My question concerns a special kind of dynamical process. The setup is the following: We are given a Matrix $\xi_{\mu}^{i} \in \mathbb{R}^{MxN}$ which is initially chosen randomly (with entries from $\mathcal{N}(0,1)$). We are also given a vector $p_{i} \in \mathbb{R}^{N}$, initialised randomly.
The process I am studying is the following: I compute the dot product $\vec{\xi}_{\mu} \cdot \vec{p}$ and check whether
\begin{equation} \vec{\xi}_{\mu} \cdot \vec{p} \geq \sigma \qquad \forall \mu, \end{equation}
with $\sigma \in \mathbb{R}$.
If the condition is not satisfied for the $\mu$th vector $\vec{\xi}_{\mu}$, I remove this vector and replace it with a new vector chosen randomly from the distribution $\mathcal{N} (0,1)$
This process is what I call "remove and replace". I would like to know whether this dynamical process can be modeled by some kind of master equation (or any other stochastic differential equation) or not. What I am interested in is to understand how long-time limit (understood as performing this remove and replace procedure for a large number of times) depends on $\sigma$.
I am not an expert on stochastic processes or master equations. So any help/pointers/criticisms are welcome. Thank you.