Math Problems Vault

Problem 11.1

Verify that s - dimensional Hausdorff measure is an outer measure.

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Problem 11.2

Show that the $d$ - dimensional Hausdorff measure of a bounded open set $U \subset \mathbb {R}^{d}$ is finite and positive: $0 < \mathscr {H}^{d}(U) < \infty$ . In particular, $\operatorname {dim} U = d$ .

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Problem 11.3

(a)(b)

Let $E \subset \mathbb {R}^{d}$ . Show that Hausdorff dimension $\operatorname {dim} E$ coincides with the numbers

\begin{array}{cll} \sup \left\{ \alpha \geqslant 0: \mathscr {H}^{\alpha }(E) = \infty \right\} , & \sup \left\{ \alpha \geqslant 0: \mathscr {H}^{\alpha }(E) > 0\right\} \\ \inf \left\{ \alpha \geqslant 0: \mathscr {H}^{\alpha }(E) < \infty \right\} , & \inf \left\{ \alpha \geqslant 0: \mathscr {H}^{\alpha }(E) = 0\right\} . \end{array}

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(a)

(b)

Problem 11.4

Let $\left(E_{j}\right)_{j \geqslant 1}$ be subsets of $\mathbb {R}^{d}$ and $E: = \bigcup_{j \geqslant 1} E_{j}$ . Show that $\operatorname {dim} E = \sup_{j \geqslant 1} \operatorname {dim} E_{j}$ .

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Problem 11.5

Let $E \subset \mathbb {R}^{d}$ . Show that $\operatorname {dim}\left(E \times \mathbb {R}^{n}\right) = \operatorname {dim} E+n$ . Does a similar formula hold for $E \times F$ where $E \subset \mathbb {R}^{d}$ and $F \subset \mathbb {R}^{n}$ ?

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Problem 11.6

Let $f: \mathbb {R}^{d} \rightarrow \mathbb {R}^{n}$ be a bi - Lipschitz map, i.e. both $f$ and $f^{ - 1}$ are Lipschitz continuous. Show that $\operatorname {dim} f(E) = \operatorname {dim} E$ . Is this also true for a Hölder continuous map with index $\gamma \in (0,1)$ ?

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Problem 11.7

Let $C$ denote Cantor's discontinuum which is obtained if we remove recursively the open middle third of any remaining interval:

[0,1] \leadsto \left[0, \frac{1}{3}\right] \cup \left[\frac{2}{3}, 1\right] \leadsto \left[0, \frac{1}{9}\right] \cup \left[\frac{2}{9}, \frac{1}{3}\right] \cup \left[\frac{2}{3}, \frac{7}{9}\right] \cup \left[\frac{8}{9}, 1\right] \leadsto \ldots

see e.g. [232, Problem 7.12] or [236, § 2.5]. Show that $C = f_{1}(C) \cup f_{2}(C)$ with $f_{1}(x) = \frac{1}{3} x$ and $f_{2}(x) = \frac{1}{3} x+\frac{2}{3}$ and $\operatorname {dim} C = \log 2 / \log 3$ .

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Problem 11.8

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{d}$ . Show that for $\lambda < d$ the expectation $\mathbb {E}\left[\left|B_{1}\right|^{ - \lambda }\right]$ is finite.

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Problem 11.9

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a one - dimensional Brownian motion. Show that

\operatorname {dim} B^{ - 1}(A) \leqslant \frac{1}{2}+\frac{1}{2} \operatorname {dim} A \quad \text{ a.s. for all } A \in \mathscr {B}(\mathbb {R})

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Problem 11.10

Let $F \subset \mathbb {R}$ be a non - void perfect set, i.e. a closed set such that each point in $F$ is an accumulation point of $F$ . Show that a perfect set is uncountable.

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Problem 11.11

(a)(b)(c)

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$ and denote by $g_{t}$ the last zero before time $t > 0$ .

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(a)

Use Theorem 11.25 to give a further proof of Lévy's arc - sine law (cf. Theorem 6.20): $\mathbb {P}\left(g_{t} < s\right) = \frac{2}{\pi } \arcsin \sqrt{\frac{s}{t}}$ for all $s \in [0, t]$ .

(b)

Find the conditional density $\mathbb {P}\left(\left|B_{t}\right| \in d y \mid g_{t} = s\right)$ .

(c)

Integrate the formula appearing in Theorem 11.25 to get

\sqrt{\frac{2}{\pi t}} \exp \left( - \frac{y^{2}}{2 t}\right) = \int _{0}^{t} \sqrt{\frac{2}{\pi s}} \frac{y}{\sqrt{2 \pi (t - s)^{3}}} \exp \left( - \frac{y^{2}}{2(t - s)}\right) d s

and give a probabilistic interpretation of this identity.

Problem 11.12

Show that Corollary 11.26 also follows from Lemma 11.23.

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Problem 11.13

Let $\left(B_{t}\right)_{t \geqslant 0}, d_{t}$ and $g_{t}$ be as in Corollary 11.26. Define

L_{t}^{ - }: = t - g_{t} \quad \text{ and } \quad L_{t}: = d_{t} - g_{t}

Find the laws of $\left(L_{t}^{ - }, L_{t}\right), L_{t}^{ - }, L_{t}$ and $L_{t}$ conditional on $L_{t}^{ - } = r$ .

Find the probability that $\mathbb {P}\left(L_{t} > r+s \mid L_{t}^{ - } = r\right)$ , i.e. the probability that an excursion continues for $s$ units of time if we know that it is already $r$ units of time under way.

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