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 sufficient 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 definitions of Markov operators is not entirely consistent in the literature. Markov operators are named after 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 [math]\displaystyle{ (E,\mathcal{F}) }[/math] be a measurable space and [math]\displaystyle{ V }[/math] a set of real, measurable functions [math]\displaystyle{ f:(E,\mathcal{F})\to (\mathbb{R},\mathcal{B}(\mathbb{R})) }[/math].

A linear operator [math]\displaystyle{ P }[/math] on [math]\displaystyle{ V }[/math] is a Markov operator if the following is true[1]

1. [math]\displaystyle{ P }[/math] maps bounded, measurable function on bounded, measurable functions.
2. Let [math]\displaystyle{ \mathbf{1} }[/math] be the constant function [math]\displaystyle{ x\mapsto 1 }[/math], then [math]\displaystyle{ P(\mathbf{1})=\mathbf{1} }[/math] holds. (conservation of mass / Markov property)
3. If [math]\displaystyle{ f\geq 0 }[/math] then [math]\displaystyle{ Pf\geq 0 }[/math]. (conservation of positivity)

#### Alternative definitions

Some authors define the operators on the Lp spaces as [math]\displaystyle{ P:L^p(X)\to L^p(Y) }[/math] and replace the first condition (bounded, measurable functions on such) with the property[2][3]

[math]\displaystyle{ \|Pf\|_Y = \|f\|_X,\quad \forall f\in L^p(X) }[/math]

### Markov semigroup

Let [math]\displaystyle{ \mathcal{P}=\\{P_t\\}_{t\geq 0} }[/math] be a family of Markov operators defined on the set of bounded, measurables function on [math]\displaystyle{ (E,\mathcal{F}) }[/math]. Then [math]\displaystyle{ \mathcal{P} }[/math] is a Markov semigroup when the following is true[4]

1. [math]\displaystyle{ P_0=\operatorname{Id} }[/math].
2. [math]\displaystyle{ P_{t+s}=P_t\circ P_s }[/math] for all [math]\displaystyle{ t,s\geq 0 }[/math].
3. There exist a σ-finite measure [math]\displaystyle{ \mu }[/math] on [math]\displaystyle{ (E,\mathcal{F}) }[/math] that is invariant under [math]\displaystyle{ \mathcal{P} }[/math], that means for all bounded, positive and measurable functions [math]\displaystyle{ f:E\to \mathbb{R} }[/math] and every [math]\displaystyle{ t\geq 0 }[/math] the following holds

[math]\displaystyle{ \int_E P_tf\mathrm{d}\mu =\int_E f\mathrm{d}\mu }[/math].

### Dual semigroup

Each Markov semigroup [math]\displaystyle{ \mathcal{P}=\\{P_t\\}_{t\geq 0} }[/math] induces a duale semigroup [math]\displaystyle{ (P^*_t)_{t\geq 0} }[/math] through

[math]\displaystyle{ \int_EP_tf\mathrm{d\mu} =\int_E f\mathrm{d}\left(P^*_t\mu\right). }[/math]

If [math]\displaystyle{ \mu }[/math] is invariant under [math]\displaystyle{ \mathcal{P} }[/math] then [math]\displaystyle{ P^*_t\mu=\mu }[/math].

### Infinitesimal generator of the semigroup

Let [math]\displaystyle{ \\{P_t\\}_{t\geq 0} }[/math] be a family of bounded, linear Markov operators on the Hilbert space [math]\displaystyle{ L^2(\mu) }[/math], where [math]\displaystyle{ \mu }[/math] is an invariant measure. The infinitesimale generator [math]\displaystyle{ L }[/math] of the Markov semigroup [math]\displaystyle{ \mathcal{P}=\\{P_t\\}_{t\geq 0} }[/math] is defined as

[math]\displaystyle{ Lf=\lim\limits_{t\downarrow 0}\frac{P_t f-f}{t}, }[/math]

and the domain [math]\displaystyle{ D(L) }[/math] is the [math]\displaystyle{ L^2(\mu) }[/math]-space of all such functions where this limit exists and is in [math]\displaystyle{ L^2(\mu) }[/math] again.[5][6]

[math]\displaystyle{ D(L)=\left\\{f\in L^2(\mu): \lim\limits_{t\downarrow 0}\frac{P_t f-f}{t}\text{ exists and is in } L^2(\mu)\right\\}. }[/math]

### Kernel representation of a Markov operator

A Markov operator [math]\displaystyle{ P }[/math] has a kernel representation

[math]\displaystyle{ (P_tf)(x)=\int_E f(y)p_t(x,\mathrm{d}y),\quad x\in E, }[/math]

with respect to some probability kernel [math]\displaystyle{ p_t(x,A) }[/math], if the underlying measurable space [math]\displaystyle{ (E,\mathcal{F}) }[/math] has sufficient topological properties:

1. Each probability measure [math]\displaystyle{ \mu:\mathcal{F}\times \mathcal{F}\to [0,1] }[/math] can be decomposed as [math]\displaystyle{ \mu(\mathrm{d}x,\mathrm{d}y)=k(x,\mathrm{d}y)\mu_1(\mathrm{d}x) }[/math], where [math]\displaystyle{ \mu_1 }[/math] is the projection onto the first component and [math]\displaystyle{ k(x,\mathrm{d}y) }[/math] is a probability kernel.
2. There exist a countable family that generates the σ-algebra [math]\displaystyle{ \mathcal{F} }[/math].

If one defines now a σ-finite measure on [math]\displaystyle{ (E,\mathcal{F}) }[/math] then it is possible to prove that ever Markov operator [math]\displaystyle{ P }[/math] admits such a kernel representation with respect to [math]\displaystyle{ k(x,\mathrm{d}y) }[/math].[7]

## 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. ↑ Bakry, Dominique; Gentil, Ivan; Ledoux, Michel. Analysis and Geometry of Markov Diffusion Operators. Springer Cham. pp. 9-12. 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. ↑ Bakry, Dominique; Gentil, Ivan; Ledoux, Michel. Analysis and Geometry of Markov Diffusion Operators. Springer Cham. pp. 12. doi:10.1007/978-3-319-00227-9.
5. ↑ Bakry, Dominique; Gentil, Ivan; Ledoux, Michel. Analysis and Geometry of Markov Diffusion Operators. Springer Cham. p. 18. doi:10.1007/978-3-319-00227-9.
6. ↑ Wang, Fengyu (2006). Functional Inequalities Markov Semigroups and Spectral Theory. Ukraine: Elsevier Science. p. 1.
7. ↑ Bakry, Dominique; Gentil, Ivan; Ledoux, Michel. Analysis and Geometry of Markov Diffusion Operators. Springer Cham. pp. 7-13. doi:10.1007/978-3-319-00227-9.

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