Interacting particles, the stochastic Fisher-Kolmogorov-Petrovsky-Piscounov equation, and duality. (English) Zbl 1025.60027

Summary: The stochastic Fisher-Kolmogorov-Petrovsky-Piskunov equation is \[ \partial_tU(x,t)=D\partial_{xx}U+\gamma U(1-U)+\varepsilon\sqrt{U(1-U)}\eta(x,t) \] for \(0\leqslant U\leqslant 1\) where \(\eta(x,t)\) is a Gaussian white noise process in space and time. Here \(D\), \(\gamma\) and \(\varepsilon\) are parameters and the equation is interpreted as the continuum limit of a spatially discretized set of Itô equations. Solutions of this stochastic partial differential equation have an exact connection to the \(A\rightleftharpoons A+A\) reaction-diffusion system at appropriate values of the rate coefficients and particles’ diffusion constant. This relationship is called “duality” by the probabilists; it is not via some hydrodynamic description of the interacting particle system. In this paper we present a complete derivation of the duality relationship and use it to deduce some properties of solutions to the stochastic Fisher-Kolmogorov-Petrovsky-Piskunov equation.


60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35K57 Reaction-diffusion equations
Full Text: DOI


[1] Fisher, R. A., The wave of advance of advantageous genes, Ann. Eugenics, 7, 353 (1937) · JFM 63.1111.04
[2] Kolmogorov, A.; Petrovsky, I.; Piscounov, N., Etude de l’equation de la diffusion avec croissance de la quantité de matière et son application a un probleme biologique, Moscow Univ. Bull. Math., 1, 1 (1937) · Zbl 0018.32106
[3] Murray, J. A., Mathematical Biology (1998), Springer: Springer Berlin
[4] van Kampen, N. G., Stochastic Processes in Physics and Chemistry (1981), North-Holland: North-Holland Amsterdam · Zbl 0511.60038
[5] Mueller, C.; Tribe, R., Stochastic PDEs arising from the long range contact and long range voter processes, Probab. Theory & Related Fields, 102, 519 (1995) · Zbl 0827.60050
[6] Liggett, T. M., Interacting Particle Systems (1985), Springer: Springer Berlin · Zbl 0832.60094
[7] Shiga, T.; Uchiyama, K., Stationary states and the stability of the stepping stone model involving mutation and selection, Probab. Theory & Related Fields, 73, 87 (1986) · Zbl 0642.92008
[8] Bramson, M., Convergence of solutions of the Kolmogorov equations to traveling waves, Mem. Am. Math. Soc., 44 (1983)
[9] Ebert, U.; van Saarloos, W., Front propagation into unstable statesuniversal algebraic convergence towards uniformly translating pulled fronts, Physica D, 146, 1 (2000) · Zbl 0984.35030
[10] Brunet, E.; Derrida, B., Shift in the velocity of a front due to a cutoff, Phys. Rev. E, 56, 2597 (1997)
[11] Kessler, D. A.; Ner, Z.; Sander, L. M., Front propagationprecursors, cutoffs and structural stability, Phys. Rev. E, 58, 107 (1998)
[12] Pechenik, L.; Levine, H., Interfacial velocity corrections due to multiplicative noise, Phys. Rev. E, 59, 3893 (1999)
[13] Brunet, E.; Derrida, B., Effect of microscopic noise on front propagation, J. Statist. Phys., 103, 269 (2001) · Zbl 1018.82020
[14] Xin, J., Front propagation in heterogeneous media, SIAM Rev., 42, 161 (2000) · Zbl 0951.35060
[15] Mueller, C.; Sowers, R. B., Random travelling waves for the KPP equation with noise, J. Funct. Anal., 128, 439 (1995) · Zbl 0820.60039
[16] Burschka, M. A.; Doering, C. R.; Ben Avraham, D., Statics and dynamics of a diffusion-limited reactionanomalous kinetics, nonequilibrium self-ordering, and a dynamic transition, J. Statist. Phys., 60, 695 (1990) · Zbl 1086.82556
[17] Doering, C. R.; Burschka, M. A.; Horsthemke, W., Fluctuations and correlations in a diffusion-reaction systemsexact hydrodynamics, J. Statist. Phys., 65, 953 (1991) · Zbl 0946.82516
[18] Ben Avraham, D., Fisher waves in the diffusion-limited coalescence process \(A+A\) ⇌ \(A\), Phys. Lett. A, 247, 53 (1998)
[19] Riordan, J.; Doering, C.; Ben-Avraham, D., Fluctuations and stability of Fisher waves, Phys. Rev. Lett., 75, 565 (1995)
[20] Tripathy, G.; van Saarloos, W., Fluctuation and relaxation properties of pulled frontsa scenario for nonstandard Kardar-Parisi-Zhang scaling, Phys. Rev. Lett., 85, 3556 (2000)
[21] Tripathy, G.; Rocco, A.; Casademunt, J.; van Saarloos, W., The universality class of fluctuating pulled fronts, Phys. Rev. Lett., 86, 5215 (2001)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.