Math Problems Vault

Problem 7.1

Show that $\mathcal{C}_{\infty }: = \left\{ u: \mathbb {R}^{d} \rightarrow \mathbb {R} \text{ continuous and } \lim_{\left|x\right| \rightarrow \infty } u(x) = 0\right\}$ equipped with the uniform topology is a Banach space. Show that in this topology $\mathcal{C}_{\infty }$ is the closure of $\mathcal{C}_{c}$ , i.e. the family of continuous functions with compact support.

?

Problem 7.2

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{d}$ with transition semigroup $\left(P_{t}\right)_{t \geqslant 0}$ . Show, using arguments similar to those in the proof of Proposition 7.3.f), that $(t, x, u) \mapsto P_{t} u(x)$ is continuous. As usual, we equip $[0, \infty ) \times \mathbb {R}^{d} \times \mathcal{C}_{\infty }\left(\mathbb {R}^{d}\right)$ with the norm $\left|t\right|+\left|x\right|+\left\| u\right\|_{\infty }$ .

?

Problem 7.3

Let $\left(X_{t}\right)_{t \geqslant 0}$ be a $d$ - dimensional Feller process and let $f, g \in \mathcal{C}_{\infty }\left(\mathbb {R}^{d}\right)$ . Show that the function $x \mapsto \mathbb {E}^{x}\left(f\left(X_{t}\right) g\left(X_{t+s}\right)\right)$ is also in $\mathcal{C}_{\infty }\left(\mathbb {R}^{d}\right)$ .

?

Problem 7.4

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{d}$ and set $u(t, x): = P_{t} u(x)$ . Adapt the proof of Proposition 7.3.g) and show that for $u \in \mathcal{B}_{b}\left(\mathbb {R}^{d}\right)$ the function $u(t, \cdot ) \in \mathcal{C}^{\infty }$ and $u \in \mathcal{C}^{1,2}$ (i.e. once continuously differentiable in $t$ , twice continuously differentiable in $x$ ).

?

Problem 7.5

(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 7.6

Let $\left(T_{t}\right)_{t \geqslant 0}$ be a Markov semigroup given by $T_{t} u(x) = \int_{\mathbb {R}^{d}} u(y) p_{t}(x, d y)$ where $p_{t}(x, C)$ is a kernel in the sense of Remark 7.6. Show that the semigroup property of $\left(T_{t}\right)_{t \geqslant 0}$ entails the Chapman - Kolmogorov equations

p_{t+s}(x, C) = \int p_{t}(y, C) p_{s}(x, d y) \quad \text{ for all } \quad x \in \mathbb {R}^{d} \quad \text{ and } \quad C \in \mathscr {B}\left(\mathbb {R}^{d}\right)

?

Problem 7.7

Show that $p_{t_{1}, \ldots , t_{n}}^{x}\left(C_{1} \times \cdots \times C_{n}\right)$ of Remark 7.6 define measures on $\mathscr {B}\left(\mathbb {R}^{d \cdot n}\right)$ .

?

Problem 7.8

(a)(b)(c)

?

(a)

Let $d(x, A): = \inf_{a \in A}\left|x - a\right|$ be the distance between the point $x \in \mathbb {R}^{d}$ and the set $A \in \mathscr {B}\left(\mathbb {R}^{d}\right)$ . Show that $x \mapsto d(x, A)$ is continuous.

(b)

Let $u_{n}$ be as in (7.7). Show that $u_{n} \in \mathcal{C}_{c}\left(\mathbb {R}^{d}\right)$ and $\inf_{n} u_{n}(x) = \; \mathbb {1}_{K}(x)$ . Draw a picture if $d = 1$ and $K = [0,1]$ .

(c)

Let $\chi_{n}$ be a sequence of type $\delta$ , i.e. $\chi_{n} \in \mathcal{C}_{c}^{\infty }\left(\mathbb {R}^{d}\right)$ , $\operatorname {supp} \chi_{n} \subset \mathbb {B}(0,1 / n), \chi_{n} \geqslant 0$ and $\int \chi_{n}(x) d x = 1$ . Show that $v_{n}: = \chi_{n} \star u_{n} \in \mathcal{C}_{c}^{\infty }\left(\mathbb {R}^{d}\right)$ are smooth functions such that $\lim_{n \rightarrow \infty } v_{n} = \; \mathbb {1}_{K}$ .

