I'm cross-posting this question from MSE. It's the first time I do this so I'm unsure of etiquette regarding how to cross-post, if this irritates anyone please vote this down and I'll delete the post. Also if any reply appears to the MSE post I'll update this one.

Suppose that $\{T(t)\}_{t\geq 0}$ is a Markov semigroup on the space of continuous bounded functions defined on $\mathbb{R}^n$ and that $\mu$ is an invariant measure of the semigroup. Then, under some technical assumptions, $\{T(t)\}_{t\geq0}$ can be extended to a strongly continuous semigroup on $L^p(\mathbb{R}^n,\mu)$ for every $p\geq1$.

Can this be generalised to Markov semigroups on more general function spaces (in particular, spaces of functions that do not take values necessarily in $\mathbb{R}^n$)? Ideally, I'm looking for a result that covers both the case of functions that are defined on $\mathbb{R}^n$ and those that are defined on $\mathbb{N}^n$. Any references to texts containing these type of results would also be great.

Edit: Typo, should have been $p\geq 1$ and not $p\geq 0$.

I'm cross-posting this question from MSE. It's the first time I do this so I'm unsure of etiquette regarding how to cross-post, if this irritates anyone please vote this down and I'll delete the post. Also if any reply appears to the MSE post I'll update this one.

Suppose that $\{T(t)\}_{t\geq 0}$ is a Markov semigroup on the space of continuous bounded functions defined on $\mathbb{R}^n$ and that $\mu$ is an invariant measure of the semigroup. Then, under some technical assumptions, $\{T(t)\}_{t\geq0}$ can be extended to a strongly continuous semigroup on $L^p(\mathbb{R}^n,\mu)$ for every $p\geq1$.

Can this be generalised to Markov semigroups on more general function spaces (in particular, spaces of functions that do not take values necessarily in $\mathbb{R}^n$)? Ideally, I'm looking for a result that covers both the case of functions that are defined on $\mathbb{R}^n$ and those that are defined on $\mathbb{N}^n$. Any references to texts containing these type of results would also be great.

Edit: Typo, should have been $p\geq 1$ and not $p\geq 0$.

I'm cross-posting this question from MSE. It's the first time I do this so I'm unsure of etiquette regarding how to cross-post, if this irritates anyone please vote this down and I'll delete the post. Also if any reply appears to the MSE post I'll update this one.

Suppose that $\{T(t)\}_{t\geq 0}$ is a Markov semigroup on the space of continuous bounded functions defined on $\mathbb{R}^n$ and that $\mu$ is an invariant measure of the semigroup. Then, under some technical assumptions, $\{T(t)\}_{t\geq0}$ can be extended to a strongly continuous semigroup on $L^p(\mathbb{R}^n,\mu)$ for every $p\geq0$.

Can this be generalised to Markov semigroups on more general function spaces (in particular, spaces of functions that do not take values necessarily in $\mathbb{R}^n$)? Ideally, I'm looking for a result that covers both the case of functions that are defined on $\mathbb{R}^n$ and those that are defined on $\mathbb{N}^n$. Any references to texts containing these type of results would also be great.

I'm cross-posting this question from MSE. It's the first time I do this so I'm unsure of etiquette regarding how to cross-post, if this irritates anyone please vote this down and I'll delete the post. Also if any reply appears to the MSE post I'll update this one.

Suppose that $\{T(t)\}_{t\geq 0}$ is a Markov semigroup on the space of continuous bounded functions defined on $\mathbb{R}^n$ and that $\mu$ is an invariant measure of the semigroup. Then, under some technical assumptions, $\{T(t)\}_{t\geq0}$ can be extended to a strongly continuous semigroup on $L^p(\mathbb{R}^n,\mu)$ for every $p\geq1$.

Can this be generalised to Markov semigroups on more general function spaces (in particular, spaces of functions that do not take values necessarily in $\mathbb{R}^n$)? Ideally, I'm looking for a result that covers both the case of functions that are defined on $\mathbb{R}^n$ and those that are defined on $\mathbb{N}^n$. Any references to texts containing these type of results would also be great.

Edit: Typo, should have been $p\geq 1$ and not $p\geq 0$.