American Institute of Mathematical Sciences

Advanced
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 and Continuous Dynamical Systems - B, 2009, 11 (2) : 421-442. doi: 10.3934/dcdsb.2009.11.421
[1]

Thomas Bogenschütz, Achim Doebler. Large deviations in expanding random dynamical systems. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 805-812. doi: 10.3934/dcds.1999.5.805
[2]

Miguel Abadi, Sandro Vaienti. Large deviations for short recurrence. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 729-747. doi: 10.3934/dcds.2008.21.729
[3]

Dongmei Zheng, Ercai Chen, Jiahong Yang. On large deviations for amenable group actions. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7191-7206. doi: 10.3934/dcds.2016113
[4]

Salah-Eldin A. Mohammed, Tusheng Zhang. Large deviations for stochastic systems with memory. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 881-893. doi: 10.3934/dcdsb.2006.6.881
[5]

Yongqiang Suo, Chenggui Yuan. Large deviations for neutral stochastic functional differential equations. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2369-2384. doi: 10.3934/cpaa.2020103
[6]

Mathias Staudigl, Srinivas Arigapudi, William H. Sandholm. Large deviations and Stochastic stability in Population Games. Journal of Dynamics and Games, 2021  doi: 10.3934/jdg.2021021
[7]

Feng Cao, Wenxian Shen. Spreading speeds and transition fronts of lattice KPP equations in time heterogeneous media. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4697-4727. doi: 10.3934/dcds.2017202
[8]

Wenxian Shen, Zhongwei Shen. Transition fronts in nonlocal Fisher-KPP equations in time heterogeneous media. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1193-1213. doi: 10.3934/cpaa.2016.15.1193
[9]

Lina Wang, Xueli Bai, Yang Cao. Exponential stability of the traveling fronts for a viscous Fisher-KPP equation. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 801-815. doi: 10.3934/dcdsb.2014.19.801
[10]

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

[11]

Tomas Godoy, Jean-Pierre Gossez, Sofia Paczka. On the principal eigenvalues of some elliptic problems with large drift. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 225-237. doi: 10.3934/dcds.2013.33.225
[12]

Renaud Leplaideur, Benoît Saussol. Large deviations for return times in non-rectangle sets for axiom a diffeomorphisms. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 327-344. doi: 10.3934/dcds.2008.22.327
[13]

Artur O. Lopes, Rafael O. Ruggiero. Large deviations and Aubry-Mather measures supported in nonhyperbolic closed geodesics. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 1155-1174. doi: 10.3934/dcds.2011.29.1155
[14]

Vesselin Petkov, Luchezar Stoyanov. Spectral estimates for Ruelle operators with two parameters and sharp large deviations. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6391-6417. doi: 10.3934/dcds.2019277
[15]

Federico Bassetti, Lucia Ladelli. Large deviations for the solution of a Kac-type kinetic equation. Kinetic and Related Models, 2013, 6 (2) : 245-268. doi: 10.3934/krm.2013.6.245
[16]

Alexander Veretennikov. On large deviations in the averaging principle for SDE's with a "full dependence,'' revisited. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 523-549. doi: 10.3934/dcdsb.2013.18.523
[17]

Martino Bardi, Annalisa Cesaroni, Daria Ghilli. Large deviations for some fast stochastic volatility models by viscosity methods. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 3965-3988. doi: 10.3934/dcds.2015.35.3965
[18]

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
[19]

Patrick W. Dondl, Martin Jesenko. Threshold phenomenon for homogenized fronts in random elastic media. Discrete and Continuous Dynamical Systems - S, 2021, 14 (1) : 353-372. doi: 10.3934/dcdss.2020329
[20]

Jean-Michel Roquejoffre, Luca Rossi, Violaine Roussier-Michon. Sharp large time behaviour in $ N $-dimensional Fisher-KPP equations. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 7265-7290. doi: 10.3934/dcds.2019303

2021 Impact Factor: 1.497

Tools

Metrics

  • PDF downloads (74)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy