Theorem 2.3.5.label Let $L$ be an essentially self-adjoint elliptic diffusion operator and $\bracs{\bp_t|t \ge 0}$ be its heat semigroup, then there exists $p \in C^{\infty}(\real \times \real^{d} \times \real^{d}; \real)$ such that
- (1)
For any $f \in L^{2}(\real^{d}; \real)$ and $x \in \real^{d}$,
\[\bp_{t} f(x) = \int p(t, x, y)f(y) \mu(dy)\] - (2)
For any $x, y \in \real^{d}$ and $t > 0$, $p(t, x, y) = p(t, y, x)$.
- (3)
Chapman-Kolmogorov Relation: For any $x, y \in \real^{d}$ and $s, t > 0$,
\[p(s + t, x, y) = \int p(t, x, z)p(s, z, y)\mu(dz)\]
Proof. (1): For each $t > 0$ and $x \in \real^{d}$, the mapping $f \mapsto \bp_{t}f(x)$ is a continuous linear functional on $L^{2}(\mu)$. Hence there exists a continuous mapping
such that
(2): Since $\bp_{t}$ is self-adjoint on $L^{2}(\mu)$, $p$ is symmetric.
(3): Since $\bracs{\bp_t|t \ge 0}$ is a semigroup, $p$ satisfies the Chapman-Kolmogorov relation.
Smoothness: For any $f \in L^{2}(\mu)$,
is smooth. Thus, viewed as a map $\real \times \real^{d} \to L^{2}(\mu)$, $(t, x) \mapsto p(t, x, \cdot)$ is differentiable with respect to the weak topology on $L^{2}(\mu)$.
Since the inner product $L^{2}(\mu) \times L^{2}(\mu) \to \real$ is smooth, by the Chapman-Kolmogorov relation, for any $t > 0$,
is continuous.$\square$