%0 Journal Article %1 MR1319963 %A Mueller, Carl %A Sowers, Richard B. %D 1995 %J J. Funct. Anal. %K Fisher-KPP SPDE travelling_wave %N 2 %P 439--498 %T Random travelling waves for the KPP equation with noise %U http://dx.doi.org/10.1006/jfan.1995.1038 %V 128 %X Consider the stochastic partial differential equation $u_t=u_xx+u-u^2+\epsilonu(1-u)W, u(0,x)=u_0(x), t>0, xınR$, where $W=W(t,x)$ is two-parameter white noise. Assume that $u_0$ is a continuous function taking values in $0,1$ such that for some constant $a>0$, we have (C1) $u_0(x)=1$ for $x<-a$, (C2) $u_0(x)=0$ for $x>a$. ``Let the wavefront $b(t)=\sup\xR\colon\ u(t,x)>0\$. We show that, for $\epsilon$ small enough and with probability 1, (1) $łim_t\toınftyb(t)/t$ exists and lies in $(0,ınfty)$ (this limit depends only on $\epsilon$), and (2) the law of $v(t,x)u(t,b(t)+x)$ tends toward a stationary limit as $t\toınfty$. We also analyze the length of the region $a(t),b(t)$, which is the smallest closed interval containing the points $x$ at which $0<u(t,x)<1$. We show that the length of this region tends toward a stationary distribution. Thus, the wavefront does not degenerate.''

@article{MR1319963, abstract = {Consider the stochastic partial differential equation $u_t=u_{xx}+u-u^2+\epsilon\sqrt{u(1-u)}\dot W, u(0,x)=u_0(x), t>0, x\inR$, where $\dot W=\dot W(t,x)$ is two-parameter white noise. Assume that $u_0$ is a continuous function taking values in $[0,1]$ such that for some constant $a>0$, we have (C1) $u_0(x)=1$ for $x<-a$, (C2) $u_0(x)=0$ for $x>a$. ``Let the wavefront $b(t)=\sup\{x\in{\bf R}\colon\ u(t,x)>0\}$. We show that, for $\epsilon$ small enough and with probability 1, (1) $\lim_{t\to\infty}b(t)/t$ exists and lies in $(0,\infty)$ (this limit depends only on $\epsilon$), and (2) the law of $v(t,x)\equiv u(t,b(t)+x)$ tends toward a stationary limit as $t\to\infty$. We also analyze the length of the region $[a(t),b(t)]$, which is the smallest closed interval containing the points $x$ at which $0<u(t,x)<1$. We show that the length of this region tends toward a stationary distribution. Thus, the wavefront does not degenerate.'' }, added-at = {2009-09-11T00:17:31.000+0200}, author = {Mueller, Carl and Sowers, Richard B.}, biburl = {https://www.bibsonomy.org/bibtex/2ddc7fd3ff2dccbdfc40e77f44a0b621b/peter.ralph}, coden = {JFUAAW}, description = {MR: Selected Matches for: Author=(mueller) AND Title=(KPP)}, fjournal = {Journal of Functional Analysis}, interhash = {1d0f08ead6fa48670c8aeee3d20d4661}, intrahash = {ddc7fd3ff2dccbdfc40e77f44a0b621b}, issn = {0022-1236}, journal = {J. Funct. Anal.}, keywords = {Fisher-KPP SPDE travelling_wave}, mrclass = {60H15 (35Q53 35R60)}, mrnumber = {MR1319963 (97a:60083)}, number = 2, pages = {439--498}, timestamp = {2013-09-12T22:23:01.000+0200}, title = {Random travelling waves for the {KPP} equation with noise}, url = {http://dx.doi.org/10.1006/jfan.1995.1038}, volume = 128, year = 1995 }