The Martin Entrance Boundary of the Galton-Watson Process

The paper provides a complete description of the Martin entrance boundary and its minimal elements for a Galton-Watson process (Zn)n≥0. Since this is easily done and known for critical processes, we deal with the noncritical case which in turn can be reduced to the subcritical one. The Martin entrance boundary consists of all quasi-invariant Radon measures. The minimal Martin entrance boundary is isomorphic to [0, 1) as a torus. Every element of the minimal Martin entrance boundary is uniquely identified through its generating function. These minimal quasi-invariant measures are the extremals in the simplex of quasiinvariant Radon measures. We provide explicitly the Martin topology in the set of potentials. All this is done via the Martin kernel approach and under no additional assumption on (Zn)n≥0. In particular, we do not require the (L logL)-condition EZ1 logZ1 <∞.