Stochastic Traveling Wave Solution to a Stochastic KPP Equation

Arnold, L.: Random Dynamical System. Springer, New York (1998) Ablowitz, M.J., Zeppetella, A.: Explicit solutions of Fisher’s equation for a special wave speed. Bull. Math. Biol. 41, 835–840 (1979) Bramson, M.: Convergence of solutions of the Kolmogorov equation to travelling waves. Mem. Am. Math. Soc. 44, 285 (1983) Bressloff, P.C., Webber, M.A.: Front propagation in stochastic neural fields. SIAM J. Appl. Dyn. Syst. 11, 708–740 (2012) Bates, P.W., Lu, K., Wang, B.: Random attractors for stochastic reaction-diffusion equations on unbounded domains. J. Differ. Equ. 246, 845–869 (2009) Conlon, J.G., Doering, C.R.: On travelling waves for the stochastic Fisher-Kolmogorov-Petrovsky-Piscunov equation. J. Stat. Phys. 144, 137–177 (2009) Davies, L.M., Truman, A., Zhao, H.: Stochastic generalized KPP equations. Proc. R. Soc. Edinb. A 126, 957–983 (1996) Elworthy, K.D., Zhao, H.: The propagation of travelling waves for stochastic generalized KPP equations. Math. Comput. Model. 20, 131–166 (1994) Ethier, S., Kurtz, T.: Markov process: characterization and covergence. Wiley, New York (2005) Fisher, R.A.: The wave of advance of advantageous gene. Ann. Eugen. 7, 355–369 (1937) Fife, P.C., McLeod, J.B.: A phase plane discussion of convergence of traveling fronts for nonlinear diffusion. Arch. Rational Mech. Anal. 75, 281–314 (1980) Han, X., Shen, W., Zhou, S.: Random attractors for stochastic lattice dynamical systems in weighted spaces. J. Differ. Equ. 250, 1235–1266 (2011) Kolmogorov, A., Petrovsky, I., Piscunov, N.: Study of adiffusion equation that is related to the growth of a quality of matter, and its application to a biological problem. Byul. Mosk. Gos. Univ. Ser. A Mat. Mekh. 1, 1–26 (1937) Keiding, N.: Extinction and exponential growth in random environments. Theor. Popul. Biol. 8, 49–63 (1975) Konno, N., Shiga, T.: Stochastic partial differential equations for some measure-valued diffusions. Probab. Theory Relat. Fields 79, 201–225 (1988) Kiester, A.R., Barakat, R.: Exact solutions to certain stochastic differential equation models of population growth. Theor. Popul. Biol. 6, 199–216 (1974) Lau, K.S.: On the nonlinear diffusion equation of Kolmogorov, Petrovsky and Piscounov. J. Differ. Equ. 59, 44–70 (1985) Li, D., He, D., Xu, D.: Mean square exponential stability of impulsive stochastic reaction-diffusion Cohen-Grossberg neural networks with delays. Math. Comp. Simu. 82, 1531–1543 (2012) Lu, K., Schmalfuß, B.: Invariant manifolds for stochastic wave equations. J. Differ. Equ. 236, 460–492 (2007) Lindvall, T.: On Strassen’s theorem on stochastic domination. Electron. Comm. Probab. 4, 51–59 (1999) Mueller, C., Sowers, R.B.: Random travelling waves for the KPP equation with noise. J. Fun. Anal. 128, 439–498 (1995) Mueller, C., Mytnik, L., Quastel, J.: Small noise asymptotics of traveling waves. Markov Proc. Relat. Fields 14, 333–342 (2008) Mueller, C., Perkins, E.A.: The compact support property for solutions to the heat equation with noise. Probab. Theory Relat. Fields 93, 325–358 (1992) May, R.M.: Stability in randomly fluctuating versus deterministic environments. Am. Nat. 10, 621–650 (1973) May, R.M.: Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton (1974) McKean, H.: Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov. Comm. Pure Appl. Math. 28, 323–331 (1975) Mohammed, S.E.A., Scheutzow, K.R.: The stable manifold theorem for stochastic differential equations. Ann. Probab. 27, 615–652 (1999) Øksendala, B., Våge, G., Zhao, H.: Asymptotic properties of the solutions to stochastic KPP equations. Proc. R. Soc. Edinb. A 130, 1363–1381 (2000) Øksendala, B., Våge, G., Zhao, H.: Two properties of stochastic KPP equations: ergodicity and pathwise property. Nonlinearity 14, 639–662 (2001) Raugel, G., Kirchgässner, K.: Stability of fronts for a KPP-system. II. The critical case. J. Differ. Equ. 146, 399–456 (1998) Shen, W.: Traveling waves in diffusive random media. J. Dyn. Differ. Equ. 16, 1011–1059 (2004) Sharpe, M.: General Theorey of Markov Processes. Academic Press, New York (1988) Strassen, V.: The existence of probability measures with given marginals. Ann. Math. Statist. 36, 423–439 (1965) Tribe, R.: A travelling wave solution to the Kolmogorov equation with noise. Stoch. Stoch. Rep. 56, 317–340 (1995) Tribe, R., Woodward, N.: Stochastic order methods applied to stochastic travelling waves. Electron. J. Probab. 16, 436–469 (2011) Woodward, N.: University of Warwick Ph.D. Thesis, Stochastic Traveling Waves (2010) Yanagida, E.: Irregular behavior of solutions for Fisher’s equation. J. Dyn. Differ. Equ. 19, 895–914 (2007) Zhou, S., Wei, L.: A random attractor for a stochastic second order lattice system with random coupled coefficients. J. Math. Anal. Appl. 395, 42–55 (2012)