Article,
Travelling waves in inhomogeneous branching Brownian motions. II
Ann. Probab., 17 (1): 116--127 (1989)
Abstract
Initially, a single particle is at the origin, and it moves according to a Brownian motion until it splits into two. The children behave independently in a similar way, and so on. The inhomogeneity of the title occurs in the splitting rate for a particle, which is $\beta(x)$ if the particle is at $x$. In Part I the authors, same journal 16 (1988), no. 3, 1051--1062; MR0942755 (89g:60259) the case where most splitting occurs near the origin (e.g., $\beta$ continuous with compact support) was considered. Here the same type of inhomogeneity is superimposed on a basic splitting rate, so $\beta(x)=1+\beta_0(x)$ with $\beta_0$ nonnegative and continuous with compact support. The main result concerns the limiting distribution, when suitably centred, of $M(t)$, the position of the rightmost particle at time $t$ in such a process. It turns out that there are three possible regimes depending on the largest eigenvalue $łambda$ of the differential operator taking $g(x)$ to $12 g''(x)+\beta(x)g(x)$. If $łambda<2$, the position of $M(t)$ is similar to the homogeneous case, studied by M. D. BramsonMem. Amer. Math. Soc. 44 (1983), no. 285; MR0705746 (84m:60098). If $łambda=2$, the centring has to increase slightly, by $(1/2)t$, whilst if $łambda>2$ the rightmost particle is much further out than in the homogeneous case. The main idea used in the study is to treat particles as of two types, depending on whether they result from the base rate splitting or the inhomogeneous component. Then at any time the process can be viewed as the superposition of a random number of homogeneous processes, and the results derive from how quickly that random number grows with time.