Problem 7.9

(a)(b)(c)

Let $\left(T_{t}\right)_{t \geqslant 0}$ be a Feller semigroup, i.e. a strongly continuous, positivity preserving sub - Markovian semigroup on $\left(\mathcal{C}_{\infty }\left(\mathbb {R}^{d}\right),\left\| \cdot \right\|_{\infty }\right)$ , cf. Remark 7.4.

?

(a)

Show that $(t, x) \mapsto T_{t} u(x)$ is jointly continuous for $t \geqslant 0$ and $x \in \mathbb {R}^{d}$ .

(b)

Conclude that for every compact set $K \subset \mathbb {R}^{d}$ the function $(t, x) \mapsto T_{t} \; \mathbb {1}_{K}(x)$ is measurable.

(c)

Show that $(t, x, u) \mapsto T_{t} u(x)$ is jointly continuous for all $t \geqslant 0, x \in \mathbb {R}^{d}$ and $u \in \mathcal{C}_{\infty }\left(\mathbb {R}^{d}\right)$ .

Problem 7.10

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

Problem 7.11

(a)(b)(c)

Let $\left(P_{t}\right)_{t \geqslant 0}$ and $\left(T_{t}\right)_{t \geqslant 0}$ be two Feller semigroups with generators $(A, \mathfrak {D}(A))$ , resp. $(B, \mathfrak {D}(B))$ .

?

(a)

If $\mathfrak {D}(B) \subset \mathfrak {D}(A)$ , then we have $\frac{d}{d s} P_{t - s} T_{s} = - P_{t - s} A T_{s}+P_{t - s} B T_{s}$ on $\mathfrak {D}(B)$ .

(b)

Is it possible that $U_{t}: = T_{t} P_{t}$ is again a Feller semigroup?

(c)

(Trotter product formula) Assume that $U_{t} f: = \lim_{n \rightarrow \infty }\left(T_{t / n} P_{t / n}\right)^{n} f$ exists strongly and locally uniformly for $t$ . Show that $U_{t}$ is a Feller semigroup and determine its generator.

Problem 7.12

(a)(b)(c)

Complete the following alternative argument for Example 7.16. Assume that $\left(u_{n}\right)_{n \geqslant 1} \subset \mathcal{C}_{\infty }^{2}(\mathbb {R})$ such that $\left(\frac{1}{2} u_{n}^{\prime \prime }\right)_{n \geqslant 1}$ is a Cauchy sequence in $\mathcal{C}_{\infty }$ and that $u_{n}$ converges uniformly to $u$ .

?

(a)

Show that $u_{n}(x) - u_{n}(0) - x u_{n}^{\prime }(0) = \int_{0}^{x} \int_{0}^{y} u_{n}^{\prime \prime }(z) d z d y \rightarrow \int_{0}^{x} \int_{0}^{y} 2 g(z) d z d y$ uniformly. Conclude that $c^{\prime }: = \lim_{n \rightarrow \infty } u_{n}^{\prime }(0)$ exists.

(b)

Show that $u_{n}^{\prime }(x) - u_{n}^{\prime }(0) \rightarrow \int_{0}^{x} 2 g(z) d z$ uniformly. Deduce from this and part a) that $u_{n}^{\prime }(x)$ converges uniformly to $\int_{0}^{x} 2 g(z) d z+c^{\prime }$ where $c^{\prime } = \int_{ - \infty }^{0} 2 g(z) d z$ .

(c)

Use a) and b) to show that $u_{n}(x) - u_{n}(0) \rightarrow \int_{0}^{x} \int_{ - \infty }^{y} 2 g(z) d z$ uniformly and argue that $u_{n}(x)$ has the uniform limit $\int_{ - \infty }^{x} \int_{ - \infty }^{y} 2 g(z) d z d y$ .

Problem 7.13

Let $U_{\alpha }$ be the $\alpha$ - potential operator of a $\mathrm{BM}^{d}$ . Give a probabilistic interpretation of $U_{\alpha } \; \mathbb {1}_{C}$ and $\lim_{\alpha \rightarrow 0} U_{\alpha } \; \mathbb {1}_{C}$ if $C \subset \mathbb {R}^{d}$ is a Borel set.

?

Problem 7.14

Let $U_{0}$ be the potential operator of a $\mathrm{BM}^{d}$ in dimension $d = 1$ or $d = 2$ . Show that every $u \in \mathfrak {D}\left(U_{0}\right)$ such that $u \geqslant 0$ is trivial, i.e. $u = 0$ .

?

Problem 7.15

Let $\left(U_{\alpha }\right)_{\alpha > 0}$ be the $\alpha$ - potential operator of a $\mathrm{BM}^{d}$ . Use the resolvent equation to prove the following formulae for $f \in \mathcal{B}_{b}$ and $x \in \mathbb {R}^{d}$ :

\frac{d^{n}}{d \alpha ^{n}} U_{\alpha } f(x) = n !( - 1)^{n} U_{\alpha }^{n+1} f(x)

and

\frac{d^{n}}{d \alpha ^{n}}\left(\alpha U_{\alpha }\right) f(x) = n !( - 1)^{n+1}\left(\mathrm{id} - \alpha U_{\alpha }\right) U_{\alpha }^{n} f(x)

?

Problem 7.16

Let $\left(f_{n}\right)_{n \geqslant 1} \subset \mathcal{C}_{\infty }\left(\mathbb {R}^{d}\right)$ be a sequence of functions such that $0 \leqslant f_{n} \leqslant f_{n+1}$ and $f: = \sup_{n} f_{n} \in \mathcal{C}_{\infty }\left(\mathbb {R}^{d}\right)$ . Show that $\lim_{n \rightarrow \infty }\left\| f - f_{n}\right\|_{\infty } = 0$ .

?

Remark.

?

Positivity is in fact not needed. If instead $f_{1} \leqslant f_{2} \leqslant \cdots \leqslant f_{n} \leqslant \cdots$ , $f_{n}\in \mathcal{C}_{\infty }\left(\mathbb {R}^{d}\right)$ of any sign, and $f=\sup_{n}f_{n}$ , then by Dini's theorem $\lim_{n\rightarrow \infty }\sup_{x\in K}\left|f(x)-f_{n}(x)\right|=0$ for any compact $K\subset \mathbb {R}^{d}$ ; fixing $\varepsilon >0$ and a compact $K_{\varepsilon }$ with $-\varepsilon \leqslant f_{1}(x)\leqslant f_{n}(x)\leqslant f(x)\leqslant \varepsilon$ on $\mathbb {R}^{d}\setminus K_{\varepsilon }$ gives $\limsup_{n}\sup_{x}\left|f_{n}(x)-f(x)\right| \leqslant \limsup_{n}\sup_{x\in K_{\varepsilon }}\left|f_{n}(x)-f(x)\right|+\limsup_{n}\sup_{x\notin K_{\varepsilon }}(f(x)-f_{n}(x)) \leqslant 0+2\varepsilon \leqslant 3\varepsilon$ , i.e. the same conclusion holds without positivity. (Alternatively: apply the positive-case argument to $g_{n}:=f_{n}-f_{1}$ , using $0\leqslant g_{n}\leqslant g_{n+1}$ and $\sup_{n}g_{n}=f-f_{1}$ .)

Problem 7.17

Let $(A, \mathfrak {D}(A))$ be the generator of a $\mathrm{BM}^{d}$ . Adapt the arguments of Example 7.25 and show that $\mathcal{C}_{\infty }^{2}\left(\mathbb {R}^{d}\right) \varsubsetneqq \mathfrak {D}(A)$ for any dimension $d \geqslant 3$ . Instructions. Use in Example $7.25 d$ - dimensional polar coordinates: $y_{1} = r \sin \phi \sin \theta_{1} \cdot \ldots \cdot \sin \theta_{d - 2}, y_{2} = r \cos \phi \sin \theta_{1} \cdot \ldots \cdot \sin \theta_{d - 2}, \ldots y_{d - 1} = r \cos \theta_{d - 3} \cdot \sin \theta_{d - 2}$ , $y_{d} = r \cos \theta_{d - 2}$ , where $\left(r, \phi , \theta_{1}, \ldots , \theta_{d - 2}\right) \in (0, \infty ) \times (0,2 \pi ) \times (0, \pi )^{d - 2}$ .

The functional determinant is

r^{d - 1}\left(\sin \theta _{1}\right)^{1}\left(\sin \theta _{2}\right)^{2} \ldots \left(\sin \theta _{d - 2}\right)^{d - 2}

The calculations will lead to the following integrals $(\left|x\right| < 1)$

\int _{0}^{\pi } \frac{(\sin \theta )^{d - 2}}{\left(1+x^{2} - 2 x \cos \theta \right)^{(d - 2) / 2}} d \theta \text{ and } \int _{0}^{\pi } \frac{(\sin \theta )^{d}}{\left(1+x^{2} - 2 x \cos \theta \right)^{(d - 2) / 2}} d \theta

which have the values $a_{d}$ and $b_{d} x^{2}+c_{d}$ , respectively, with constants $a_{d}, b_{d}, c_{d} \in \mathbb {R}$ depending on the dimension, cf. Gradshteyn - Ryzhik [101, 3.665.2, p. 384].

Remark. A simple evaluation of the integrals ${ }^{\star }$ ) is possible only in $d = 3$ . For $d > 3$ these integrals are most elegantly evaluated using Gegenbauer polynomials, cf. [101, 8.93, pp. 1029 - 1031] and the discussion on stack exchange http://math.stackexchange.com/questions/395336.

?

Problem 7.18

(a)(b)(c)

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$ and consider the two - dimensional process $X_{t}: = \left(t, B_{t}\right), t \geqslant 0$ .

?

(a)

Show that $\left(X_{t}\right)_{t \geqslant 0}$ is a Feller process.

(b)

Determine its transition semigroup, resolvent and generator.

(c)

State and prove Theorem 7.30 for this process and compare the result with Theorem 5.6

Problem 7.19

Show that Dynkin's formula (7.30) in Proposition 7.31 follows from Theorem 7.30.

?

Problem 7.20

Let $t \mapsto X_{t}$ be a right continuous stochastic process. Show that for closed sets $F$

\mathbb {P}\left(X_{t} \in F \quad \forall t \in \mathbb {R}^{+}\right) = \mathbb {P}\left(X_{q} \in F \quad \forall q \in \mathbb {Q}^{+}\right)

?

Problem 7.21

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

Assume that $L: \mathcal{C}_{c}^{\infty }\left(\mathbb {R}^{d}\right) \rightarrow \mathcal{C}\left(\mathbb {R}^{d}\right)$ is a local operator (cf. Definition 7.36) which is almost positive, i.e. for all $\phi \in \mathcal{C}_{c}^{\infty }$ such that $\phi \geqslant 0$ and $\phi \left(x_{0}\right) = 0$ we have $L \phi \left(x_{0}\right) \geqslant 0$ .

?

(a)

Show that an operator that satisfies the positive maximum principle is almost positive. (This does not require locality).

(b)

Show that $\operatorname {supp} L \phi \subset \operatorname {supp} \phi$ .

(c)

Show that the operator $L$ is of order two, i.e. $\left\| L \phi \right\|_{\infty } \leqslant C\left\| \phi \right\|_{(2)}$ holds for all $\phi \in \mathcal{C}_{c}^{\infty }(K)$ and any compact set $K \subset \mathbb {R}^{d}$ . Here, $\left\| \phi \right\|_{(2)} = \sum_{0 \leqslant \left|\alpha \right| \leqslant 2}\left\| \partial^{\alpha } \phi \right\|_{\infty }$ and the constant $C$ depends only on $K$ .

(d)

Use Theorem 7.37 to conclude that $L$ is a differential operator of order $\leqslant 2$ .