Branching processes for the fragmentation equation

We investigate branching properties of the solution of a fragmentation equation for the mass distribution and we properly associate a continuous time càdlàg Markov process on the space S↓ of all fragmentation sizes, introduced by J. Bertoin. A binary fragmentation kernel induces a specific class of integral type branching kernels and taking as base process the solution of the initial fragmentation equation for the mass distribution, we construct a branching process corresponding to a rate of loss of mass greater than a given strictly positive threshold d. It turns out that this branching process takes values in the set of all finite configurations of sizes greater than d. The process on S↓ is then obtained by letting d tend to zero. A key argument for the convergence of the branching processes is given by the Bochner–Kolmogorov theorem. The construction and the proof of the path regularity of the Markov processes are based on several newly developed potential theoretical tools, in terms of excessive functions and measures, compact Lyapunov functions, and some appropriate absorbing sets.