In probability theory and ergodic theory, a Markov operator is an operator on a certain function space that conserves the mass (the so-called Markov property). If the underlying measurable space is topologically sufficiently rich enough, then the Markov operator admits a kernel representation. Markov operators can be linear or non-linear. Closely related to Markov operators is the Markov semigroup. [1]
The definition of Markov operators is not entirely consistent in the literature. Markov operators are named after the Russian mathematician Andrey Markov.
Let be a measurable space and a set of real, measurable functions .
A linear operator on is a Markov operator if the following is true [1]: 9–12
Some authors define the operators on the Lp spaces as and replace the first condition (bounded, measurable functions on such) with the property [2] [3]
Let be a family of Markov operators defined on the set of bounded, measurables function on . Then is a Markov semigroup when the following is true [1]: 12
Each Markov semigroup induces a dual semigroup through
If is invariant under then .
Let be a family of bounded, linear Markov operators on the Hilbert space , where is an invariant measure. The infinitesimal generator of the Markov semigroup is defined as
and the domain is the -space of all such functions where this limit exists and is in again. [1]: 18 [4]
The carré du champ operator measuers how far is from being a derivation.
A Markov operator has a kernel representation
with respect to some probability kernel , if the underlying measurable space has the following sufficient topological properties:
If one defines now a σ-finite measure on then it is possible to prove that ever Markov operator admits such a kernel representation with respect to . [1]: 7–13
In probability theory and ergodic theory, a Markov operator is an operator on a certain function space that conserves the mass (the so-called Markov property). If the underlying measurable space is topologically sufficiently rich enough, then the Markov operator admits a kernel representation. Markov operators can be linear or non-linear. Closely related to Markov operators is the Markov semigroup. [1]
The definition of Markov operators is not entirely consistent in the literature. Markov operators are named after the Russian mathematician Andrey Markov.
Let be a measurable space and a set of real, measurable functions .
A linear operator on is a Markov operator if the following is true [1]: 9–12
Some authors define the operators on the Lp spaces as and replace the first condition (bounded, measurable functions on such) with the property [2] [3]
Let be a family of Markov operators defined on the set of bounded, measurables function on . Then is a Markov semigroup when the following is true [1]: 12
Each Markov semigroup induces a dual semigroup through
If is invariant under then .
Let be a family of bounded, linear Markov operators on the Hilbert space , where is an invariant measure. The infinitesimal generator of the Markov semigroup is defined as
and the domain is the -space of all such functions where this limit exists and is in again. [1]: 18 [4]
The carré du champ operator measuers how far is from being a derivation.
A Markov operator has a kernel representation
with respect to some probability kernel , if the underlying measurable space has the following sufficient topological properties:
If one defines now a σ-finite measure on then it is possible to prove that ever Markov operator admits such a kernel representation with respect to . [1]: 7–13