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  • Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Volume 129, Issue 3
  • January 1999, pp. 503-517

Travelling-waves for the FKPP equation via probabilistic arguments

  • Simon C. Harris (a1)
  • DOI:
  • Published online: 14 November 2011

We outline a completely probabilistic study of travelling-wave solutions of the FKPP reaction-diffusion equation that are monotone and connect 0 to 1. The necessary asymptotics of such travelling-waves are proved using martingale and Brownian motion techniques. Recalling the connection between the FKPP equation and branching Brownian motion through the work of McKean and Neveu, we show how the necessary asymptotics and results about branching Brownian motion combine to give the existence and uniqueness of travelling waves of all speeds greater than or equal to the critical speed.

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2K. Athreya and P. Ney . Branching processes (Springer, 1972).

3J. D. Biggins . Martingale convergence in the branching random walk. J. Appl. Prob. 14 (1977), 2537.

4J. D. Biggins . Growth rates in the branching random walk. Z. Wahr. 48 (1979), 1734.

5J. D. Biggins . Uniform convergence of martingales in the branching random walk. Ann. Prob. 20 (1992), 137151.

6M. D. Bramson . Maximal displacement of branching Brownian motion. Commun. Pure Appl. Math. 31 (1978), 531581.

8B. Chauvin . Product martingales and stopping lines for branching Brownian motion. Ann. Prob. 19 (1991), 11961205.

9K. D. Elworthy , A. Truman , H. Z. Zhao and J. G. Gaines . Approximate travelling waves for generalised KPP equations and classical mechanics. Proc. R. Soc. Lond. A 446 (1994), 529554.

10R. A. Fisher . The wave of advance of an advantageous gene. Ann. Eugenics 7 (1937), 353369.

11M. Friedlin . Functional integration and partial differential equations. Annals of Mathematics Series, vol. 109 (Princeton University Press, 1985).

14H. P. McKean . Application of Brownian motion to the equation of Kolmogorov–Petrovskii–Piskunov. Commun. Pure Appl. Math. 28 (1975), 323331.

15H. P. McKean . Correction to the above. Commun. Pure Appl. Math. 29 (1976), 553554.

19T. Shiga and S. Watanabe . Bessel diffusions as a one-parameter family of diffusion processes. Z. Wahr. verw. Geb. 27 (1973), 3746.

20K. Uchiyama . Brownian first exit and sojourn over a one-sided moving boundary and applications. Z. Wahr. 54 (1981), 75116.

21K. Uchiyama . Spatial growth of a branching process of particles living in ℝd. Ann. Prob. 10 (1982), 896918.

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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
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