Product martingales and stopping lines for branching Brownian motion (Q1176366)

It is considered a supercritical branching Brownian motion, where the particles have the \(\exp(\alpha)\)-distributed living time and reproduce according to a Galton-Watson process with law having the mean value \(m\) and the generating function \(f\), and an increasing family \(\{\tau_ x, x\in R_ +\}\) of stopping lines tending to infinity. By using a solution of the Kolmogorov-Petrosvkij-Piskunov equation \[ {1 \over 2}\psi''- a\psi'+\alpha(f(\psi)-\psi)=0, \qquad \psi(-\infty)=1, \quad \psi(+\infty)=0, \quad a\geq \sqrt{2\alpha(m-1)}, \] the related family of product martingales \(\{M_{\tau^ x}, x\in R_ +\}\) is defined. Conditions are found, when \(\lim_{x\to\infty} M_{\tau^ x}=W_ \psi\) does not depend on the family \(\{\tau^ x, x\in R_ +\}\) and \[ E_ 0(W_ \psi)^{e^{-\ell z}}=\psi(z),\qquad z\geq 0, \qquad \ell=a-\sqrt{a^ 2-2\alpha(m-1)}. \] Some examples are given.