A spatially-dependent fragmentation process

We define a fragmentation process which involves rectangles breaking up into progressively smaller pieces at rates that depend on their shape. Long, thin rectangles are more likely to break quickly, whereas squares break more slowly. Each rectangle is also more likely to split along its longest side. We are interested in how the system evolves over time: how many fragments are there of different shapes and sizes, and how did they reach that state? Using a standard transformation this fragmentation process with shape-dependent rates is equivalent to a two-dimensional branching random walk in continuous time in which the branching rate and the direction of each jump depend on the particles' position. Our main theorem gives an almost sure growth rate along paths for the number of particles in the branching random walk, which in turn gives the number of fragments with a fixed shape as the solution to an optimisation problem. This is a result of interest in the context of spatial branching systems and provides an example of a multitype branching process with a continuum of types.