# American Institute of Mathematical Sciences

June  2011, 6(2): 167-194. doi: 10.3934/nhm.2011.6.167

## A central limit theorem for pulled fronts in a random medium

 1 Department of Mathematics, Duke University, Box 90320, Durham, NC, 27708-0320, United States

Received  August 2010 Revised  February 2011 Published  May 2011

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.
Citation: James Nolen. A central limit theorem for pulled fronts in a random medium. Networks and Heterogeneous Media, 2011, 6 (2) : 167-194. doi: 10.3934/nhm.2011.6.167
##### References:
 [1] M. Bages, P. Martinez and J.-M. Roquejoffre, How traveling waves attract the solutions of KPP-type equations, preprint 2010. [2] H. Berestycki and F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55 (2002), 949-1032. doi: 10.1002/cpa.3022. [3] 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. [4] P. Billingsley, "Convergence of Probability Measures," John Wiley and Sons, New York, 1968. [5] 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. [6] S. Chatterjee, A new method of normal approximation, Ann. Probab., 36 (2008), 1584-1610. doi: 10.1214/07-AOP370. [7] R. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x. [8] M. Freidlin, "Functional Integration and Partial Differential Equations," Ann. Math. Stud. 109, Princeton University Press, Princeton, NJ, 1985. [9] J. Gärtner and M. I. Freidlin, The propagation of concentration waves in periodic and random media, Dokl. Acad. Nauk SSSR, 249 (1979), 521-525. [10] P. Hall and C. C. Heyde, "Martingale Limit Theory and its Application," Academic Press, New York, 1980. [11] 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. [12] 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. [13] 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. [14] 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. [15] P. Martinez and J.-M. Roquejoffre, Convergence to critical waves in KPP-type equations, Preprint 2010. [16] 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. [17] 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. [18] J. Nolen, An invariance principle for random traveling waves in one dimension, SIAM J. Math. Anal., 43 (2011), 153-188. doi: 10.1137/090746513. [19] J. Nolen and L. Ryzhik, Traveling waves in a one-dimensional heterogeneous medium, AIHP - Analyse Non Linéaire, 26 (2009), 1021-1047. [20] 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. [21] 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. [22] 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. [23] W. Shen, Traveling waves in diffusive random media, J. Dynamics and Diff. Eqns., 16 (2004), 1011-1060. doi: 10.1007/s10884-004-7832-x. [24] R. Tribe, A travelling wave solution to the Kolmogorov equation with noise, Stochastics Stochastics Rep., 56 (1996), 317-340. [25] W. van Saarloos, Front propagation into unstable states, Physics Reports, 386 (2003), 29-222. doi: 10.1016/j.physrep.2003.08.001. [26] 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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##### References:
 [1] M. Bages, P. Martinez and J.-M. Roquejoffre, How traveling waves attract the solutions of KPP-type equations, preprint 2010. [2] H. Berestycki and F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55 (2002), 949-1032. doi: 10.1002/cpa.3022. [3] 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. [4] P. Billingsley, "Convergence of Probability Measures," John Wiley and Sons, New York, 1968. [5] 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. [6] S. Chatterjee, A new method of normal approximation, Ann. Probab., 36 (2008), 1584-1610. doi: 10.1214/07-AOP370. [7] R. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x. [8] M. Freidlin, "Functional Integration and Partial Differential Equations," Ann. Math. Stud. 109, Princeton University Press, Princeton, NJ, 1985. [9] J. Gärtner and M. I. Freidlin, The propagation of concentration waves in periodic and random media, Dokl. Acad. Nauk SSSR, 249 (1979), 521-525. [10] P. Hall and C. C. Heyde, "Martingale Limit Theory and its Application," Academic Press, New York, 1980. [11] 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. [12] 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. [13] 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. [14] 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. [15] P. Martinez and J.-M. Roquejoffre, Convergence to critical waves in KPP-type equations, Preprint 2010. [16] 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. [17] 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. [18] J. Nolen, An invariance principle for random traveling waves in one dimension, SIAM J. Math. Anal., 43 (2011), 153-188. doi: 10.1137/090746513. [19] J. Nolen and L. Ryzhik, Traveling waves in a one-dimensional heterogeneous medium, AIHP - Analyse Non Linéaire, 26 (2009), 1021-1047. [20] 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. [21] 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. [22] 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. [23] W. Shen, Traveling waves in diffusive random media, J. Dynamics and Diff. Eqns., 16 (2004), 1011-1060. doi: 10.1007/s10884-004-7832-x. [24] R. Tribe, A travelling wave solution to the Kolmogorov equation with noise, Stochastics Stochastics Rep., 56 (1996), 317-340. [25] W. van Saarloos, Front propagation into unstable states, Physics Reports, 386 (2003), 29-222. doi: 10.1016/j.physrep.2003.08.001. [26] 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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