# 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 & Heterogeneous Media, 2011, 6 (2) : 167-194. doi: 10.3934/nhm.2011.6.167
##### References:

show all references

##### References:
 [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

2018 Impact Factor: 0.871