An exponential estimate for the extinction time of the branching random walk on a cube

We prove the exponential estimate \begin{equation*} P \{ s < τ< \infty \} \leq C e^{-q s}, \quad s \geq 0, \end{equation*} where $C, q >0$ are constants and $τ$ is the extinction time of the supercritical branching random walk (BRW) on a cube. We cover both the discrete-space and continuous-space BRWs.

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