Applications of the Markov properties

Suppose $d=2,\left(X_{t}, \mathbb {P}[x]\right)$ is a two-dimensional Brownian motion, and $r>0$. Imitate the argument of Proposition 21.1 but with $h(x)=\log \left(\left|x\right|\right)$ to show that $\mathbb {P}_{x}\left(X_{t} \text{ hits } B(0, r)\right)=1$ when $\left|x\right|>r$. Then use the strong Markov property to show that there are times $T_{i} \rightarrow \infty$ with $X_{T_{i}} \in B(0, r)$. That is, two-dimensional Brownian motion is neighborhood recurrent.

In the proof of Lemma 21.4, justify each inequality in (21.6).

Lemma 21.4: Let $\left(X_{t}, \mathbb {P}[x]\right)$ be a strong Markov process with state space $\mathcal{S}$. For all $x \in \mathcal{S}$ and all $\lambda \geq 0$,

From the proof: writing $F(t, \lambda )=\sup_{s \leq t} \sup_{y \in \mathcal{S}} \mathbb {P}_{y}\left(d\left(X_{s}, X_{0}\right) \geq \lambda \right)$ and $S=\inf \left\{ t: d\left(X_{t}, X_{0}\right) \geq \lambda \right\}$, then by the strong Markov property,

Let $\left(X_{t}, \mathbb {P}[x]\right)$ be a Poisson process with parameter $a$ and let $F$ be defined by (21.5). Show $F(t, 1 / 2) / t$ does not converge to 0 as $t \rightarrow 0$.

Suppose $d \geq 3,\left(X_{t}, \mathbb {P}[x]\right)$ is a $d$-dimensional Brownian motion, and

Show that if $f$ is bounded and measurable with compact support, then $U f$ is continuous and $\left|U f(x)\right| \rightarrow 0$ as $\left|x\right| \rightarrow \infty$. Show that if $f \in C^{2}$ with compact support, then $U f$ is $C^{2}$. Show that $\frac{1}{2} \Delta U f=-f$.

Let $W_{t}$ be a Brownian motion and $f$ a continuous function. Prove that if $f\left(W_{t}\right)$ is a submartingale, then $f$ must be convex.

Prove the maximum principle for harmonic functions. This says that if $h$ is harmonic in a bounded domain $D$, then

If $W$ is a $d$-dimensional Brownian motion started at 0 , find $\mathbb {E}\left[T\right]$, where $T$ is the first time $W$ exits the ball of radius $r$ centered at the origin.

Hint: Use the fact that $\left|W_{t}\right|^{2}-d t$ is a martingale.

Let $f: \mathbb {R} \rightarrow \mathbb {R}$ be a bounded function with $\left|f(x)-f(y)\right| \leq \left|x-y\right|$ for all $x, y \in \mathbb {R}$. Let $D_{\varepsilon }=\left\{ (x, y) \in \mathbb {R}^{2}: f(x)<y<f(x)+\varepsilon \right\}$ for $\varepsilon \in (0,1)$. Let $\left(X_{t}, \mathbb {P}[x]\right)$ be a two-dimensional Brownian motion and let $\tau_{\varepsilon }=\inf \left\{ t: X_{t} \notin D_{\varepsilon }\right\}$. Prove that there exists a constant $c$ not depending on $\varepsilon$ such that $\mathbb {E}_{0}\left[\tau_{\varepsilon }\right] \leq c \varepsilon^{2}$.

Hint: By Exercise 21.7 the expected time for two-dimensional Brownian motion to leave a ball of radius $2 \varepsilon$ is less than $c \varepsilon^{2}$. Then use the strong Markov property repeatedly at the times $S_{i}$, where $S_{i}$ is the first time after time $S_{i-1}$ that Brownian motion has moved at least $2 \varepsilon$ from $X_{S_{i-1}}$.