Exact time-dependent solution for general multi-type branching diffusions

Derive an exact time-dependent solution for the multi-type branching diffusion generated by Eq. (1) for arbitrary dimension d>1, general mutation-rate matrix [200~\gamma, and specified initial condition.

Background

The paper studies a neutral multi-type branching diffusion whose coordinates represent population sizes of d types and whose mutation dynamics are governed by a general rate matrix \gamma. For d=1, the process reduces to the classical Feller diffusion and has a known explicit solution.

For d>1, coupling through the mutation terms prevents the direct extension of the one-type solution. The paper therefore develops asymptotic approximations for small mutation rates rather than an exact solution valid for general parameters and initial conditions.