×

zbMATH — the first resource for mathematics

Right marker speeds of solutions to the KPP equation with noise. (English) Zbl 1432.60063
Summary: We consider the one-dimensional KPP-equation driven by space-time white noise. We show that for all parameters above the critical value for survival, there exist stochastic wavelike solutions which travel with a deterministic positive linear speed. We further give a sufficient condition on the initial condition of a solution to attain this speed. Our approach is in the spirit of corresponding results for the nearest-neighbor contact process respectively oriented percolation. Here, the main difficulty arises from the moderate size of the parameter and the long range interaction. Stopping times and averaging techniques are used to overcome this difficulty.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] Aronson, D. G. and Weinberger, H. F. (1975). Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In Partial Differential Equations and Related Topics (Program, Tulane Univ., New Orleans, LA, 1974). Lecture Notes in Math. 446 5-49. Springer, Berlin. · Zbl 0325.35050
[2] Bessonov, M. and Durrett, R. (2017). Phase transitions for a planar quadratic contact process. Adv. in Appl. Math. 87 82-107. · Zbl 1370.60187
[3] Durrett, R. (1980). On the growth of one-dimensional contact processes. Ann. Probab. 8 890-907. · Zbl 0457.60082
[4] Durrett, R. (1984). Oriented percolation in two dimensions. Ann. Probab. 12 999-1040. · Zbl 0567.60095
[5] Durrett, R. (1995). Ten lectures on particle systems. In Lectures on Probability Theory (Saint-Flour, 1993). Lecture Notes in Math. 1608 97-201. Springer, Berlin. · Zbl 0840.60088
[6] Ethier, S. N. and Kurtz, T. G. (2005). Markov Processes: Characterization and Convergence. Wiley, Hoboken, NJ. · Zbl 1089.60005
[7] Griffeath, D. (1981). The basic contact processes. Stochastic Process. Appl. 11 151-185. · Zbl 0463.60085
[8] Grimmett, G. R. and Stirzaker, D. R. (2001). Probability and Random Processes, 3rd ed. Oxford Univ. Press, New York. · Zbl 1015.60002
[9] Horridge, P. and Tribe, R. (2004). On stationary distributions for the KPP equation with branching noise. Ann. Inst. Henri Poincaré Probab. Stat. 40 759-770. · Zbl 1058.60049
[10] Iscoe, I. (1986). A weighted occupation time for a class of measure-valued branching processes. Probab. Theory Related Fields 71 85-116. · Zbl 0555.60034
[11] Iscoe, I. (1988). On the supports of measure-valued critical branching Brownian motion. Ann. Probab. 16 200-221. · Zbl 0635.60094
[12] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin. · Zbl 1018.60002
[13] Klenke, A. (2014). Probability Theory, 2nd ed. Universitext. Springer, London. · Zbl 1295.60001
[14] Kliem, S. (2017). Travelling wave solutions to the KPP equation with branching noise arising from initial conditions with compact support. Stochastic Process. Appl. 127 385-418. · Zbl 1354.60071
[15] Liggett, T. M. (1999). Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 324. Springer, Berlin. · Zbl 0949.60006
[16] Liggett, T. M. (2005). Interacting Particle Systems. Classics in Mathematics. Springer, Berlin. · Zbl 1103.82016
[17] Mueller, C. and Tribe, R. (1994). A phase transition for a stochastic PDE related to the contact process. Probab. Theory Related Fields 100 131-156. · Zbl 0809.60072
[18] Mueller, C. and Tribe, R. (1995). Stochastic p.d.e.’s arising from the long range contact and long range voter processes. Probab. Theory Related Fields 102 519-545. · Zbl 0827.60050
[19] Perkins, E. (2002). Dawson-Watanabe superprocesses and measure-valued diffusions. In Lectures on Probability Theory and Statistics (Saint-Flour, 1999). Lecture Notes in Math. 1781 125-324. Springer, Berlin. · Zbl 1020.60075
[20] Shiga, T. (1994). Two contrasting properties of solutions for one-dimensional stochastic partial differential equations. Canad. J. Math. 46 415-437. · Zbl 0801.60050
[21] Tribe, R. (1996). A travelling wave solution to the Kolmogorov equation with noise. Stoch. Stoch. Rep. 56 317-340. · Zbl 1002.60555
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.