Proposition 2.3.4 | Brownian Motion and Stochastic Calculus

Proposition 2.3.4.label Let $L$ be an essentially self-adjoint elliptic diffusion operator with smooth coefficients, and $\bracs{\bp_t|t \ge 0}$ be its heat semigroup, then:

(1)
For any precompact open set $U \subset \subset \real^{d}$ and $k \in \nat$,
\[\norm{P_tf}_{H^k(K)}\le C_{K, k, \mu}\paren{1 + \frac{1}{t^{k}}}\norm{f}_{L^2(\real^d)}\]

In particular, by the Sobolev Embedding theorem,
\[\sup_{x \in K}|\bp_{t}f(x)| \le C_{K}\paren{1 + \frac{1}{t^{\lfloor d/2 \rfloor + 1}}}\norm{f}_{L^2(\real^d)}\]
(2)
For any $f \in L^{2}(\real^{d}; \real)$, the mapping $(t, x) \mapsto \bp_{t}f(x)$ is smooth on $(0, \infty) \times \real^{d}$.

Proof. (1): For any $k \in \nat$,

\[L^{k}\bp_{t} = \int z^{k} e^{-tz}P(dz)\]

Hence

\[\normn{L^k \bp_t}_{L(L^2(\mu), L^2(\mu))}\le \sup_{z \ge 0}z^{k}e^{-z t}\le \frac{k^{k}}{t^{k}}e^{-k}\le \frac{C_{k}}{t^{k}}\]

By the Elliptic Regularity Theorem,

\begin{align*}\norm{P_tf}_{H^k(K)}&\le C_{K}\sum_{\ell = 0}^{k} \normn{L^k\bp_tf}_{L^2(\real^d)}\\&\le C_{K, k, \mu}\paren{1 + \frac{1}{t^{k}}}\norm{f}_{L^2(\real^d)}\end{align*}

(2): By (1), $\bp_{t} f \in H^{k}_{\text{loc}}$ for all $k \in \nat$, so $\bp_{t} f \in C^{\infty}$. For any $s, t > 0$ and $k \in \nat$,

\[\norm{L^k\bp_t - L^k\bp_s}_{L(L^2(\mu), L^2(\mu))}\le \sup_{z \ge 0}|{z^ke^{-z}(e^{-t} - e^{-s})}|\]

so $(t, x) \mapsto \bp_{t} f(x)$ is jointly continuous. By a very wasteful chain of Sobolev shenanigans, $(t, x) \mapsto \bp_{t} f(x)$ is smooth.$\square$