Exercise on a hitting time for a Brownian Motion

Question

I'm following Chapter 3 of "Brownian Motion", by Peres and Mörters, about The Dirichlet Problem(DP). As it is known, in order to obtain existence and uniqueness of a solution for DP it is necessary to ask some smoothness to the boundary of the region $\ U$ where the DP is posed. The following lemma enables to prove that under the "Zaremba condition" there is such unique solution, but I'm having trouble with both parts of the proof of such lemma

Let $(B(t))$ be a Brownian Motion (BM) in $\mathbb{R}^d$, and let us denote a cone centered at $z\in\mathbb{R}^d$ with opening angle $\alpha$ as $C_{\alpha}(z)$. If centered at the origin, let us note $C_{\alpha}$, and its hiting time as $\tau$.

Lemma 3.11 states that there's an uniform positive probability for a BM of reaching $C_{\alpha}$ before reaching the boundary of the unit ball (also centered at the origin), starting from any $x\in\partial B_{1/2}\ $ i.e.

$$ \sup_{x\in\partial B_{1/2} }\mathbb{P}_x(\tau_{\partial B_1}<\tau ):=a < 1 $$

Moreover, for all $k\in \mathbb{N}$, we have,

$$ \sup_{x\in\partial B_{2^{-k}} }\mathbb{P}_x(\tau_{\partial B_1}<\tau )\leqslant a^k $$

a) For the first statement, the book says it is "easy"(sic) to prove with the following property: if $f:[0,1]\rightarrow \mathbb{R}$ continuous with $f(0)=0$, then for a Standard BM in $\mathbb{R}$ and $\varepsilon >0$ we have $$ \mathbb{P}(\sup_{[0,1]}|B(t)-f(t)|<\varepsilon) > 0 $$

I don't have any idea how to apply this property to this problem, the only thing that occurred to me is that $f$ could represent something related to the distance to the cone (there must be some projection in the middle, as the property is about a BM in $\mathbb{R}$).

b) the second statement is clearly true by the use of the Strong Mkv Property on $\tau_{k-1}$ (where $\tau_j$ is the hitting time of $\partial B_{2^{-k+j}}$, induction and the use of the first statement, but I can't seem to write this rigorously. I would really appreciate if someone could help with this task.

Answer 1

Here's one way of proving your first claim.

Let $z$ be the center of the ball. Choose an $\epsilon < 1/4$ and a finite set of points $\{x_1, \dots, x_n\} \in B_{3/4}(z)$ such that for every cone $C$ with angle $\alpha$ and vertex at $z$, there is some $x_i$ such that $B_\epsilon(x_i) \subset C$. (This is possible by a compactness argument which I leave as an exercise.) Now fix an arbitrary $x_0 \in \partial B_{1/2}$ and let $\gamma : [0,1] \to \mathbb{R}^n$ be a continuous curve starting at $x_0$, staying inside $B_{3/4}$, and passing through all of the points $x_1, \dots, x_n$. Let $1-a$ be the probability that a Brownian motion started at $x_0$ stays within $\epsilon$ of $\gamma$ up to time 1, which by the quoted property is nonzero.

Now fix a particular cone $C$. Given any $x \in \partial B_{1/2}$, let $T$ be a rotation of $\mathbb{R}^n$ about $z$ which maps $x_0$ to $x$. Observe that if $$\mathcal{H}_{\alpha}:=\{\text{cones with angle } \alpha \text{ and vertex at }z\} $$ then $T\mathcal{H}_{\alpha} =\mathcal{H}_{\alpha}$ and $\{Tx_1,...,Tx_n \}$ preserves the property of $\{x_1,...,x_n \}$. Moreover, the the probability that a Brownian motion started at $x$ stays within $\epsilon$ of $T\gamma$ is $1-a$, by rotational invariance of Brownian motion. Consequently, on this event, the Brownian motion passes through every cone of angle $\alpha$ with vertex at $z$ before time 1, so in particular it passes through $C$. And it remains inside the ball $B_{3/4 + \epsilon}$ before time 1, so necessarily it meets $C$ before $\partial B_1$. Hence on this event we have $\tau \le \tau_{\partial B_1}$, so $P_x(\tau_{\partial B_1} < \tau) \le a$. And $x$ was arbitrary. So the first claim is proved.