Markov operator

Let $(E,\backslash mathcal\{F\})$ be a measurable space and $V$ a set of real, measurable functions $f:(E,\backslash mathcal\{F\})\backslash to\; (\backslash mathbb\{R\},\backslash mathcal\{B\}(\backslash mathbb\{R\}))$.

A linear operator $P$ on $V$ is a Markov operator if the following is true{{rp|9-12}}

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

Some authors define the operators on the L^p spaces as $P:L^p(X)\backslash to\; L^p(Y)$ and replace the first condition (bounded, measurable functions on such) with the property{{cite book|first1=Tanja|last1=Eisner|first2=Bálint|last2=Farkas|first3=Markus|last3=Haase|first4=Rainer|last4=Nagel|date=2015|title=Operator Theoretic Aspects of Ergodic Theory|chapter=Markov Operators|series=Graduate Texts in Mathematics|volume=2727|publisher=Springer|place=Cham|doi=10.1007/978-3-319-16898-2|pages=249}}{{cite book|first=Fengyu|last=Wang|title=Functional Inequalities Markov Semigroups and Spectral Theory|place=Ukraine|publisher=Elsevier Science|date=2006|page=3}}

:$\backslash |Pf\backslash |\_Y\; =\; \backslash |f\backslash |\_X,\backslash quad\; \backslash forall\; f\backslash in\; L^p(X)$

Let $\backslash mathcal\{P\}=\backslash \{P\_t\backslash \}\_\{t\backslash geq\; 0\}$ be a family of Markov operators defined on the set of bounded, measurables function on $(E,\backslash mathcal\{F\})$. Then $\backslash mathcal\{P\}$ is a Markov semigroup when the following is true{{rp|12}}

1. $P\_0=\backslash operatorname\{Id\}$.
2. $P\_\{t+s\}=P\_t\backslash circ\; P\_s$ for all $t,s\backslash geq\; 0$.
3. There exist a σ-finite measure $\backslash mu$ on $(E,\backslash mathcal\{F\})$ that is invariant under $\backslash mathcal\{P\}$, that means for all bounded, positive and measurable functions $f:E\backslash to\; \backslash mathbb\{R\}$ and every $t\backslash geq\; 0$ the following holds

:::$\backslash int\_E\; P\_tf\backslash mathrm\{d\}\backslash mu\; =\backslash int\_E\; f\backslash mathrm\{d\}\backslash mu$.

Each Markov semigroup $\backslash mathcal\{P\}=\backslash \{P\_t\backslash \}\_\{t\backslash geq\; 0\}$ induces a dual semigroup $(P^*\_t)\_\{t\backslash geq\; 0\}$ through

:$\backslash int\_EP\_tf\backslash mathrm\{d\backslash mu\}\; =\backslash int\_E\; f\backslash mathrm\{d\}\backslash left(P^*\_t\backslash mu\backslash right).$

If $\backslash mu$ is invariant under $\backslash mathcal\{P\}$ then $P^*\_t\backslash mu=\backslash mu$.

Let $\backslash \{P\_t\backslash \}\_\{t\backslash geq\; 0\}$ be a family of bounded, linear Markov operators on the Hilbert space $L^2(\backslash mu)$, where $\backslash mu$ is an invariant measure. The infinitesimal generator $L$ of the Markov semigroup $\backslash mathcal\{P\}=\backslash \{P\_t\backslash \}\_\{t\backslash geq\; 0\}$ is defined as

:$Lf=\backslash lim\backslash limits\_\{t\backslash downarrow\; 0\}\backslash frac\{P\_t\; f-f\}\{t\},$

and the domain $D(L)$ is the $L^2(\backslash mu)$-space of all such functions where this limit exists and is in $L^2(\backslash mu)$ again.{{rp|18}}{{cite book|first=Fengyu|last=Wang|title=Functional Inequalities Markov Semigroups and Spectral Theory|place=Ukraine|publisher=Elsevier Science|date=2006|page=1}}

:$D(L)=\backslash left\backslash \{f\backslash in\; L^2(\backslash mu):\; \backslash lim\backslash limits\_\{t\backslash downarrow\; 0\}\backslash frac\{P\_t\; f-f\}\{t\}\backslash text\{\; exists\; and\; is\; in\; \}\; L^2(\backslash mu)\backslash right\backslash \}.$

The carré du champ operator $\backslash Gamma$ measuers how far $L$ is from being a derivation.

A Markov operator $P\_t$ has a kernel representation

:$(P\_tf)(x)=\backslash int\_E\; f(y)p\_t(x,\backslash mathrm\{d\}y),\backslash quad\; x\backslash in\; E,$

with respect to some probability kernel $p\_t(x,A)$, if the underlying measurable space $(E,\backslash mathcal\{F\})$ has the following sufficient topological properties:

1. Each probability measure $\backslash mu:\backslash mathcal\{F\}\backslash times\; \backslash mathcal\{F\}\backslash to\; [0,1]$ can be decomposed as $\backslash mu(\backslash mathrm\{d\}x,\backslash mathrm\{d\}y)=k(x,\backslash mathrm\{d\}y)\backslash mu\_1(\backslash mathrm\{d\}x)$, where $\backslash mu\_1$ is the projection onto the first component and $k(x,\backslash mathrm\{d\}y)$ is a probability kernel.
2. There exist a countable family that generates the σ-algebra $\backslash mathcal\{F\}$.

If one defines now a σ-finite measure on $(E,\backslash mathcal\{F\})$ then it is possible to prove that ever Markov operator $P$ admits such a kernel representation with respect to $k(x,\backslash mathrm\{d\}y)$.{{rp|7-13}}

• {{cite book|title=Analysis and Geometry of Markov Diffusion Operators|first1=Dominique|last1=Bakry|first2=Ivan|last2=Gentil|first3=Michel|last3=Ledoux|publisher=Springer Cham|doi=10.1007/978-3-319-00227-9}}
• {{cite book| first1=Tanja|last1=Eisner|first2=Bálint|last2=Farkas|first3=Markus|last3=Haase|first4=Rainer|last4=Nagel|date=2015|title=Operator Theoretic Aspects of Ergodic Theory|chapter=Markov Operators|series=Graduate Texts in Mathematics|volume=2727|publisher=Springer|place=Cham|doi=10.1007/978-3-319-16898-2}}
• {{cite book|first=Fengyu|last=Wang|title=Functional Inequalities Markov Semigroups and Spectral Theory|place=Ukraine|publisher=Elsevier Science|date=2006}}

Category:Probability theory

Category:Ergodic theory

Category:Linear operators