Markov operator

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.

## Contents

* 1 Definitions
* 1.1 Markov operator
* 1.1.1 Alternative definitions
* 1.2 Markov semigroup
* 1.3 Dual semigroup
* 1.4 Infinitesimal generator of the semigroup
* 1.5 Kernel representation of a Markov operator
* 2 Literature
* 3 References

## Definitions

### Markov operator

Let (E,ℱ) be a measurable space and V a set of real, measurable functions f:(E,ℱ)->(ℝ,ℬ(ℝ)).

A linear operator P on V is a Markov operator if the following is true[1]: 9–12

1. P maps bounded, measurable function on bounded, measurable functions.
2. Let 𝟏 be the constant function x↦1, then P(𝟏)=𝟏 holds. (conservation of mass / Markov property)
3. If f≥0 then Pf≥0. (conservation of positivity)

#### Alternative definitions

Some authors define the operators on the Lp spaces as P:Lp(X)->Lp(Y) and replace the first condition (bounded, measurable functions on such) with the property[2][3]

‖Pf‖Y=‖f‖X,∀f∈Lp(X)

### Markov semigroup

Let 𝒫={Pt}t≥0 be a family of Markov operators defined on the set of bounded, measurables function on (E,ℱ). Then 𝒫 is a Markov semigroup when the following is true[1]: 12

1. P0=Id.
2. Pt+s=Pt∘Ps for all t,s≥0.
3. There exist a σ-finite measure μ on (E,ℱ) that is invariant under 𝒫, that means for all bounded, positive and measurable functions f:E->ℝ and every t≥0 the following holds

∫EPtfdμ=∫Efdμ.

### Dual semigroup

Each Markov semigroup 𝒫={Pt}t≥0 induces a dual semigroup (Pt*)t≥0 through

∫EPtfdμ=∫Efd(Pt*μ).

If μ is invariant under 𝒫 then Pt*μ=μ.

### Infinitesimal generator of the semigroup

Let {Pt}t≥0 be a family of bounded, linear Markov operators on the Hilbert space L2(μ), where μ is an invariant measure. The infinitesimale generator L of the Markov semigroup 𝒫={Pt}t≥0 is defined as

Lf=lim⁡t↓0Ptf−ft,

and the domain D(L) is the L2(μ)-space of all such functions where this limit exists and is in L2(μ) again.[1]: 18 [4]

D(L)={f∈L2(μ):lim⁡t↓0Ptf−ft exists and is in L2(μ)}.

The carré du champ operator Γ measuers how far L is from being a derivation.

### Kernel representation of a Markov operator

A Markov operator Pt has a kernel representation

(Ptf)(x)=∫Ef(y)pt(x,dy),x∈E,

with respect to some probability kernel pt(x,A), if the underlying measurable space (E,ℱ) has the following sufficient topological properties:

1. Each probability measure μ:ℱ×ℱ->[0,1] can be decomposed as μ(dx,dy)=k(x,dy)μ1(dx), where μ1 is the projection onto the first component and k(x,dy) is a probability kernel.
2. There exist a countable family that generates the σ-algebra ℱ.

If one defines now a σ-finite measure on (E,ℱ) then it is possible to prove that ever Markov operator P admits such a kernel representation with respect to k(x,dy).[1]: 7–13

## Literature

* Bakry, Dominique; Gentil, Ivan; Ledoux, Michel. Analysis and Geometry of Markov Diffusion Operators. Springer Cham. doi:10.1007/978-3-319-00227-9.
* Eisner, Tanja; Farkas, Bálint; Haase, Markus; Nagel, Rainer (2015). "Markov Operators". Operator Theoretic Aspects of Ergodic Theory. Graduate Texts in Mathematics. 2727. Cham: Springer. doi:10.1007/978-3-319-16898-2.
* Wang, Fengyu (2006). Functional Inequalities Markov Semigroups and Spectral Theory. Ukraine: Elsevier Science.

## References

1. ↑ 1.0 1.1 1.2 1.3 1.4 Bakry, Dominique; Gentil, Ivan; Ledoux, Michel. Analysis and Geometry of Markov Diffusion Operators. Springer Cham. doi:10.1007/978-3-319-00227-9.
2. ↑ Eisner, Tanja; Farkas, Bálint; Haase, Markus; Nagel, Rainer (2015). "Markov Operators". Operator Theoretic Aspects of Ergodic Theory. Graduate Texts in Mathematics. 2727. Cham: Springer. pp. 249. doi:10.1007/978-3-319-16898-2.
3. ↑ Wang, Fengyu (2006). Functional Inequalities Markov Semigroups and Spectral Theory. Ukraine: Elsevier Science. p. 3.
4. ↑ Wang, Fengyu (2006). Functional Inequalities Markov Semigroups and Spectral Theory. Ukraine: Elsevier Science. p. 1.

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