Markov operator

 ( Probability Theory ) 

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 operator 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.

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Definitions

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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

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)

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Alternative definitions Some authors define the operators on the Lp space as $P:L^p(X)\backslash to\; L^p(Y)$ and replace the first condition (bounded, measurable functions on such) with the property

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

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Markov semigroup 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

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

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

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Dual semigroup

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 $P^*\_t\backslash mu=\backslash mu$.

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Infinitesimal generator of the semigroup

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.

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

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

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Kernel representation of a Markov operator

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 $k(x,\backslash mathrm\{d\}y)$.

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Literature

Categories: Probability Theory, Ergodic Theory, Linear Operators

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