Could anyone show me a way to directly compute the generator of a strongly continuous semigroup $S(t)$ in $L^p$ $$\lim_{t\to 0}\frac{S(t)f-f}{t}$$ where the limit shall lies in $L^p$ Here $f\in C^2(\mathbb{R}^d)\cap L^p(\mathbb{R}^d)$, and on $L^p(\mathbb{R}^d)$, $p\in [1,\infty)$, we have the expression $$(S(t)f)(x):=\frac{1}{\sqrt{4\pi t}}\left(\int_{\mathbb{R}^d}f(y)e^{-\frac{|x-y|^2}{4t}}dy\right)$$
I meant to start with $\frac{d}{dt}\int f(y)n_t(x-y)dy$ and make use of $\frac{d}{dt} n_t(x-y)= \Delta_x n_t(x-y)$.
Thanks for any help