Kolmogorov Differential Equations governing random dynamical system

Question

I am reading the original paper (Translation) by Kolmogorov "On Analytical Methods in Probability Theory". I do not quite understand some of the conclusions in the paper.

This is also on the wiki page,

https://en.wikipedia.org/wiki/Kolmogorov_equations_(Markov_jump_process)

I do not understand the condition $\sum_k A_{jk} (t) = 0$. This is supposed to follow from the fact that $\sum_j P_{ij}(t,u) = 1$.

Answer 1

Renaming indices you get to

$$ A_{jk} = \left. \frac{\partial P_{jk}}{\partial u} \right|_{u=t} $$

so that

\begin{eqnarray} \sum_k A_{jk} &=& \sum_k\left. \frac{\partial P_{jk}}{\partial u} \right|_{u=t} \\ &=& \left.\frac{\partial}{\partial u}\left(\color{blue}{\sum_k P_{jk} }\right)\right|_{u=t} \\ &=& \left.\frac{\partial}{\partial u}\left(\color{blue}{1 }\right)\right|_{u=t} \\ &=& 0 \end{eqnarray}

so that