521–542 (2013)

    MathSciNet  MATH  Google Scholar 

  19. Groisman, P., Jonckheere, M., Martínez, J.: Another branching-selection particle system associated to the f-kpp equation. In preparation

    Google Scholar 

  20. Groisman, P., Jonckheere, M.: A note on front propagation and quasi-stationary distributions for one-dimensional Lévy processes. Preprint arXiv:1609.09338 (2013)

  21. Iglehart, D.L.: Random walks with negative drift conditioned to stay positive. J. Appl. Probab. 11, 742–751 (1974)

    Article  MathSciNet  Google Scholar 

  22. Kessler, D.A., Levine, H.: Fluctuation-induced diffusive instabilities. Nature 394, 556–558 (1998)

    Article  Google Scholar 

  23. Kolb, M., Wübker, A.: Geometric ergodicity of a Fleming–Viot interaction particle process with constant drift. Personal communication

    Google Scholar 

  24. Kolmogorov, A., Petrovsky, I., Piscounov, N.: Etude de l’equation de la diffusion avec croissance de la quantite de matiere et son application a un probleme biologique. Bull. Univ. Etat Moscou 1, 1–25 (1937)

    Google Scholar 

  25. Kyprianou, A.E.: A note on branching Lévy processes. Stochast. Process. Appl. 82(1), 1–14 (1999)

    Article  MathSciNet  Google Scholar 

  26. Maillard, P.: Branching brownian motion with selection. Preprint arXiv:1210.3500 (2012)

  27. Martinez, S., Picco, P., San Martín, J.: Domain of attraction of quasi-stationary distributions for the Brownian motion with drift. Adv. Appl. Probab. 30(2), 385–408 (1998)

    Article  MathSciNet  Google Scholar 

  28. Martínez, S., San Martín, J.: Quasi-stationary distributions for a Brownian motion with drift and associated limit laws. J. Appl. Probab. 31(4), 911–920 (1994)

    Article  MathSciNet  Google Scholar 

  29. McKean, H.P.: Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov. Commun. Pure Appl. Math. 28(3), 323–331 (1975)

    Article  MathSciNet  Google Scholar 

  30. McKean, H.P.: A correction to: application of Brownian motion to the equation of Kolmogorov-Petrovskiĭ-Piskonov. Comm. Pure Appl. Math. 28(3), 323–331 (1975)

    Article  MathSciNet  Google Scholar 

  31. McKean, H.P.: A correction to: application of Brownian motion to the equation of Kolmogorov-Petrovskiĭ-Piskonov. Comm. Pure Appl. Math. 29(5), 553–554 (1976)

    Article  MathSciNet  Google Scholar 

  32. Seneta, E., Vere-Jones, D.: On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J. Appl. Probab. 3, 403–434 (1966)

    Article  MathSciNet  Google Scholar 

  33. Villemonais, D.: Interacting particle systems and Yaglom limit approximation of diffusions with unbounded drift. Electron. J. Probab. 16(61), 1663–1692 (2011)

    Article  MathSciNet  Google Scholar 

  34. Villemonais, D.: Minimal quasi-stationary distribution approximation for a birth and death process. Electron. J. Probab. 20(30), 1–18 (2015)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

We would like to thank the projects UBACyT 2013–2016 20020120100151BA, PICT 2012-2744 “Stochastic Processes and Statistical Mechanics”, and MathAmSud 777/2011 “Stochastic Structure of Large Interactive Systems” for financial support.

Author information

Authors and Affiliations

  1. Departamento de Matemática, FCEN, Universidad de Buenos Aires, IMAS-CONICET, Buenos Aires, Argentina

    Pablo Groisman

  2. NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, Shanghai, China

    Pablo Groisman

  3. Instituto de Cálculo, FCEN, Universidad de Buenos Aires and IMAS-CONICET, Buenos Aires, Argentina

    Matthieu Jonckheere

Authors
  1. Pablo Groisman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Matthieu Jonckheere
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pablo Groisman .

Editor information

Editors and Affiliations

  1. NYU Shanghai, Shanghai, China

    Vladas Sidoravicius

Additional information

Dedicated to Chuck Newman on the occasion of his 70th birthday.

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Singapore Pte Ltd.

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Groisman, P., Jonckheere, M. (2019). Front Propagation and Quasi-Stationary Distributions: Two Faces of the Same Coin. In: Sidoravicius, V. (eds) Sojourns in Probability Theory and Statistical Physics - III. Springer Proceedings in Mathematics & Statistics, vol 300. Springer, Singapore. https://doi.org/10.1007/978-981-15-0302-3_9

Download citation

Publish with us

Policies and ethics

Search

Navigation