Probability that planar Brownian motion doesn't "encircle" 0

Question

Suppose $B_t$ is a standard Brownian motion in $\mathbb{R}^2$ and $T = \text{inf}\{t : |B_t| = 1\}$. Let $E$ denote the event that $0$ is contained in the unbounded component of $\mathbb{R}^2 \setminus B[0, T]$. Do there exist $\lambda < \infty$ and $\alpha > 0$ such that for all $|x| < 1$, we have$$\mathbb{P}^x(E) \le \lambda |x|^\alpha?$$

Answer 1

The probability is asymptotic to $\lambda |x|^{1/4}$. This was proved by Schramm, Werner and myself using the Schramm-Loewner evolution. The exponent is called the disconnection exponent for Brownian motion.

Answer 2

Divide the interval $[|x|,1]$ into $-\log_2 |x|$ intervals $[r_i,r_{i+1}]$. Radially, the BM performs a (simple) random walk between the circles of these radii, and there will be at least $-\log_2 |x| $ steps to this random walk before it hits $1$. During each step the tangential component of the BM has probability $>\delta$ to encircle the origin, some $\delta>0$ (the tangential component is a time changed BM on the circle with time change bounded within a constant factor of $r_i^2$). So $$P(E)\leq (1-\delta)^{-\log_2 |x|}$$

This computation does not give the correct exponent $\alpha$.

Answer 3

Take any $s$ and $r$ such that $0\le s\le r\le1$, and let $p(s,r)$ denote the probability that the Brownian motion in $\mathbb R^2$ starting at a point at distance $s$ from the origin $O$ will encircle $O$ before hitting the circle of radius $r$ centered at $O$. Then for any $u\in[s,r]$ $$(1)\qquad p(s,r)\ge p(s,u)+[1-p(s,u)]p(u,r). $$ The other crucial observation is that $p(s,r)=f(s/r)$ for some function $f\colon[0,1]\to[0,1]$ and all $s$ and $r$ such that $0\le s\le r\le1$ and $r>0$. This observation follows because, for the Brownian motion, space rescaling is equivalent to appropriate time rescaling. So, with $a:=s/u$ and $b:=u/r$, (1) yields $$f(ab)\ge f(a)+[1-f(a)]f(b) $$ and then
$$g(ab)\le g(a)g(b), $$ where $g:=1-f$ and $0<a,b\le1$. Note that $f(1)=0$ and hence $g(1)=1$. Also, $g$ is nonnegative and nondecreasing, since $p(s,r)$ is obviously nondecreasing in $r$. Using ``path corridors'' that are (say) unions of rectangles, one can show that $0<g(1/2)<1$. Take now any $a\in(0,1/2]$ and let $k:=k_a:=\lfloor\log_2\frac1a\rfloor$, so that $a^{1/k}\le1/2$ and $k\ge1$, and hence $k\ge\frac12\,\log_2\frac1a$. Then $$g(a)\le g(a^{1/k})^k\le g(1/2)^k\le g(1/2)^{\frac12\,\log_2\frac1a}=a^\alpha, $$ where $\alpha:=-\frac12\,\log_2 g(1/2)\in(0,\infty)$. That is, $$P^x(E)=1-p(|x|,1)=g(|x|)\le|x|^\alpha$$ for all $x$ with $|x|\le1/2$. If now $|x|\in[1/2,1]$, then $P^x(E)\le1\le(2|x|)^\alpha$. So, $$P^x(E)\le2^\alpha |x|^\alpha$$ for all $x$ with $|x|\le1$, as desired.

Answer 4

This is an extended comment to observe that $E$ is measurable.

Let $U$ be the set of $\omega \in C([0,1]; \mathbb{R}^2\}$ that have not encircled the origin by time 1; i.e. such that $0$ is in the unbounded component of $\mathbb{R}^2 \setminus \omega([0,1])$. I claim $U$ is open (with respect to the uniform norm $\|\cdot\|_\infty$).

Suppose $\omega \in U$, so that $0$ is in the unbounded component. Choose any $y \in \mathbb{R}^2$ with $|y| > \|\omega\|_{\infty} + 1$, so $y$ is also in the unbounded component. Now the unbounded component is a connected open set, so it is path connected; let $\gamma \in C([0,1]; \mathbb{R}^2)$ be a path joining $0$ to $y$ which does not intersect $\omega$. Set $\epsilon = \inf\{|\omega(t) - \gamma(s)| : s,t \in [0,1]\}$ which is strictly positive by compactness. If $\|\omega - \tilde{\omega}\| < 1 \wedge \epsilon$ then $\gamma$ does not intersect $\tilde{\omega}$ either, so both $0$ and $y$ are in the same component of $\mathbb{R}^2 \setminus \tilde{\omega}([0,1])$. Since $\|\tilde{\omega}\| \le \|\omega\| + \|\omega - \tilde{\omega}\| < \|\omega\|+1$, $y$ is also in the unbounded component of $\mathbb{R}^2 \setminus \tilde{\omega}([0,1])$, hence so is 0. So $\tilde{\omega} \in U$.

By the same argument, for any $t \ge 0$, the set $U_t$, consisting of all those $\omega \in C([0,\infty) ; \mathbb{R}^2)$ which have not encircled the origin by time $t$, is open.

So let $\tau(\omega) = \inf\{t : \omega \notin U_t\}$ be the first time at which $\omega$ has encircled the origin. Since $U_s \supset U_t$ for $s < t$, we can take the infimum over the rationals instead, and see that $\tau : C([0,\infty); \mathbb{R}^\infty) \to [0,\infty]$ is Borel. Indeed, $\tau$ is a stopping time. And $E$ is simply the event $\{\tau > T\}$.

Likewise, the function $p(s,r)$ in Iosif Pinelis's answer does make sense, since we simply have $p(s,r) = P^{x_0}(\tau < T_r)$ where $T_r = \inf\{t : |B_t| = r\}$ and $x_0$ is any point with $|x_0|=s$ (by symmetry it does not matter which such $x_0$ is chosen).