A large deviation theorem for a supercritical super-Brownian motion with absorption

We consider a one-dimensional superprocess with a supercritical local branching mechanism $\psi$ , where particles move as a Brownian motion with drift $-\rho$ and are killed when they reach the origin. It is known that the process survives with positive probability if and only if $\rho<\sqrt{2\alpha}$ , where $\alpha=-\psi'(0)$ . When $\rho<\sqrt{2 \alpha}$ , Kyprianou et al. [18] proved that $\lim_{t\to \infty}R_t/t =\sqrt{2\alpha}-\rho$ almost surely on the survival set, where $R_t$ is the rightmost position of the support at time t. Motivated by this work, we investigate its large deviation, in other words, the convergence rate of $\mathbb{P}_{\delta_x} (R_t >\gamma t+\theta)$ as $t \to \infty$ , where $\gamma >\sqrt{2 \alpha} -\rho$ , $\theta \ge 0$ . As a by-product, a related Yaglom-type conditional limit theorem is obtained. Analogous results for branching Brownian motion can be found in Harris et al. [13].