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March  2009, 11(2): 421-442. doi: 10.3934/dcdsb.2009.11.421

KPP fronts in a one-dimensional random drift

James Nolen 1, and  Jack Xin 2,

1. 

Department of Mathematics, Stanford University, Stanford, CA 94305, United States
2. 

Department of Mathematics/Mathematics, University of California at Irvine, Irvine, CA 92697, United States

Received  November 2007 Revised  April 2008 Published  December 2008

We establish the variational principle of Kolmogorov-Petrovsky-Piskunov (KPP) front speeds in a one dimensional random drift which is a mean zero stationary ergodic process with mixing property and local Lipschitz continuity. To prove the variational principle, we use the path integral representation of solutions, hitting time and large deviation estimates of the associated stochastic flows. The variational principle allows us to derive upper and lower bounds of the front speeds which decay according to a power law in the limit of large root mean square amplitude of the drift. This scaling law is different from that of the effective diffusion (homogenization) approximation which is valid for front speeds in incompressible periodic advection.
Keywords: fronts, random drift, KPP, large deviations.
Mathematics Subject Classification: Primary: 35K57; Secondary: 35K15.
Citation: James Nolen, Jack Xin. KPP fronts in a one-dimensional random drift. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 421-442. doi: 10.3934/dcdsb.2009.11.421
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