Set $\{G(t);t>0\}$ the semigroup defined as $G(t):L^2 \to L^2$ for every $t>0$, where
$$G(t)u=g_t*u$$
and $g_t(x)=(4\pi t)^{-\frac{n}{2}}\exp(-|x|²/4t)$, $\forall x \in \mathbb{R}^n$. On the other hand I have seen some notations where $G(t)=e^{t \Delta}$. Exist some relation between $e^{t\Delta}$ and
$e^{tA}=\displaystyle \sum_{n=0}^{\infty}\frac{(t A)^n}{n!}?$ The sign $e^{t \Delta}$ even represent the heat semigroup?