The Annals of Probability

Wavefront propagation for reaction-diffusion systems and backward SDEs

Frédéric Pradeilles

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Abstract

We first show a large deviation principle for degenerate diffusion-transmutation processes and study the Riemannian metric associated with the action functional under a Hörmander-type assumption. Then we study the behavior of the solution $u^{\varepsilon}$ of a system of strongly coupled scaled KPP equations. Using backward stochastic differential equations and the theory of Hamilton-Jacobi equations, we show that, when the parabolic operator satisfies a Hörmander-type hypothesis or when the nonlinearity depends on the gradient, the wavefront location is given by the same formula as that in Freidlin and Lee or Barles, Evans and Souganidis. We obtain the exact logarithmic rates of convergence to the unstable equilibrium state in the general case and to the stable equilibrium state when the equations are uniformly positively coupled.

Article information

Source
Ann. Probab., Volume 26, Number 4 (1998), 1575-1613.

Dates
First available in Project Euclid: 31 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022855874

Digital Object Identifier
doi:10.1214/aop/1022855874

Mathematical Reviews number (MathSciNet)
MR1675043

Zentralblatt MATH identifier
0933.35098

Subjects
Primary: 35K57: Reaction-diffusion equations 35K65: Degenerate parabolic equations 49L25: Viscosity solutions 60F10: Large deviations 60H30: Applications of stochastic analysis (to PDE, etc.)

Keywords
Large deviations degenerate diffusion-transmutation process sub-Riemannian metric backward stochastic differential equations systems of reaction-diffusion equations Hamilton-Jacobi equations viscosity solutions

Citation

Pradeilles, Frédéric. Wavefront propagation for reaction-diffusion systems and backward SDEs. Ann. Probab. 26 (1998), no. 4, 1575--1613. doi:10.1214/aop/1022855874. https://projecteuclid.org/euclid.aop/1022855874


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References

  • 1 ARONSON, D. G. and WEINBERGER, H. F. 1978. Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30 33 76.
  • 2 AZENCOTT, R. 1980. Ecole d'ete de Probabilites de St. Flour. Lecture Notes in Math. 774. ´ ´ ´ Springer, Berlin.
  • 3 BARLES, G. 1994. Solutions de viscosite des equations de Hamilton Jacobi. Math. Appl. 17. ´ ´
  • 4 BARLES, G. and SOUGANIDIS, P. E. 1994. A remark on the asymptotic behavior of the solution of the KPP equation. C. R. Acad. Sci. Paris Ser. I 319 679 684. ´
  • 5 BARLES, G., EVANS, L. C. and SOUGANIDIS, P. E. 1990. Wavefront propagation for reaction diffusion systems of PDE. Duke Math. J. 61 835 858.
  • 8 CHAMPNEYS, A., HARRIS, S., TOLAND, J., WARREN, J. and WILLIAMS, D. 1995. Algebra, analysis and probability for a coupled system of reaction diffusion equations. Philos. Trans. Roy. Soc. London 305 69 112.
  • 9 ELWORTHY, K. D., TRUMAN, A., ZHAO, H.and GAINES, J. G. 1994. Approximate traveling waves for generalized KPP equations and classical mechanics. Proc. Roy. Soc. London Ser. A 446 529 554.
  • 10 EVANS, L. C. and SOUGANIDIS, P. E. 1989. A P.D.E. approach to geometric optics for certain semilinear parabolic equations. Indiana Univ. Math. J. 38 141 172.
  • 11 EVANS, L. C. and SOUGANIDIS, P. E. 1989. A PDE approach to certain large deviation problems for systems of parabolic equations. Ann. Inst. H. Poincare Anal. Non ´ Lineaire 6 229 258.
  • 12 FREIDLIN, M. I. 1985. Limit theorems for large deviations and reaction diffusion equations. Ann. Probab. 13 639 675.
  • 13 FREIDLIN, M. I. 1990. Ecole d'ete de Probabilites de St. Flour. Lecture Notes in Math. 1527. ´ ´ ´ ´ Springer, Berlin.
  • 14 FREIDLIN, M. I. 1991. Coupled reaction diffusion equations. Ann. Probab. 19 29 57.
  • 15 FREIDLIN, M. I. and LEE, T. Y. 1992. Large deviation principle for the diffusion transmutation processes and Dirichlet problem for PDE systems with small parameter. Probab. Theory Related Fields. To appear.
  • 16 FREIDLIN, M. I. and LEE, T. Y. 1992. Wave front propagation and large deviations for diffusion transmutation process. Probab. Theory Related Fields. To appear.
  • 17 FREIDLIN, M. I. and WENTZEL, A. D. 1984. Random perturbations of dynamical systems. Grundlehren Math. Wiss. 260.
  • 18 KOLMOGOROV, A., PETROVSKII, I. and PISKUNOV, N. 1937. Etude de l'equation de la diffusion ´ avec croissance de la matiere et son application a un probleme biologique. Moscow Univ. Bull. Math. 1 1 25.
  • 19 LEANDRE, R. 1987. Minoration en temps petit de la densite d'une diffusion degeneree. ´ ´ ´ ´ ´ ´ J. Funct. Anal. 74 399 414.
  • 20 LEANDRE, R. 1987. Integration dans la fibre associee a une diffusion degeneree. Probab. ´ ´ ´ ´ ´ ´ ´ Theory Related Fields 76 341 358.
  • 21 MCKEAN, H. 1975. Application of Brownian motion to the equations of Kolmogorov Petrovskii Piskunov. Comm. Pure Appl. Math. 28 323 331.
  • 22 MCKEAN, H. 1975. A correction to: Application of Brownian motion to the equations of Kolmogorov Petrovskii Piskunov. Comm. Pure Appl. Math. 29 553 554.
  • 23 PARDOUX, E. and PENG, S. 1992. Backward stochastic differential equations and quasilinear parabolic differential equations. Stochastic Partial Differential Equations and Their Applications. Lecture Notes in Control and Inform. Sci. 176. Springer, Berlin.
  • 24 PARDOUX, E., PRADEILLES, F. and RAO,1997. Probabilistic interpretation for a system of nonlinear parabolic PDE's. Ann. Inst. H. Poincare Probab. Statist. 33 467 490. ´
  • 25 PRADEILLES, F. 1995. Une methode probabiliste pour l'etude des equations et systemes ´ ´ ´ d'equation de reaction diffusion. Ph.D. thesis. ´ ´
  • 26 PRIOURET, P. 1973. Ecole d'ete de Probabilites de St. Flour. Lecture Notes in Math. 390. ´ ´ ´ Springer, Berlin.
  • 27 PRIOURET, P. 1982. Remarques sur les petites perturbations de systemes dynamiques. Seminaire de Probabilites XVI. Lecture Notes in Math. 920. Springer, Berlin. ´ ´
  • 28 SMOLLER, J. 1983. Shock waves and reaction diffusion equations. Grundlehren Math. Wiss. 258.
  • 29 ZHAO, H. 1994. The travelling wave fronts of nonlinear reaction diffusion systems via Freidlin's stochastic approaches. Proc. Roy. Soc. Edinburgh Sect. A 124 273 299.