Math Problems Vault

Problem 23.1

Show that if $f$ is excessive, then $1-e^{-f}$ is excessive. Thus, for some purposes it is enough to look at bounded excessive functions.

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Problem 23.2

Show that if $f$ and $g$ are excessive, then $f \wedge g$ is excessive.

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Problem 23.3

Let $A_{t}$ be an additive functional (defined in (22.4)) and let $f(x)=\mathbb {E}_{x}\left[A_{\infty }\right]$ . Show that $f$ is excessive.

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Problem 23.4

Let $f$ be an excessive function for a strong Markov process $\left(X_{t}, \mathbb {P}[x]\right)$ . Let $\varepsilon >0$ and $S_{1}=\inf \left\{ t : \left|f\left(X_{t}\right)-f\left(X_{0}\right)\right| \geq \varepsilon \right\}$ . Let $S_{i+1}=S_{i}+S_{1} \circ \theta_{S_{i}}$ . Prove that $f\left(X_{S_{i}}\right)$ is a supermartingale with respect to the $\sigma$ -fields $\mathcal{F}_{S_{i}}$ and with respect to $\mathbb {P}[x]$ for each $x$ .

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Problem 23.5

For each $n$ , let $A_{t}^{n}$ be an additive functional with continuous paths and suppose that $f_{n}(x)=\mathbb {E}_{x}\left[A_{\infty }^{n}\right]$ is finite for every $x$ . Suppose $A_{t}$ is a continuous additive functional with $f(x)=\mathbb {E}_{x}\left[A_{\infty }\right]$ also finite for each $x$ . Suppose $f_{n}$ converges to $f$ uniformly. Prove that for each $x$ , with $\mathbb {P}[x]$ -probability one, $A_{t}^{n}$ converges to $A_{t}$ , uniformly over $t \geq 0$ .

Hint: Use Proposition 9.11.

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Problem 23.6

Suppose $f$ is bounded and excessive, $\lambda \geq 0$ , and $A=\left\{ y: f(y) \leq \lambda \right\}$ . Prove that if $x \in A^{r}$ (i.e., $x$ is regular for $A$ ), then $f(x) \leq \lambda$ .

Hint: Use the optional section theorem (Theorem 16.12) to find stopping times $T_{m}$ whose graphs are contained in $\left\{ (t, \omega ): t \leq 1 / m, f\left(X_{t}\right) \leq \lambda \right\}$ with $\mathbb {P}[x]$ -probability at least $1-(1 / m)$ .

Lemma 23.1: If $f$ is excessive, there exist functions $g_{n} \geq 0$ such that $U g_{n}$ increases up to $f$ , where $U g_{n}(x) = \mathbb {E}_{x}\left[\int_0^\infty g_n(X_s)\, ds\right]$ .

If the $g_{n}$ are as in Lemma 23.1, write

\begin{aligned} U g_{n}(x) & =\mathbb {E}_{x}\left[\int _{0}^{T_{m}} g_{n}\left(X_{s}\right) d s\right]+\mathbb {E}_{x}\left[U g_{n}\left(X_{T_{m}}\right)\right] \\ & \leq \mathbb {E}_{x}\left[\int _{0}^{T_{m}} g_{n}\left(X_{s}\right) d s\right]+\mathbb {E}_{x}\left[f\left(X_{T_{m}}\right)\right] \\ & \leq \mathbb {E}_{x}\left[\int _{0}^{T_{m}} g_{n}\left(X_{s}\right) d s\right]+\lambda +\left\| f\right\| _{\infty } / m \end{aligned}

Let $m \rightarrow \infty$ , then $n \rightarrow \infty$ .

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Problem 23.7

Suppose $f$ is bounded and excessive, $\lambda \geq 0$ , and $B=\left\{ y: f(y) \geq \lambda \right\}$ . Prove that if $x \in B^{r}$ , then $f(x) \leq \lambda$ .

Hint: Use the optional stopping theorem as in Exercise 23.6 to find stopping times $R_{m}$ analogous to the $T_{m}$ . Write

f(x) \geq \mathbb {E}_{x}\left[f\left(X_{R_{m}}\right)\right] \geq \lambda -\left\| f\right\| _{\infty } / m

and then let $m \rightarrow \infty$ .

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Problem 23.8

(1)(2)(3)(4)

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(1)

Suppose $f$ is bounded and excessive, $x \in \mathcal{S}, \varepsilon >0$ , and $C=\left\{ y:\left|f(y)-f(x)\right| \geq \varepsilon \right\}$ . Use Exercises 23.6 and 23.7 to show that if $z \in C^{r}$ , then $\left|f(z)-f(x)\right| \geq \varepsilon$ .

(2)

Let $f, \varepsilon$ , and $x$ be as in (1) and set $S=\inf \left\{ t>0:\left|f\left(X_{t}\right)-f(x)\right| \geq \varepsilon \right\}$ . Use Exercise 20.9 to show that $\left|f\left(X_{S}\right)-f(x)\right| \geq \varepsilon$ with $\mathbb {P}[x]$ -probability one.

(3)

Let $f, \varepsilon , x$ , and $S$ be as in (2). Define $S=0$ and $S_{i+1}=S_{i}+S \circ \theta_{S_{i}}$ . By Exercise 23.4, $f\left(X_{S_{i}}\right)$ is a positive supermartingale. Use Corollary A. 36 to show $S_{i} \rightarrow \infty , \mathbb {P}[x]$ -a.s. Deduce that with $\mathbb {P}[x]$ -probability one, $f\left(X_{t}\right)$ has paths that are right continuous with left limits.

(4)

Use Exercise 23.1 to show that if $f$ is excessive but not necessarily bounded, then $f\left(X_{t}\right)$ has paths that are right continuous with left limits.

Problem 23.9

(1)(2)(3)

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(1)

Show that every continuous function is lower semicontinuous.

(2)

Show that if $f$ is lower semicontinuous and $x \in \mathcal{S}$ , then

\liminf _{y \rightarrow x} f(y) \geq f(x)

(3)

Show that if $f_{n}$ is a sequence of continuous functions increasing to $f$ , then $f$ is lower semicontinuous.

Problem 23.10

Suppose $g$ is non-negative, bounded, and continuous, and Assumption 20.1 holds. Let $g_{0}=g$ and define $g_{n}(x)=\sup_{t \geq 0} P_{t} g_{n-1}(x)$ for $n \geq 1$ . Prove that $g_{n}$ increases to the least excessive majorant of $g$ .

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