Chapman-Kolmogorov equation

1. Sablon:Matematika Chapman-Kolmogorov-egyenlet

The Chapman-Kolmogorov equations are fundamental in the theory of stochastic processes, particularly in the context of Markov processes. They provide a way to relate the transition probabilities of a Markov process at different times.

For a discrete-time Markov chain, if $P(X_n=j|X_0=i)$ denotes the transition probability from state $i$ to state $j$ in $n$ steps, the Chapman-Kolmogorov equations state:

$$P(X_n=j|X_0=i)=\sum _{k}P(X_m=k|X_0=i)P(X_{n-m}=j|X_0=k)$$

for $m=0,1,2,\dots ,n$.

In simpler terms, this equation expresses the idea that the probability of transitioning from state $i$ to state $j$ in $n$ steps can be computed by summing over all possible intermediate states $k$ that the process could be in after $m$ steps, and then transitioning from $k$ to $j$ in the remaining $n-m$ steps.

For continuous-time Markov processes, the Chapman-Kolmogorov equations can also be expressed in terms of transition probability densities, which can involve the infinitesimal generator of the process.

These equations are crucial for deriving many results in probability theory and statistical mechanics, especially in the study of Markov chains and their long-term behavior.