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.

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

A linear operator ${\displaystyle P}${\displaystyle P} on ${\displaystyle V}${\displaystyle V} is a Markov operator if the following is true^{[1] : 9–12}

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

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

${\displaystyle \|Pf\|_{Y}=\|f\|_{X},\quad \forall f\in L^{p}(X)}${\displaystyle \|Pf\|_{Y}=\|f\|_{X},\quad \forall f\in L^{p}(X)}

Let ${\displaystyle {\mathcal {P}}=\{P_{t}\}_{t\geq 0}}${\displaystyle {\mathcal {P}}=\{P_{t}\}_{t\geq 0}} be a family of Markov operators defined on the set of bounded, measurables function on ${\displaystyle (E,{\mathcal {F}})}${\displaystyle (E,{\mathcal {F}})}. Then ${\displaystyle {\mathcal {P}}}${\displaystyle {\mathcal {P}}} is a Markov semigroup when the following is true^{[1] : 12}

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

${\displaystyle \int _{E}P_{t}f\mathrm {d} \mu =\int _{E}f\mathrm {d} \mu }${\displaystyle \int _{E}P_{t}f\mathrm {d} \mu =\int _{E}f\mathrm {d} \mu }.

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

${\displaystyle \int _{E}P_{t}f\mathrm {d\mu } =\int _{E}f\mathrm {d} \left(P_{t}^{*}\mu \right).}${\displaystyle \int _{E}P_{t}f\mathrm {d\mu } =\int _{E}f\mathrm {d} \left(P_{t}^{*}\mu \right).}

If ${\displaystyle \mu }${\displaystyle \mu } is invariant under ${\displaystyle {\mathcal {P}}}${\displaystyle {\mathcal {P}}} then ${\displaystyle P_{t}^{*}\mu =\mu }${\displaystyle P_{t}^{*}\mu =\mu }.

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

${\displaystyle Lf=\lim \limits _{t\downarrow 0}{\frac {P_{t}f-f}{t}},}${\displaystyle Lf=\lim \limits _{t\downarrow 0}{\frac {P_{t}f-f}{t}},}

and the domain ${\displaystyle D(L)}${\displaystyle D(L)} is the ${\displaystyle L^{2}(\mu )}${\displaystyle L^{2}(\mu )}-space of all such functions where this limit exists and is in ${\displaystyle L^{2}(\mu )}${\displaystyle L^{2}(\mu )} again.^{[1] : 18 [4]}

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

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

A Markov operator ${\displaystyle P_{t}}${\displaystyle P_{t}} has a kernel representation

${\displaystyle (P_{t}f)(x)=\int _{E}f(y)p_{t}(x,\mathrm {d} y),\quad x\in E,}${\displaystyle (P_{t}f)(x)=\int _{E}f(y)p_{t}(x,\mathrm {d} y),\quad x\in E,}

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

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

If one defines now a σ-finite measure on ${\displaystyle (E,{\mathcal {F}})}${\displaystyle (E,{\mathcal {F}})} then it is possible to prove that ever Markov operator ${\displaystyle P}${\displaystyle P} admits such a kernel representation with respect to ${\displaystyle k(x,\mathrm {d} y)}${\displaystyle k(x,\mathrm {d} y)}.^{[1] : 7–13}

• 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. Vol. 2727. Cham: Springer. doi:10.1007/978-3-319-16898-2.
• Wang, Fengyu (2006). Functional Inequalities Markov Semigroups and Spectral Theory. Ukraine: Elsevier Science.

• Probability theory
• Ergodic theory
• Linear operators