Markov property

The Markov property says that, conditional on the present state, the future evolution is independent of the past.

The operator that averages a function over the next-state conditional law of a time-homogeneous Markov process. For a random walk with increments of law $\nu$ independent of the past including the initial state, $(Ph)(s)=\int h(s+y)\nu (dy)$. This additive operator preserves convexity when the integrals exist.

For an ordered Markov sequence, the joint density factors into an initial density and one-step transition densities:

The time-homogeneous Markov property says that the conditional law of a future state depends only on the current state and the elapsed time, not on the current calendar time or earlier history.