Concrete semigroups and their infinitesimal generators

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Problem 3.13.22 from Rudin W. "Functional Analysis" [Rud91]More problems in Chapter 3.13 "Unbounded Operators"

If $f \in H^{2}$ (see Exercise 9) and $f(z) = \sum c_{n} z^{n}$ , define

[Q(t) f](z) = \sum _{n = 0}^{\infty }(n+1)^{-t} c_{n} z^{n} \quad (0 \leq t < \infty )

Show that each $Q(t)$ is self-adjoint (and positive). Find the infinitesimal generator $A$ of the semigroup $\left\{ Q(t)\right\}$ . Is $A$ self-adjoint? Show that $A$ has pure point spectrum, at the points $\log 1, \log \frac{1}{2}, \log \frac{1}{3}, \ldots$ .

Problem 3.13.23 from Rudin W. "Functional Analysis" [Rud91]More problems in Chapter 3.13 "Unbounded Operators"

For $f \in L^{2}(R), x \in R, 0 < y < \infty$ , define

[Q(y) f](x) = \frac{1}{\pi } \int _{-\infty }^{\infty } \frac{y}{(x-\xi )^{2}+y^{2}} f(\xi ) d \xi

and put $Q(0) f = f$ . Show that $\left\{ Q(y): 0 \leq y < \infty \right\}$ satisfies the conditions imposed in Definition 13.34 and that $\left\| Q(y)\right\| = 1$ for all $y$ .

Find the domain of the infinitesimal generator $A$ of $\left\{ Q(y)\right\}$ , and prove that

A f = -H f^{\prime }

where $H$ is the Hilbert transform (Chapter 7, Exercise 24). Prove that $-A$ is positive and self-adjoint.

Remark.

The integral represents a harmonic function in the upper half-plane, with boundary values $f$ . The semigroup property of $\left\{ Q(y)\right\}$ can be deduced from this, as well as from a look at the Fourier transforms of the functions $Q(y) f$ .

Problem 7.5 from Schilling R.L. "Brownian Motion: A Guide to Random Processes and Stochastic Calculus" [Sch21]More problems in Chapter 7 "Brownian motion and transition semigroups"

(a)(b)(c)(d)

Let $\left(P_{t}\right)_{t \geqslant 0}$ be the transition semigroup of a $\mathrm{BM}^{d}$ and denote by $L^{p}, 1 \leqslant p < \infty$ , the space of $p$ th power integrable functions with respect to Lebesgue measure on $\mathbb {R}^{d}$ . Set $u_{n}: = ( - n) \vee u \wedge n$ for every $u \in L^{p}$ . Verify that

(a)

$u_{n} \in \mathcal{B}_{b}\left(\mathbb {R}^{d}\right) \cap L^{p}\left(\mathbb {R}^{d}\right)$ and $L^{p} - \lim_{n \rightarrow \infty } u_{n} = u$ .

(b)

$\widetilde{P}_{t} u: = \lim_{n \rightarrow \infty } P_{t} u_{n}$ extends $P_{t}$ and gives a contraction semigroup on $L^{p}$ .

(c)

$\widetilde{P}_{t}$ is sub - Markovian, i.e. if $u \in L^{p}$ and $0 \leqslant u \leqslant 1$ a.e., then $0 \leqslant \widetilde{P}_{t} u \leqslant 1$ a. e.

(d)

$\widetilde{P}_{t}$ is strongly continuous, i.e. $\lim_{t \rightarrow 0}\left\| \widetilde{P}_{t} u - u\right\|_{L^{p}} = 0$ for all $u \in L^{p}$ .

Problem 7.10 from Schilling R.L. "Brownian Motion: A Guide to Random Processes and Stochastic Calculus" [Sch21]More problems in Chapter 7 "Brownian motion and transition semigroups"

(a)(b)(c)(d)

Let $A, B \in \mathbb {R}^{d \times d}$ and set $P_{t}: = \exp (t A): = \sum_{j = 0}^{\infty }(t A)^{j} / j$ !.

(a)

Show that $P_{t}$ is a strongly continuous semigroup. Is it contractive?

(b)

Show that $\frac{d}{d t} e^{t A}$ exists and that $\frac{d}{d t} e^{t A} = A e^{t A} = e^{t A} A$ .

(c)

Show that $e^{t A} e^{t B} = e^{t B} e^{t A} \forall t \Longleftrightarrow A B = B A$ .

(d)

(Trotter product formula) Show that $e^{A+B} = \lim_{k \rightarrow \infty }\left(e^{A / k} e^{B / k}\right)^{k}$ .

References

[Rud91] Rudin W. Functional Analysis. 2nd ed., McGraw-Hill, 1991.

[Sch21] Schilling R.L. Brownian Motion: A Guide to Random Processes and Stochastic Calculus. 3rd ed., De Gruyter, 2021.