Let $\lambda_i$ be the sojourn rate of state i, $q_{ij}$ be the transition rate form i to j, and $p_{ij}$ be the transition probability from i to j.
The Kolmogorov Forward and backwards equation are defined as such:
Kolmogorov Forward equations $$p_{ij} (t)' = \sum_{k \neq j}q_{ik} p_{kj} (t) - \lambda_j p_{ij} (t) $$
Kolmogorov Backward equations $$p_{ij} (t)' = \sum_{k \neq j}q_{kj} p_{ik} (t) - \lambda_j p_{ij} (t) $$
What is the differences in interpretation of the forward and backwards equation and relationship between the forward and backwards equation?