Definition 2.3.3.label Let $L$ be an essentially self-adjoint diffusion operator and $P$ be the spectral measure associated with $-L$. For each $t \ge 0$, let
\[\bp_{t} = \int_{\complex} e^{-tz}P(dz)\]
then:
- (1)
For any $f \in L^{2}(\real^{d})$, $\norm{\bp_tf}_{L^2(\real^d, \mu)}\le \norm{f}_{L^2(\real^d, \mu)}$.
- (2)
For each $t \ge 0$, $\bp_{t}$ is self-adjoint.
- (3)
$\bracs{\bp_t|t \ge 0}$ is a $C_{0}$-semigroup with $L$ as its generator.
The family $\bracs{\bp_t|t \ge 0}$ is the heat semigroup associated with $L$.