I'm revising for an exam, so I'm working through some problems but I've got stuck on a particular one.
Let $D \subset \mathbb{R}^d$ be open, bounded and connected, and let $T_D = \inf\{t > 0: B_t + x \in D\},$ where $B$ is a standard $d$-dimensional Brownian motion.
$(i)$ Show that $\{T_D < \infty\}$ is measurable with respect to $\sigma(B_t: t \geq 0).$
$(ii)$ Show that $\mathbb{P}[T_D < \infty] = \mathbb{P}[T_{cD} < \infty],$ where $cD = \{cd: d \in D\}.$
$(iii)$ Suppose that $u \in C^2(\mathbb{R}^d)$ and satisfies $$\Delta u = 0, x \notin \overline{D}$$ $$u =1, x \in \partial D$$ $$u(x) \rightarrow 0, |x| \rightarrow \infty.$$ Show that $u(x) = \mathbb{P}[T_D < \infty].$
For part $(ii)$ I used the scaling property of the Brownian motion and I'm pretty sure I got it right. I'm not really sure how to approach part $(i)$. Regarding part $(iii),$ I know there is some result which says that this kind of PDE has a unique solution which is given by $$u(x) = \mathbb{E}[x+B_{T_D}]$$ but I was unable to apply this expression in the case of my problem.
Any help would be much appreciated! Thank you!
[I assume $D$ is non-empty, and that $d\ge 3$. If $d=1$ or $2$, then $\Bbb P[T_D<\infty]$ is identically $1$.]
(i) $\{T_D<\infty\}=\cup_{0<t\in\Bbb Q}\{ B_t+x\in D\}$. ($\Bbb Q$ denotes the rational numbers.)
(ii) Take care to include the "initial condition" $x$ in your calculation — it changes under scaling.
(iii) If $X_t:=u(x+B_{t\wedge T_D})$, $t\ge 0$, then $(X_t)$ is a bounded martingale with $\lim_{t\to\infty} X_t = 1$ on $\{T_D<\infty\}$, while $\lim_{t\to\infty} X_t = 0$ on $\{T_D=\infty\}$ because $\lim_{t\to\infty}|B_t|=0$ when $d\ge 3$. Now use the martingale convergence theorem to check that $u(x) =\Bbb P[T_D<\infty]$.