Product Martingales and Stopping Lines for Branching Brownian Motion

For a branching Brownian motion, a probability space of trees is defined. By analogy with stopping times on R, stopping lines are defined to get a general branching property. We exhibit an intrinsic class of martingales which are products indexed by the elements of a stopping line. We prove that all these martingales have the same limit which we identify. Two particular cases arise: the line of particles living at time t and the first crossings of a straight line whose equation is $y = at - x$ in the plane $(y,t)$.