Planar coordinate hitting does not imply point hitting

Fix $R>\mathrm{\mid }x-y\mathrm{\mid }$. For every $n$ with $1\mathrm{/}n<\mathrm{\mid }x-y\mathrm{\mid }$, the event $\{T_y<T_R\}$ is contained in $\{S_{1\mathrm{/}n}<T_R\}$: a path that reaches $y$ before leaving the disc of radius $R$ passes through the circle of radius $1\mathrm{/}n$ about $y$ first, by continuity. Hence by [F1], $P_x(T_y<T_R)\le {\lim }_n\frac{\log R-\log \mathrm{\mid }x-y\mathrm{\mid }}{\log R-\log (1\mathrm{/}n)}=0$, the denominator tending to $+\mathrm{\infty }$.

For a given $\delta >0$ choose $0<\epsilon <\min \{\delta ,\mathrm{\mid }x-y\mathrm{\mid }\}$; then $P_x(S_{\epsilon }<T_R)=\frac{\log R-\log \mathrm{\mid }x-y\mathrm{\mid }}{\log R-\log \epsilon }\to 1$ as $R\to \mathrm{\infty }$, so the $\epsilon$-circle about $y$ is hit almost surely, hence the disc of radius $\delta$ about $y$ is hit almost surely. Applying this to the countably many discs with rational centre and rational radius and intersecting the resulting probability-one events via [F3], while every nonempty open disc contains such a rational disc by [F4], gives the almost-sure statement of assertion 3.

By [F2] each coordinate increment $W^{(i)}-x_i$ is a standard one-dimensional Brownian motion, so it visits every neighbourhood of the level $y_i-x_i$ at arbitrarily large times; equivalently, $W^{(i)}$ visits every neighbourhood of $y_i$ at arbitrarily large times almost surely. This is assertion 1, and it does not synchronize the two coordinates.

The event $\{T_y<\mathrm{\infty }\}$ is the union over the countably many integers $R>\mathrm{\mid }x-y\mathrm{\mid }$ of the increasing events $\{T_y<T_R\}$: if $T_y<\mathrm{\infty }$ then the path on $[0,T_y]$ is a compact subset of ${\mathbb{R}}^2$, hence stays in some disc of integer radius about $y$, and conversely $T_y<T_R<\mathrm{\infty }$ implies $T_y<\mathrm{\infty }$. By [F3] and step 1.1, $P_x(T_y<\mathrm{\infty })\le {\sum }_R{P}_x(T_y<T_R)=0$.

By step 2.1 there is a probability-one event on which the planar path never equals $y$; on that event no time can satisfy both coordinate equations simultaneously, so $\{t:W_t^{(1)}=y_1\}\cap \{t:W_t^{(2)}=y_2\}=\mathrm{\varnothing }$, even though each of the two sets is almost surely unbounded by step 1.3. This is assertion 2.

The cases are consistent: the point $x=y$ is excluded, so $T_y=0$ is not possible; the dimension is two, and the polarity statement is not asserted in dimension one, where the same computation fails because the logarithm is replaced by a bounded harmonic function; the radius parameter $\delta >0$ and the inner radius $\epsilon$ are chosen strictly positive. AC is used only through [F5].