On laws of large numbers in \(L^2\) for supercritical branching Markov processes beyond \(\lambda \)-positivity
The processes considered are supercritical Markov branching processes \((\xi_t)_{t\ge 0}\) with a general type space \((J,\mathcal{B}_J)\), a constant, i.e., type-independent branching rate, a constant local branching law with finite second moment and with possible absorbing states on \(\overline{J}\backslash J\). The aim is to provide conditions, covering as many different examples as possible, for \(L^2\)-convergence, as \(t\to\infty\), of \(\xi_t(B)/\mathbf{E}\,\xi_t(B')\) to \(\nu(B)D_\infty/\nu(B')\), where \(\xi_t(B)\) is the number of particles in \(B\) at time \(t\), \(\nu\) a measure on \((J,\mathcal{B}_J)\), \(D_\infty\) a nonnegative random variable and \(B\), \(B'\) are sets in some class \(\mathcal{C}\subseteq \mathcal{B}_J\), \(\nu(B')\ne 0\). Proceeding probabilistically and using spatial decomposition techniques, it is shown that whenever the distribution of the ``immortal particle'' process is regularly varying, as \(t\to\infty\), \(L^2\)-convergence holds if and only if a specific additive martingale associated with the branching process is bounded in \(L^2\). Given that boundedness, \(D_\infty\) is the \(L^2\)-limit of the martingale. An explicit formula for the asymptotic variance of the martingale is obtained, so that boundedness can be checked by direct computation. Conditions for \(\mathbf{P}(D_\infty> 0\mid\text{survival})=1\) are investigated and several examples discussed, among them classical \(\lambda\)-positive processes and, in particular, the non-\(\lambda\)-positive branching Brownian motion with drift and absorption at 0 introduced by \textit{H. Kesten} [Stochastic Processes Appl. 7, 9--47 (1978; Zbl 0383.60077)].
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