It is shown that a certain functional of a branching process has representations in terms of both a maximisation problem and a minimisation problem. A consequence of these representations is that upper and lower bounds on the functional can be found easily, yielding a non-asymptotic Trotter product formula. As an application, the speed of the right-most particle of a branching Lévy process is calculated.
"Optimisation-based representations for branching processes." Electron. J. Probab. 25 1 - 15, 2020. https://doi.org/10.1214/20-EJP548