Chapman-Kolmogorov equations

The Chapman-Kolmogorov equations are a fundamental concept in probability theory and stochastic processes. They describe how the probability of transitioning from one state to another over a certain period can be broken down into intermediate steps. Essentially, they state that the probability of moving from state A to state C in a given time can be found by summing (or integrating) over all possible intermediate states B, multiplying the probability of going from A to B and then B to C. This helps analyze complex systems by understanding short-term behaviors and how they combine over time.