American Institute of Mathematical Sciences

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

A central limit theorem for pulled fronts in a random medium

James Nolen 1,

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.
Keywords: Front propagation, reaction-diffusion., random media.
Mathematics Subject Classification: Primary: 35R60; Secondary: 35K57, 60H9.
Citation: James Nolen. A central limit theorem for pulled fronts in a random medium. Networks & 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., (2010).   Google Scholar

[2]

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

[3]

H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations,, In:, 446 (2007), 101.   Google Scholar

[4]

P. Billingsley, "Convergence of Probability Measures,", John Wiley and Sons, (1968).   Google Scholar

[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).  doi: 10.1103/PhysRevE.73.056126.  Google Scholar

[6]

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

[7]

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

[8]

M. Freidlin, "Functional Integration and Partial Differential Equations,", Ann. Math. Stud. 109, (1985).   Google Scholar

[9]

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

[10]

P. Hall and C. C. Heyde, "Martingale Limit Theory and its Application,", Academic Press, (1980).   Google Scholar

[11]

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

[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.   Google Scholar

[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.  doi: 10.1081/PDE-200050077.  Google Scholar

[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.  doi: 10.1002/(SICI)1097-0312(199811/12)51:11/12<1337::AID-CPA4>3.0.CO;2-B.  Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[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.   Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

show all references

References:
[1]

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

[2]

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

[3]

H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations,, In:, 446 (2007), 101.   Google Scholar

[4]

P. Billingsley, "Convergence of Probability Measures,", John Wiley and Sons, (1968).   Google Scholar

[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).  doi: 10.1103/PhysRevE.73.056126.  Google Scholar

[6]

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

[7]

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

[8]

M. Freidlin, "Functional Integration and Partial Differential Equations,", Ann. Math. Stud. 109, (1985).   Google Scholar

[9]

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

[10]

P. Hall and C. C. Heyde, "Martingale Limit Theory and its Application,", Academic Press, (1980).   Google Scholar

[11]

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

[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.   Google Scholar

[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.  doi: 10.1081/PDE-200050077.  Google Scholar

[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.  doi: 10.1002/(SICI)1097-0312(199811/12)51:11/12<1337::AID-CPA4>3.0.CO;2-B.  Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[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.   Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[1]

Mikhail Kuzmin, Stefano Ruggerini. Front propagation in diffusion-aggregation models with bi-stable reaction. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 819-833. doi: 10.3934/dcdsb.2011.16.819
[2]

Luisa Malaguti, Cristina Marcelli, Serena Matucci. Continuous dependence in front propagation of convective reaction-diffusion equations. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1083-1098. doi: 10.3934/cpaa.2010.9.1083
[3]

Matthieu Alfaro, Thomas Giletti. Varying the direction of propagation in reaction-diffusion equations in periodic media. Networks & Heterogeneous Media, 2016, 11 (3) : 369-393. doi: 10.3934/nhm.2016001
[4]

Yana Nec, Vladimir A Volpert, Alexander A Nepomnyashchy. Front propagation problems with sub-diffusion. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 827-846. doi: 10.3934/dcds.2010.27.827
[5]

Tzong-Yow Lee and Fred Torcaso. Wave propagation in a lattice KPP equation in random media. Electronic Research Announcements, 1997, 3: 121-125.

[6]

Elena Trofimchuk, Manuel Pinto, Sergei Trofimchuk. On the minimal speed of front propagation in a model of the Belousov-Zhabotinsky reaction. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1769-1781. doi: 10.3934/dcdsb.2014.19.1769
[7]

Shangbing Ai, Wenzhang Huang, Zhi-An Wang. Reaction, diffusion and chemotaxis in wave propagation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 1-21. doi: 10.3934/dcdsb.2015.20.1
[8]

Yuri Latushkin, Roland Schnaubelt, Xinyao Yang. Stable foliations near a traveling front for reaction diffusion systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3145-3165. doi: 10.3934/dcdsb.2017168
[9]

Mohar Guha, Keith Promislow. Front propagation in a noisy, nonsmooth, excitable medium. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 617-638. doi: 10.3934/dcds.2009.23.617
[10]

D. G. Aronson, N. V. Mantzaris, Hans Othmer. Wave propagation and blocking in inhomogeneous media. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 843-876. doi: 10.3934/dcds.2005.13.843
[11]

Chang-Yeol Jung, Alex Mahalov. Wave propagation in random waveguides. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 147-159. doi: 10.3934/dcds.2010.28.147
[12]

Stephen Coombes, Helmut Schmidt, Carlo R. Laing, Nils Svanstedt, John A. Wyller. Waves in random neural media. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2951-2970. doi: 10.3934/dcds.2012.32.2951
[13]

Tomás Caraballo, José A. Langa, James C. Robinson. Stability and random attractors for a reaction-diffusion equation with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 875-892. doi: 10.3934/dcds.2000.6.875
[14]

Yuncheng You. Random attractors and robustness for stochastic reversible reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 301-333. doi: 10.3934/dcds.2014.34.301
[15]

Benoît Perthame, P. E. Souganidis. Front propagation for a jump process model arising in spacial ecology. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1235-1246. doi: 10.3934/dcds.2005.13.1235
[16]

Emeric Bouin. A Hamilton-Jacobi approach for front propagation in kinetic equations. Kinetic & Related Models, 2015, 8 (2) : 255-280. doi: 10.3934/krm.2015.8.255
[17]

Bo Su and Martin Burger. Global weak solutions of non-isothermal front propagation problem. Electronic Research Announcements, 2007, 13: 46-52.

[18]

Margarita Arias, Juan Campos, Cristina Marcelli. Fastness and continuous dependence in front propagation in Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 11-30. doi: 10.3934/dcdsb.2009.11.11
[19]

Jong-Shenq Guo, Chang-Hong Wu. Front propagation for a two-dimensional periodic monostable lattice dynamical system. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 197-223. doi: 10.3934/dcds.2010.26.197
[20]

C.B. Muratov. A global variational structure and propagation of disturbances in reaction-diffusion systems of gradient type. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 867-892. doi: 10.3934/dcdsb.2004.4.867

2019 Impact Factor: 1.053

Tools

Metrics

  • PDF downloads (23)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences