Infinitesimal generators

Let $\mathcal{B}$ be either the space $L^{2}$ with respect to a finite measure or else the continuous functions vanishing at infinity for some locally compact separable metric space $S$. In the former case, we say $f \geq 0$ if $f(x) \geq 0$ for almost every $x$, in the latter case if $f(x) \geq 0$ for all $x$. A semigroup is non-negative if $f \geq 0$ implies $P_{t} f \geq 0$ for all $t \geq 0$. Suppose that $P_{t}$ is a semigroup, the space $\mathcal{B}$ contains the constant functions, and $P_{t} 1=1$ for all $t$. Show that $P_{t}$ is a contraction if and only if $P_{t}$ is non-negative.

Show that $P_{t}$ and $R_{\lambda }$ commute and that

Show that for any $a<b$ we have

Hint: Approximate $R_{\lambda } f$ by a Riemann sum.

Show that if $P_{t}$ is a contraction semigroup and $R_{\lambda }$ is the resolvent, then

Show that if $A$ is a bounded operator and $T_{t}=e^{t A}$, then $T_{t}$ is a strongly continuous semigroup of operators with infinitesimal generator $A$. (We cannot assert that the $T_{t}$ are contractions.)

Prove that if $\mathcal{L}$ is dissipative, the domain of $\mathcal{L}$ is dense in $\mathcal{B}$, and the range of $\lambda -\mathcal{L}$ is dense in $\mathcal{B}$, then $\overline{\mathcal{L}}$ defined in (37.12) is a closed extension of $\mathcal{L}$ that is dissipative and the range of $\lambda -\overline{\mathcal{L}}$ is equal to $\mathcal{B}$. Show there is only one such closed extension of $\mathcal{L}$.

If the range of $\lambda -\mathcal{L}$ equals $\mathcal{B}$ for a single value of $\lambda$, then the range of $\lambda -\mathcal{L}$ equals $\mathcal{B}$ for every value of $\lambda$.

Hint: Define $R_{\lambda }$ as the inverse of $\lambda -\mathcal{L}$, then use (37.3) to define $R_{a}$ for other values of $a$.

Let $\left(X_{t}, \mathbb {P}[x]\right)$ be a Markov process with transition probabilities given by $P_{t} f(x)=f(x+t)$. Determine $\mathcal{L}$ and $\mathcal{D}(\mathcal{L})$.

Let $P_{t}$ be a strongly continuous semigroup of contraction operators and let $\mathcal{L}$ be the infinitesimal generator. Show that $\mathcal{D}\left(\mathcal{L}^{n}\right)$ is dense in $\mathcal{B}$ for every positive integer $n$.

This is a continuation of Exercise 36.5. Prove that if $f \in C^{2}$ with compact support, $P_{t}$ is the semigroup given in Exercise 36.5, and $\mathcal{L}$ is the infinitesimal generator, then the Fourier transform of $\mathcal{L} f$ is $\widehat{f}(u) \psi (u)$.

Suppose that $P_{t}$ is a strongly continuous semigroup, but not necessarily of contractions. Thus $P_{t+s}=P_{t} P_{s}$ and $P_{t} f \rightarrow f$ in norm if $f \in \mathcal{B}$, but we do not assume $\left\| P_{t}\right\| \leq 1$. Prove that there exist constants $K, b>0$ such that $\left\| P_{t}\right\| \leq K e^{b t}$ for all $t \geq 0$.

Hint: Use the uniform boundedness principle from functional analysis to prove there exists $c, t_{0}$ such that $\left\| P_{t}\right\| \leq c$ if $t \leq t_{0}$. Then use the semigroup property.