A 2D Brownian Motion will hit any given line on the plane with probability one.

Question

I recently read about an interesting fact involving $2$-D Brownian Motions. If $B(t) = (B_1(t), B_2(t))$ is a two-dimensional standard Brownian motion, then $B(t)$ will hit any given line on the plane with probability one.

As an useful reference I believe https://www2.math.upenn.edu/~pemantle/papers/coverage.pdf can help. But I am not sure how to prove this!

Answer 1

Remark: We notice that a uni-dimensional Brownian Motion $W_t$ will reach the value $a\in \mathbb{R}$ almost surely. The proof can be found here (the question and the accepted answer). Indeed, as $W_t$ goes to $\pm \infty$ almost surely, $W_t$ will reach any value $a$ before a time $T_a<+\infty$.

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Now, in the bi-dimensional case, we will prove that the bi-dimensional BM $(B_{x}(t),B_y(t))$ will hit almost surely (which is stronger than with probability $1$) the line $(d)$ passing through the point $(x_0,y_0)$ and having the direction vector $p\vec{e}_x+q\vec{e}_y$ (with $p^2 +q^2 = 1$).

Indeed, instead of using the canonical coordonnate system $Oxy$, we use the coordinates $Ouv$ defined by $$\begin{cases} \vec{e}_u=p\vec{e}_x+q\vec{e}_y\\ \vec{e}_v=-q\vec{e}_x+p\vec{e}_y\\ \end{cases} \iff \begin{cases} \vec{e}_x&=p\vec{e}_u-q\vec{e}_v\\ \vec{e}_y&=q\vec{e}_u+p\vec{e}_v\\ \end{cases} $$ In this new coordinate system $Ouv$,

• the line $(d)$ is in parallel with the axis $Ou$ and perpendicular to the axis $Ov$. The direction of the line in the new plan is $ 0\cdot \vec{e}_u+ \color{blue}{(-x_0q+y_0p)}\cdot\vec{e}_v $
• The bi-dimensional $(B_{x}(t),B_y(t))$ can be expressed in the plan $Ouv$ as follows $$\begin{align} B_x(t)\vec{e}_x+B_y(t)\vec{e}_y&= (pB_x(t)+qB_y(t))\vec{e}_u +\underbrace{(-qB_x(t)+pB_y(t))}_{=:\color{red}{W(t)}}\vec{e}_v \end{align}$$

The process $\color{red}{W(t)}:=-qB_x(t)+pB_y(t))$ is Brownian Motion because $B_x(t)$ and $B_y(t)$ are two independent BM and $p^2+q^2 = 1$.

According to the Remark, the Brownian motion $W(t)$ reaches the value $\color{blue}{(-x_0q+y_0p)}$ almost surely at the time $T<+\infty$. Then, the bi-dimensional $(B_{x}(t),B_y(t))$ in the new coordinate system $Oxy$ hit the line $(d)$.

We conclude that the process $(B_{x}(t),B_y(t))$ hit almost surely the line $(d)$.

Q.E.D