Markov operator explained

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.

Definitions

Markov operator

Let

(E, l{F}) be a measurable space and V a set of real, measurable functions f:(E, l{F})\to ( R , l{B}( R )) .

A linear operator

P on V is a Markov operator if the following is trueP maps bounded, measurable function on bounded, measurable functions.

1. Let

1 be the constant function x\mapsto 1 , then P( 1 ) = 1 holds. (conservation of mass / Markov property)

1. If

f\geq 0 then Pf\geq 0 . (conservation of positivity)

Alternative definitions

Some authors define the operators on the L^p spaces as

P:L ^{p(X)\to} L ^{p(Y)} and replace the first condition (bounded, measurable functions on such) with the property^{[2] [3]} \|Pf\| _{Y} = \|f\| _{X,} \forall f\in L ^{p(X)}

Markov semigroup

Let

l{P}=\{P _{t\}} _{t\geq} be a family of Markov operators defined on the set of bounded, measurables function on (E, l{F}) . Then l{P} is a Markov semigroup when the following is trueP _{0=\operatorname{Id}} .P _{t+s} = P _{t\circ} P _{s} for all t,s\geq 0 .\mu on (E, l{F}) that is invariant under l{P} , that means for all bounded, positive and measurable functions f:E\to R and every t\geq 0 the following holds\int _{E} P _{tfd\mu} = \int _{E} f d \mu .

Dual semigroup

Each Markov semigroup

l{P}=\{P _{t\}} _{t\geq} induces a dual semigroup (P ^{*} _{t)} _{t\geq} through\int _{EP} _{tfd\mu} = \int _{E} f d \left(P ^{*} _{t\mu\right).} If \mu is invariant under l{P} then P ^{*} _{t\mu = \mu} .

Infinitesimal generator of the semigroup

Let

\{P _{t\}} _{t\geq} be a family of bounded, linear Markov operators on the Hilbert space L ^{2(\mu)} , where \mu is an invariant measure. The infinitesimal generator L of the Markov semigroup l{P}=\{P _{t\}} _{t\geq} is defined asLf = \lim\limits _{t\downarrow} (P _{t} f - f)/(t) , and the domain D(L) is the L ^{2(\mu)} -space of all such functions where this limit exists and is in L ^{2(\mu)} again.^[4] D(L) = \left\{f\in L ^{2(\mu):} \lim\limits _{t\downarrow} (P _{t} f - f)/(t) existsandisin L ^{2(\mu)\right\}.} \Gamma measuers how far L is from being a derivation.

Kernel representation of a Markov operator

A Markov operator

P _{t} has a kernel representation(P _{tf)(x)=\int} _{E} f(y)p _{t(x,dy),} x\in E, with respect to some probability kernel p _{t(x,A)} , if the underlying measurable space (E, l{F}) has the following sufficient topological properties:\mu: l{F} x l{F}\to [ 0 , 1 ] can be decomposed as \mu( d x, d y) = k(x, d y)\mu _{1(dx)} , where \mu _{1} is the projection onto the first component and k(x, d y) is a probability kernel.l{F} .If one defines now a σ-finite measure on (E, l{F}) then it is possible to prove that ever Markov operator P admits such a kernel representation with respect to k(x, d y) .

Literature

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

References

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