A Markov process $(X_t)_{t \geq 0}$ in continuous time on $\mathbb{R}^d$ can be described by a semigroup of Markov kernels $(p_t(x,A))_{t \geq 0}$ with $p_0(x,A) = 1_{A}(x)$ and which fulfill the Chapman-Kolmogorov equation.
One can define the operator $$P_t f (x) := \mathbb{E}^x[f(X_t)] = \int_{\mathbb{R}^d} f(y) \, p_t(x,dy)$$ on $L^\infty$ (set of all bounded, measurable functions) and it then follows that $P_0 = Id$ and $P_t P_s = P_{t+s}$ hold.
My question is if $(P_t)$ is a $C_0$-semigroup. In other words, does $$|| P_t f - f || \to 0 $$ for any $f\in L^\infty$ when $t \to 0^+$ holds?
In general: no, $|| P_t f- f|| \to 0$ does not hold for all $f \in L^\infty$ and a general semigroup $(P_t)_{t\geq 0}$ on the Banach space $L^\infty$.
If e.g. $P_t f(x) = f(x+ct)$ for some $c \neq 0$, then if $\hat{x}$ is a discontinuity point of $f$ problems appear: $P_t f(\hat{x}) - f(\hat{x}) = f(\hat{x}+ct) - f(\hat{x})$. This will not converge to $0$ as $t\to 0$. So norm convergence will even be more difficult.
It is often preferred to use other Banach spaces than $L^\infty$. Examples are $C_0(\mathbb{R}^d)$, $C_b(\mathbb{R}^d)$ or also $L^2(\mathbb{R}^d)$. I provide the link to the first two of these examples: https://almostsure.wordpress.com/2010/07/14/feller-processes/ and Section 6 in http://www.math.ist.utl.pt/~czaja/ISEM/internetseminar200607.pdf