Math Problems Vault

Problem 8.1

(a)(b)(c)

Let $f$ be the initial value in the problem considered in Lemma 8.1 and write $u(t, x): = P_{t} f(x) = \mathbb {E} f\left(B_{t}+x\right)$ where $\left(B_{t}\right)_{t \geqslant 0}$ is a BM ${ }^{d}$ .

?

(a)

Assume that $f \in \mathcal{C}_{\infty }\left(\mathbb {R}^{d}\right)$ . Show that $\partial_{t} u(t, x), \partial_{j} u(t, x)$ and $\partial_{j} \partial_{k} u(t, x)$ exist and are functions in $\mathcal{C}\left((0, \infty ) \times \mathbb {R}^{d}\right)$ .

(b)

Assume that $f: \mathbb {R}^{d} \rightarrow \mathbb {R}$ is measurable and exponentially bounded, i.e. $\left|f(x)\right| \leqslant C e^{C\left|x\right|}$ for all $x \in \mathbb {R}^{d}$ and some constant $C$ . Show that $\partial_{t} u(t, x), \partial_{j} u(t, x)$ and $\partial_{j} \partial_{k} u(t, x), j, k = 1, \ldots , d$ , exist and are exponentially bounded functions in $\mathcal{C}\left((0, \infty ) \times \mathbb {R}^{d}\right)$ .

(c)

What happens, if we assume in Part b) that $f(x) \leqslant C e^{C\left|x\right|^{2}}$ ?

Problem 8.2

Assume that in Lemma 8.1 the initial datum satisfies $f \in \mathcal{C}_{\infty }\left(\mathbb {R}^{d}\right)$ but not necessarily $f \in \mathfrak {D}(A)$ . Consider (8.3) with $P_{\epsilon } f$ instead of $f$ . Discuss the limit $\epsilon \rightarrow 0$ .

?

Problem 8.3

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{d}, f: \mathbb {R}^{d} \rightarrow \mathbb {R}$ be a continuous function such that $\int_{0}^{t} f\left(B_{s}\right) d s = 0$ for all $t > 0$ . Show that $f\left(B_{s}\right) = 0$ for all $s > 0$ , and conclude that $f \equiv 0$ .

?

Problem 8.4

(a)(b)(c)

Complete the approximation argument for Lévy's arc - sine law from $\S 8.9$ :

?

(a)

Show, by a direct calculation, that $v_{n, \lambda }(x)$ converges as $n \rightarrow \infty$ . Conclude from (8.16) that $v_{n, \lambda }^{\prime \prime }$ converges.

(b)

Integrate the ODE (8.16) to see that $v_{n, \lambda }(x) - v_{n, \lambda }^{\prime }(0)$ has a limit. Integrating once again shows that $v_{n, \lambda }^{\prime }(0)$ converges, too.

(c)

Identify the limits of $v_{n, \lambda }, v_{n, \lambda }^{\prime }$ and $v_{n, \lambda }^{\prime \prime }$ .

Problem 8.5

Show Theorem 8.5 with semigroup methods.

?

Problem 8.6

Find the solution to the Dirichlet problem in dimension $d = 1: u^{\prime \prime }(x) = 0$ for all $x \in (0,1), u(0) = a, u(1) = b$ and $u$ is continuous in [0,1]. Compare your findings with Wald's identities, cf. Theorem 5.10 and Corollary 5.11.

?

Problem 8.7

Use Lemma 7.33 and give an alternative derivation of the result of Lemma 8.10.

?

Problem 8.8

Show that Lemma 8.10 remains true for any $d$ - dimensional Feller process $X_{t}$ with continuous paths and generator $L = \sum_{j, k = 1}^{d} a_{j k}(x) \partial_{j} \partial_{k}+\sum_{j = 1}^{d} b_{j}(x) \partial_{j}$ such that $a_{11}(x) \geqslant a_{0} > 0$ and $\left|b_{1}(x)\right| \leqslant b_{0} < \infty$ .

?

Problem 8.9

Table 8.1 No . 2 records the solution to the initial value problem (8.6). Explain how this result follows from our considerations regarding (8.3) and (8.7).

?

Problem 8.10

Let $g:[0, \infty ) \times \mathbb {R}^{d} \rightarrow \mathbb {R}$ be a bounded continuous function such that $g(t, \cdot )$ is $\kappa$ - Hölder continuous with a Hölder constant which does not depend on $t$ or $x$ . Mimic the proof of Lemma 8.7 to prove that $v(t, x): = \mathbb {E}^{x}\left(\int_{0}^{t} g\left(t - s, B_{s}\right) d s\right)$ is in $\mathcal{C}^{1,2}\left((0, \infty ) \times \mathbb {R}^{d}\right) \cap \mathcal{C}_{b}\left([0, \infty ) \times \mathbb {R}^{d}\right)$ and satisfies (8.1). Explain why this proves, under suitable assumptions on $g(t, x)$ , No. 3. in Table 8.1.

?

Problem 8.11

Use the LIL (Corollary 12.2) to give an alternative proof for the fact that a one - dimensional Brownian motion oscillates in every time interval $[0, \epsilon ]$ infinitely often around its starting point.

?

Problem 8.12

Let $d \geqslant 2$ . A flat cone in $\mathbb {R}^{d}$ is a cone in $\mathbb {R}^{d - 1}$ . Adapt the argument of Example 8.18.e) to show the following useful regularity criterion for a $\mathrm{BM}^{d}$ : The boundary point $x_{0}$ is regular for $D^{c}$ if there is a truncated flat cone with vertex at $x_{0}$ and lying entirely in $D^{c}$ .

?

Problem 8.13

Let $(b, \beta )$ be a $\mathrm{BM}^{2}$ and $\sigma_{n} = \inf \left\{ t > 1 / n: b(t) = 0\right\}$ . Show that the random variable $\beta \left(\sigma_{n}\right)$ has a probability density.

?