KPP fronts in a one-dimensional random drift

Pages: 421 - 442, Volume 11, Issue 2, March 2009

doi:10.3934/dcdsb.2009.11.421       Abstract        Full Text (269.8K)       Related Articles

James Nolen - Department of Mathematics, Stanford University, Stanford, CA 94305, United States (email)
Jack Xin - Department of Mathematics, University of California at Irvine, Irvine, CA 92697, United States (email)

Abstract: 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, large deviations, KPP
Mathematics Subject Classification:  Primary: 35K57; Secondary: 35K15

Received: November 2007;      Revised: April 2008;      Published: December 2008.