Consider a particle $p_1$ moving at unit speed along a straight line in $\mathbf{R}^2$, directed by some vector $v_1 \in \mathbf{S}^1$. Equid this particle with a Poisson clock $\tau_1$, with parameter $\lambda =1$ say. When this rings, the particle decays into $N = N_1$ new particles: $p_{11},\dots,p_{1N}$. The number $N$ would be random; for example $N \sim \mathrm{Pois}(\lambda = 2)$.
Each of these again moves at unit speed along a straight line, respectively directed by the unit vectors $v_{11},\dots,v_{1N} \in \mathbf{S}^1$. In this decay the 'momentum' is conserved in the sense that \begin{equation} v_1 = \sum_{i} v_{1i}. \end{equation} (The analogy with physical particle decay is only partial, as there the masses of the $p_{11},\dots,p_{1N}$ would be smaller than that of $p_1$. For us they all have 'unit mass'.)
As to the distribution of the momenta $v_{11},\dots,v_{1N}$, I would suggest picking a solution from among the compact set \begin{equation} \{ (x_1,\dots,x_n) \in (\mathbf{S}^1)^N \mid x_1 + \cdots + x_n = v_1 \} \end{equation} uniformly at random. (However I would be open to other sensible ways of choosing them.) Note that when $N$ is small, say $N \leq 3$, there is a unique solution, but there are a continuum of possible momenta when $p_1$ decays into a large number of particles.
These particles decay independently of one another, in the same way that $p_1$ did. Picking some tuple $\alpha = (1,\alpha_2,\dots) \in \mathbf{N}^k$, the particle $p_\alpha$ would decay at time $\tau_\alpha$ in a way that $v_\alpha = \sum_i v_{\alpha i}$, with the sum leading up to $N_\alpha \sim \mathrm{Pois}(\lambda = 2)$. The created particles $p_{\alpha1},\dots$ would in turn decay at (independent) times $\tau_{\alpha1},\dots$ etc.
Question. Does this (or a related) process have a name in the literature? Where could I find more information about it, or the tools required to study it?
Specifically I would like to consider the process in some bounded, regular domain $\Omega \subset \mathbf{R}^2$, where the clocks would be stopped when the particles hit $\partial \Omega$. This could be a large disc, say $\Omega = D_{1000}$. I am interested in a 'post-mortem analysis' once all the particles have reached the boundary.
I hope the implementation as I describe it is sensible; I am not so well-versed in stochastic processes. Most important to me is the 'conserved momentum' condition. The details, such as the distribution of the clocks, or the number of particles created in the decay, I would be happy to change.
