Limit theorems for the frontier of a one-dimensional branching diffusion
Let \(R(t)\) be the position of the rightmost particle at time \(t\) in a time homogeneous and space nonhomogeneous one-dimensional branching diffusion process. Let \(\gamma(\alpha,t)\) be the \(\alpha\)-th quantile of \(R(t)\) of the process starting with a single particle at position 0. The authors show that \(\gamma(\alpha,t)\) is also a limiting \(\beta\)-th quantile of \(R(t-s)\) as \(t\to\infty\) of the process starting with a single particle at arbitrary position \(x\), where \(\beta=g(\alpha,S,X)\). In the last case the authors show that, after appropriate rescaling of space, the distribution of \(R(t)-t\) converges weakly to a nontrivial limiting distribution \(w_ x\) if the underlying diffusion is recurrent.
scientific article; zbMATH DE number 97397