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.

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