I have a question regarding Kolmogorovs forward equation. What I know is: If we have N particles moving in $\Re^2$ under the influence of a pair interaction force and a brownian motion we have (considering the 1st order system) the following macroscopic (Kolmogorov forward) equation for the 1-particle density in phase space.
$$ \partial_t \rho = \vartriangle \rho + \nabla (k \ast \rho)\rho) $$
where $k: \Re^2 \rightarrow \Re^2 $ is the pair interaction force, $\rho: \Re_0^+ \times \Re^2 \rightarrow \Re_0^+ $ is the particle density for an initial value $\rho_0$ and $ \ast$ is the convolution.
I can get this equation from the corresponding Stochastic differential equation, which is:
$$ dY_t^i = - (k \ast \rho_t) (Y_t^i) dt + \sqrt{2}dB_t^i$$
for $ i = 1,...,N$ and $ \rho_t = \mathcal{L}(Y_t^i) $ is the probability distribution of any of the i.i.d. $Y_t^i$.
My question now is as follows. If I have a time dependent growing particle number $N(t)$ instead of $N$ I know that the macroscopic equation changes to a reaction-diffusion equation as follows (if we assume the particle number grows like $f(\rho) = \alpha \rho $
$$ \partial_t \rho = \vartriangle \rho + \nabla (k \ast \rho)\rho) + \alpha \rho $$
But what I dont know is how the stochastic differential equation changes. It may sound simple but I dont get it. I tried do adapt the SDE to get this equation by Ito´s formula but it doesnt work. I also tried the simpler case when one particle divides at a given time t into two, so having $N+1$ particles instead of $N$. But then the $Y_i$ are not independent anymore. Especially the particle $Y_{N+1}$ which was born from (assume) $Y_N$ are not independent. Maybe someone can help me to adjust the SDE. Thanks and best regards, Jack.