A positive linear operator on bounded measurable functions that preserves the constant function one. A Markov semigroup is a time-indexed family of such operators with the semigroup law.
The transition operators of a time-homogeneous Markov process satisfy by the Markov property. They preserve constants and positivity. When the generator and function domain permit differentiation, is the Kolmogorov backward equation. Transition semigroup of a continuous-time Markov chain is the discrete-state special case.
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A **Markov operator** is a mathematical construct that is used primarily in the context of Markov processes, which are stochastic processes characterized by their memoryless property. In simple terms, a Markov operator is a linear operator that describes the evolution of probability distributions over states in a Markov chain or Markov process.