Speaker: Yan-Xia Ren
Title: Extremes of branching Levy processes
Abstract: I will talk about recent results on the extremes of supercritical branching L\'evy processes $\{\mathbb{X}_t, t \ge0\}$
whose spatial motions are L\'evy processes.
In supercritical case, if the L\'evy processe has regularly varying tails, when properly renormalized, $\mathbb{X}_t$ converges weakly. As a consequence, we obtain a limit theorem for the order statistics of $\mathbb{X}_t$. The result is drastically different from
the case of branching Brownian motions.
In critical case, we study asymptotic behaviors of the tails of extinction time and maximal displacement of a critical branching killed L\'{e}vy process $(\mathbb{X}_t^{(0,\infty)})_{t\ge 0}$ in $\R$, in which all particles (and their descendants) are killed upon exiting $(0, \infty)$. Let $\zeta^{(0,\infty)}$ and $M_t^{(0,\infty)}$ be the extinction time and maximal position of all the particles alive at time $t$ of this branching killed L\'{e}vy process and define $M^{(0,\infty)}: = \sup_{t\geq 0} M_t^{(0,\infty)}$. Under the assumption that the offspring distribution belongs to the domain of attraction of an $\alpha$-stable distribution, $\alpha\in (1, 2]$, and some moment conditions on the spatial motion, we give the decay rates of the survival probabilities
$$
\P_{y}(\zeta^{(0,\infty)}>t), \quad \P_{\sqrt{t}y}(\zeta^{(0,\infty)}>t)
$$
and the tail probabilities
$$
\P_{y}(M^{(0,\infty)}\geq x), \quad \P_{xy}(M^{(0,\infty)}\geq x).
$$
In subcritical case, we study asymptotic behaviors of a branching killed Brownian motion with drift $-\rho$. Let $\tilde{\zeta}^{-\rho}$ be the extinction time, $\tilde{M}_t^{-\rho}$ the maximal position of all the particles alive at time $t$ and $\tilde{M}^{-\rho}:=\max_{t\ge 0}\tilde{M}_t^{-\rho}$ the all time maximal position.
We establish the decay rates of $\P_x(\tilde{\zeta}^{-\rho}>t)$ and $\P_x(\tilde{M}^{-\rho}>y)$ as $t$ and $y$ tend to $\infty$ respectively. We also give a Yaglom-type limit theorem.