%0 Journal Article %1 MR942755 %A Lalley, S. %A Sellke, T. %D 1988 %J Ann. Probab. %K Fisher-KPP branching_Brownian_motion travelling_wave %N 3 %P 1051--1062 %T Traveling waves in inhomogeneous branching Brownian motions. I %U http://www.jstor.org/stable/2244109 %V 16 %X The process considered starts from an initial ancestor at the origin of the real line, which then moves according to a Brownian motion. This particle in due course splits into two particles which perform independent Brownian motions, each starting at the parent particle's final position, and so on. The inhomogeneity of the title is introduced by allowing a particle's instantaneous rate of splitting depend on its position, being given by $\beta(x)$ at $x$. Here it is assumed that particles far from the origin have little chance of splitting, in the sense that $ınt\beta(x)\,dx<ınfty$ and $\beta(x)\rightarrow0$ as $|x|\rightarrowınfty$. Let $N(t;J)$ be the number of particles in the bounded interval $J$ at time $t$. There is then a $łambda_0>0$, a finite measure $\nu$ and a random variable $Z>0$ such that $e^-łambda_0tN(t;J)Z\nu(J)$ S. Watanabe\en, in Markov processes and potential theory (Madison, WI, 1967), 205--232, Wiley, New York, 1967; MR0237007 (38 #5300). Let $R(t)$ be the position of the rightmost particle at time $t$; then K. B. EricksonZ. Wahrsch. Verw. Gebiete \bf66 (1984), no. 1, 129--140; MR0743089 (85m:60143) has shown that $R(t)/tłambda_0/2$ a.s. The main result here is that $P(R(t)-tłambda_0/2x)$ converges to $E \exp(-Ze^-(2łambda_0)x)$, where $\gamma$ is $(2łambda_0)^-1e^y2łambda_0 \beta(y)\nu(dy)$. The method of proof is by coupling the process to ``Poisson tidal wave'' processes, for which the result is fairly easy to establish. In a ``Poisson tidal wave'' a particle is born in $(x,x+dx)$ at time $(t,t+dt)$ with probability $Ce^t µ(dx)\,dt$ (where $łambda,C>0$ and $µ$ is a probability measure) and from time $t$ on it moves from $x$ according to an independent Brownian motion.

@article{MR942755, abstract = {The process considered starts from an initial ancestor at the origin of the real line, which then moves according to a Brownian motion. This particle in due course splits into two particles which perform independent Brownian motions, each starting at the parent particle's final position, and so on. The inhomogeneity of the title is introduced by allowing a particle's instantaneous rate of splitting depend on its position, being given by $\beta(x)$ at $x$. Here it is assumed that particles far from the origin have little chance of splitting, in the sense that $\int\beta(x)\,dx<\infty$ and $\beta(x)\rightarrow0$ as $|x|\rightarrow\infty$. Let $N(t;J)$ be the number of particles in the bounded interval $J$ at time $t$. There is then a $\lambda_0>0$, a finite measure $\nu$ and a random variable $Z>0$ such that $e^{-\lambda_0t}N(t;J)\rightarrow Z\nu(J)$ [\n S. Watanabe\en, in Markov processes and potential theory (Madison, WI, 1967), 205--232, Wiley, New York, 1967; MR0237007 (38 #5300)]. Let $R(t)$ be the position of the rightmost particle at time $t$; then \n K. B. Erickson\en [Z. Wahrsch. Verw. Gebiete {\bf66} (1984), no. 1, 129--140; MR0743089 (85m:60143)] has shown that $R(t)/t\rightarrow \sqrt{\lambda_0/2}$ a.s. The main result here is that $P(R(t)-t\sqrt{\lambda_0/2}\le x)$ converges to $E \exp(-Z\gamma e^{-(\sqrt{2\lambda_0})x})$, where $\gamma$ is $(2\lambda_0)^{-1}\int e^{y\sqrt{2\lambda_0}} \beta(y)\nu(dy)$. The method of proof is by coupling the process to ``Poisson tidal wave'' processes, for which the result is fairly easy to establish. In a ``Poisson tidal wave'' a particle is born in $(x,x+dx)$ at time $(t,t+dt)$ with probability $Ce^{\lambda t} µ(dx)\,dt$ (where $\lambda,C>0$ and $µ$ is a probability measure) and from time $t$ on it moves from $x$ according to an independent Brownian motion. }, added-at = {2009-10-15T23:11:34.000+0200}, author = {Lalley, S. and Sellke, T.}, biburl = {https://www.bibsonomy.org/bibtex/284e5a0114df8ee76f065de32cff46448/peter.ralph}, coden = {APBYAE}, description = {Traveling waves in inhomogeneous branching {B}rownian motions. {I}}, fjournal = {The Annals of Probability}, interhash = {ba8a134d714d49a05a915d003a567d3c}, intrahash = {84e5a0114df8ee76f065de32cff46448}, issn = {0091-1798}, journal = {Ann. Probab.}, keywords = {Fisher-KPP branching_Brownian_motion travelling_wave}, mrclass = {60J80 (60J65)}, mrnumber = {MR942755 (89g:60259)}, mrreviewer = {J. D. Biggins}, number = 3, pages = {1051--1062}, timestamp = {2013-09-12T22:23:01.000+0200}, title = {Traveling waves in inhomogeneous branching {B}rownian motions. {I}}, url = {http://www.jstor.org/stable/2244109}, volume = 16, year = 1988 }