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A central limit theorem for pulled fronts in a random medium

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  • We consider solutions to a nonlinear reaction diffusion equation when the reaction term varies randomly with respect to the spatial coordinate. The nonlinearity is the KPP type nonlinearity. For a stationary and ergodic medium, and for certain initial condition, the solution develops a moving front that has a deterministic asymptotic speed in the large time limit. The main result of this article is a central limit theorem for the position of the front, in the supercritical regime, if the medium satisfies a mixing condition.
    Mathematics Subject Classification: Primary: 35R60; Secondary: 35K57, 60H99.


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  • [1]

    M. Bages, P. Martinez and J.-M. Roquejoffre, How traveling waves attract the solutions of KPP-type equations, preprint 2010.


    H. Berestycki and F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55 (2002), 949-1032.doi: 10.1002/cpa.3022.


    H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, In: "Perspectives in Nonlinear Partial Differential Equations," Contemp. Math. 446, Amer. Math. Soc., (2007), 101-123.


    P. Billingsley, "Convergence of Probability Measures," John Wiley and Sons, New York, 1968.


    E. Brunet, B. Derrida, A. H. Mueller and S. Munier, Phenomenological theory giving the full statistics of the position of fluctuating pulled fronts, Phys. Rev. E, 73 (2006), 05126.doi: 10.1103/PhysRevE.73.056126.


    S. Chatterjee, A new method of normal approximation, Ann. Probab., 36 (2008), 1584-1610.doi: 10.1214/07-AOP370.


    R. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 355-369.doi: 10.1111/j.1469-1809.1937.tb02153.x.


    M. Freidlin, "Functional Integration and Partial Differential Equations," Ann. Math. Stud. 109, Princeton University Press, Princeton, NJ, 1985.


    J. Gärtner and M. I. Freidlin, The propagation of concentration waves in periodic and random media, Dokl. Acad. Nauk SSSR, 249 (1979), 521-525.


    P. Hall and C. C. Heyde, "Martingale Limit Theory and its Application," Academic Press, New York, 1980.


    F. Hamel and L. Roques, Uniqueness and stability properties of monostable pulsating fronts, J. European Math. Soc., 13 (2011), 345-390.doi: 10.4171/JEMS/256.


    A. N. Kolmogorov, I. G. Petrovskii and N. S. Piskunov, Étude de l'équation de la chaleurde matiére et son application à un problème biologique, Bull. Moskov. Gos. Univ. Mat. Mekh., 1 (1937), 1-25.


    P.-L. Lions and P. E. Souganidis, Homogenization of viscous Hamilton-Jacobi equations in stationary ergodic media, Comm. Partial Diff. Eqn., 30 (2005), 335-375.doi: 10.1081/PDE-200050077.


    A. Majda and P. E. Souganidis, Flame fronts in a turbulent combustion model with fractal velocity fields, Comm. Pure Appl. Math., 51 (1998), 1337-1348.doi: 10.1002/(SICI)1097-0312(199811/12)51:11/12<1337::AID-CPA4>3.0.CO;2-B.


    P. Martinez and J.-M. Roquejoffre, Convergence to critical waves in KPP-type equations, Preprint 2010.


    A. Mellet, J. Nolen, J.-M. Roquejoffre and L. Ryzhik, Stability of generalized transition fronts, Communications in PDE, 34 (2009), 521-552.doi: 10.1080/03605300902768677.


    C. Mueller and R. Sowers, Random travelling waves for the KPP equation with noise, J. Funct. Anal., 128 (1995), 439-498.doi: 10.1006/jfan.1995.1038.


    J. Nolen, An invariance principle for random traveling waves in one dimension, SIAM J. Math. Anal., 43 (2011), 153-188.doi: 10.1137/090746513.


    J. Nolen and L. Ryzhik, Traveling waves in a one-dimensional heterogeneous medium, AIHP - Analyse Non Linéaire, 26 (2009), 1021-1047.


    J. Nolen and J. Xin, Asymptotic spreading of KPP reactive fronts in incompressible space-time random flows, AIHP - Analyse Non Linéaire, 26 (2008), 815-839.


    J. Nolen and J. Xin, KPP fronts in 1D random drift, Discrete and Continuous Dynamical Systems B, 11 (2009), 421-442.doi: 10.3934/dcdsb.2009.11.421.


    A. Rocco, U. Ebert and W. van Saarloos, Subdiffusive fluctuations of "pulled" fronts with multiplicative noise, Phys. Rev. E, 62 (2000), R13-R16.doi: 10.1103/PhysRevE.62.R13.


    W. Shen, Traveling waves in diffusive random media, J. Dynamics and Diff. Eqns., 16 (2004), 1011-1060.doi: 10.1007/s10884-004-7832-x.


    R. Tribe, A travelling wave solution to the Kolmogorov equation with noise, Stochastics Stochastics Rep., 56 (1996), 317-340.


    W. van Saarloos, Front propagation into unstable states, Physics Reports, 386 (2003), 29-222.doi: 10.1016/j.physrep.2003.08.001.


    J. Xin, "An Introduction to Fronts in Random Media," Springer, New York, 2009.doi: 10.1007/978-0-387-87683-2.

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