A question about extensions of Markov semigroups

Question

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 $p\geq 0$.

Answer 1

I think this holds in quite some generality by the following simple argument. Let $S$ be a Polish space, let's say. If $T(t)$ is a Markov-Feller semigroup on $C_b(S)$ with kernel $p_t(x,dy)$, then note that for $f \in C_b(S) \cap L^p(\mu)$, $1 \leq p < \infty$, \begin{equation} ||T(t)f||_{L^p(\mu)}^p = \int_S \left| \int_S p_t(x, d y) f(y) \right|^p \mu(dx) \leq \int_S \left(\int_S p_t(x,dy) |f(y)|^p\right) \mu(dx) = ||f||_{L^p(\mu)}^p, \end{equation} which shows that $(T(t))$ is contracting with respect to the $L^p(\mu)$-norm. Approximation of $f \in L^p(\mu)$ by continuous functions should then give you the extension you are looking for.

Note, $L^p(\mu)$ for $p < 1$ is not a Banach space so talking about strongly continuous semigroups does not seem to make sense.

Update: simplified argument.