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Chapman-Kolmogorov Equation

The identity showing that a Markov chain's $(m+n)$-step transition probability equals the sum, over all intermediate states, of the probability of reaching that intermediate state in $m$ steps times the probability of continuing on to the destination in $n$ more steps.

Stochastic ProcessesIntermediateFree

In Plain English

To get from state $i$ to state $j$ in $m + n$ steps, the chain has to pass through some intermediate state $k$ after the first $m$ steps, whichever one it happens to be. The Chapman-Kolmogorov equation just adds up that logic formally: sum, over every possible intermediate state $k$ , the probability of getting from $i$ to $k$ in $m$ steps times the probability of then getting from $k$ to $j$ in the remaining $n$ steps. It's the Markov property's consequence written as an algebraic identity, and it's exactly the rule matrix multiplication already follows.

Definition

For a Markov chain with transition probabilities $p_{ij}^{(n)}$ , the Chapman-Kolmogorov equation states that for any non-negative integers $m, n$ , $p_{ij}^{(m + n)} = \sum_{k \in S} p_{ik}^{(m)} p_{kj}^{(n)}$ . In matrix form, letting $P^{(n)}$ denote the matrix of $n$ -step transition probabilities, this is simply $P^{(m + n)} = P^{(m)} P^{(n)}$ , i.e. $P^{n} = P \cdot P \dots P$ ( $n$ times). The equation follows directly from the Markov property applied to the law of total probability, conditioning on the state $X_{m} = k$ at the intermediate time. An analogous continuous-time version holds for continuous-time Markov chains and general Markov processes, with sums replaced by integrals over the state space where the state space is continuous.

Formula

p_{ij}^{(m + n)} = k \in S \sum p_{ik}^{(m)} p_{kj}^{(n)}

Notation

p_{ij}^{(m + n)}

—Probability of moving from state $i$ to state $j$ in exactly $m+n$ steps

k

—An intermediate state reached after the first $m$ steps, summed over all states

Properties

The equation is less a new assumption than a restatement, in algebraic form, of what the Markov property already implies: because the future depends only on the present, the path can be split at any intermediate time without losing information.
This identity is precisely why computing multi-step transition probabilities for a Markov chain reduces to ordinary matrix multiplication, an enormous practical simplification once probabilities are organized into a transition matrix.
The continuous-time version of the equation underlies the Kolmogorov forward and backward differential equations, which describe how a continuous-time Markov chain's state distribution evolves instantaneously rather than in discrete jumps.

At a Glance

In wordsthe (m+n)-step probability sums over all ways of passing through an intermediate state

Matrix form

P^{(m + n)} = P^{(m)} P^{(n)}

, i.e.

n

-step probabilities are just

P^{n}

Derived fromthe Markov property plus the law of total probability

Continuous-time analogueholds with sums replaced by integrals for general Markov processes

Last updated September 8, 2026← Back to the Dictionary