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Issue ESAIM: PS
Volume 1, 1997
Page(s) 225 - 258
DOI 10.1051/ps:1997109

ESAIM: P&S, 1997, Vol. 1, pp. 225-258
DOI: 10.1051/ps:1997109

Laplace asymptotics for generalized K.P.P. equation

Jean-Philippe Rouquès

rouques@math.uvsq.fr


Abstract
Consider a one dimensional nonlinear reaction-diffusion equation (KPP equation) with non-homogeneous second order term, discontinuous initial condition and small parameter. For points ahead of the Freidlin-KPP front, the solution tends to 0 and we obtain sharp asymptotics (i.e. non logarithmic). Our study follows the work of Ben Arous and Rouault who solved this problem in the homogeneous case. Our proof is probabilistic, and is based on the Feynman-Kac formula and the large deviation principle satisfied by the related diffusions. We use the Laplace method on Wiener space. The main difficulties come from the nonlinearity and the possibility for the endpoint of the optimal path to lie on the boundary of the support of the initial condition.


Résumé
On considère une équation de réaction-diffusion non linéaire en dimension 1 (équation KPP) avec petit paramètre, terme du second ordre non-homogène et condition initiale discontinue. En des points situés en avant du front de Freidlin-KPP, la solution tend vers 0, et nous en obtenons un équivalent précis (par opposition à logarithmique). Notre travail fait suite à celui de Ben Arous et Rouault qui ont résolu ce problème dans le cas homogène. La preuve est probabiliste, et s'appuie sur la formule de Feynman-Kac et le principe de grandes déviations vérifié par les diffusions correspondantes. Nous utilisons la méthode de Laplace sur l'espace de Wiener. Les difficultés principales viennent de la non-linéarité et de la possibilité pour le point terminal du chemin optimal de se trouver sur la frontière du support de la condition initiale.


Key words: Generalized KPP equation / Feynman-Kac formula / diffusion / large deviations / Laplace method / stochastic Taylor expansion / Skorokhod integral.


© EDP Sciences, SMAI 1997


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