Markov operator

In probability theory and ergodic theory, a Markov operator is an operator on a certain function space that conserves the mass. 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.
The definition of Markov operators is not entirely consistent in the literature. Markov operators are named after the Russian mathematician Andrey Markov.

Definitions

Markov operator

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

maps bounded, measurable function on bounded, measurable functions.
Let be the constant function, then holds.
If then.

Alternative definitions

Some authors define the operators on the L^p spaces as and replace the first condition with the property

Markov semigroup

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

.
for all.
There exist a σ-finite measure on that is invariant under, that means for all bounded, positive and measurable functions and every the following holds

Dual semigroup

Each Markov semigroup induces a dual semigroup through
If is invariant under then.

Infinitesimal generator of the semigroup

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.
The carré du champ operator measures how far is from being a derivation.

Kernel representation of a Markov operator

A Markov operator has a kernel representation
with respect to some probability kernel, if the underlying measurable space has the following sufficient topological properties:

Each probability measure can be decomposed as, where is the projection onto the first component and is a probability kernel.
There exist a countable family that generates the σ-algebra.

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.