On the boundedness of inhomogeneous branching diffusion on a straight line (Q1385949)

A branching diffusion process on the real line with state-dependent drift and diffusion coefficients is considered. A necessary and sufficient condition is stated for the almost sure finiteness of the all time maximum position of any of the particles, in terms of the existence of an appropriate solution of a second order ordinary differential equation. This extends work of the authors and \textit{M. Ya. Kelbert}, most recently in [Stochastic Processes Appl. 53, No. 1, 65-96 (1994; Zbl 0840.60084)].