Consider the closed system of transport equations
$$ \frac{\partial G_p}{\partial x_3} + \frac{2p}{\bar c}\frac{\partial G_p}{\partial\tau} = (\mathcal{L}G)_p,$$
where
$$\quad x_3 \geq -L,\quad p\in\mathbb{N},\quad \tau\in\mathbb{R},\quad Q,\bar c \in \mathcal{R}^+ $$
with
$$ (\mathcal{L}G)_p = \frac{p^2}{Q}\bigg(G_{p+1} + G_{p-1} - 2G_p\bigg), $$
The paper I'm reading then presents a probabilistic representation of the transport equations in the form of a Jump Markov process. Let the jump Markov process $(N_{x_3})_{x_3\geq -L}$ have state space $\mathbb{N}$ and infinitesimal generator $\mathcal{L}$ which is given above. The jump process is crafted as follows.
When it reaches the state $n>0$ (we change $p\rightarrow n$) a random clock with exponential distribution and parameter $2n^2/Q$ starts to tick. When the clock strikes, the process jumps to $n+1$ or $n-1$ with equal probability, zero is an absorbing state. They then define the process
$$\frac{\partial T}{\partial x_3} = \frac{-2}{\bar c}N_{x_3},$$ where $T_{x_3 = -L} = \tau.$ The pair $(N_{x_3},T_{x_3})_{x_3\geq -L}$ is Markovian with generator $$ \mathcal{L} = -\frac{2n}{\bar c}\frac{\partial}{\partial\tau}. $$ Where does this equation for $T$ come from? $T$ has dimensions of time in this equation. Also why does the clock have parameter $2n^2/Q$, the prefactor of $n^2$ (not $n$) seems to appear?
So the Kolmogorov equation is then $$\frac{\partial u}{\partial x_3} = \bigg(\mathcal{L} - \frac{2n}{\bar c}\frac{\partial}{\partial\tau}\bigg)u,\quad x_3 > -L, \quad u(n,\tau,x_3 = -L) = U(n,\tau).$$