36 — Scaling of Hitting Times, Stability, and Zero Set Structure
1. Review: Distribution of the First Hitting Time $T_a$
Recall \(T_a = \inf\{t>0 : B_t = a\},\quad a>0.\)
From the reflection principle, \(P_0(T_a < t) = 2 P_0(B_t > a) = 2P\!\left(Z > \frac{a}{\sqrt t}\right),\) where $Z\sim N(0,1)$.
Differentiate to obtain the density: \(f_{T_a}(t) = \frac{\partial}{\partial t} P_0(T_a < t) = \frac{a}{\sqrt{2\pi t^3}} \exp\!\left(-\frac{a^2}{2t}\right),\qquad t>0.\)
As $t\to\infty$,
$\exp(-a^2/(2t))\to 1$ but $t^{-3/2}\to 0$, so $f_{T_a}(t)\to 0$.
The expectation diverges: \(E[T_a] = \int_0^\infty t\, f_{T_a}(t)\,dt = \infty.\)
2. Scaling Argument for $T_a \overset{D}{=} a^2 T_1$
Instead of differentiating the reflection-principle expression, use Brownian scaling.
Define \(X_t = a\, B_t.\) Then \(\mathrm{cov}(X_{t_1}, X_{t_2}) = a^2 (t_1 \wedge t_2).\)
Define the hitting time for $X_t$: \(S = \inf\{t : X_t = a\}.\) But $X_t = a B_t$, so \(S = \inf\{t : B_t = 1\} = T_1.\)
Now scale time: \(Y_t = B(a^2 t),\qquad t\ge 0.\)
Then the covariance of $Y_t$ is \(E[Y_{t_1}Y_{t_2}] = a^2 (t_1\wedge t_2),\) the same as $X_t$. By uniqueness of Gaussian processes, the finite-dimensional distributions match.
Let \(\tilde S = \inf\{t : Y_t = a\}.\)
Since $Y_t = B(a^2 t)$, \(Y_{\tilde S} = a \iff B(a^2\tilde S) = a \iff a^2 \tilde S = T_a.\)
Thus \(T_a = a^2 \tilde S, \qquad \tilde S \overset{D}{=} T_1,\) so \(T_a \overset{D}{=} a^2 T_1.\)
3. Independent and Stationary Increments of Hitting Times
Let $0 < a < b$. Then:
-
Independence of increments \(T_b - T_a \ \perp\!\!\!\perp\ \mathcal F_{T_a}.\)
-
Stationarity of increments \(T_b - T_a \overset{D}{=} T_{\,b-a}.\)
This follows from the Strong Markov Property and the continuity of Brownian motion.
4. Hitting Times Form a Stable Law
Consider the decomposition \(T_n = T_1 + (T_2 - T_1) + \cdots + (T_n - T_{n-1}) = X_1 + X_2 + \cdots + X_n,\) where \(X_k \overset{D}{=} T_1,\qquad X_k\ \text{i.i.d.}\)
From scaling, \(T_n \overset{D}{=} n^2 T_1.\)
Thus \(X_1 + \cdots + X_n \overset{D}{=} n^2 T_1.\)
This is the defining property of a stable distribution: \(\sum_{k=1}^n X_k \overset{D}{=} n^{1/\alpha} X_1.\)
Here the exponent satisfies \(n^{1/\alpha} = n^2 \quad\Rightarrow\quad \alpha = \frac12.\)
So $T_1$ is stable with index $1/2$.
This parallels the Gaussian case: \(\sum_{k=1}^n Z_k \overset{D}{=} \sqrt{n}\, Z,\) which corresponds to stability index $\alpha = 2$.
Stable distributions only exist for $0<\alpha\le 2$.
Thus Brownian hitting times produce the $\alpha=\tfrac12$ stable law.
5. The Zero Set and Last Exit Time
Let \(L = \sup\{t\le 1 : B_t = 0\}\) be the last exit time from 0 before time 1.
The handwritten plot on page 2 of the PDF shows multiple BM paths, illustrating that the last zero time occurs near but not necessarily at $1$.
:contentReference[oaicite:1]{index=1}
Using the representation from hitting-time densities,
\[\begin{aligned} P_0(L \le s) &= 2 \int_{x=0}^{\infty} (2\pi s)^{-1/2} e^{-x^2/(2s)} \, P_0(T_x > 1-s)\, dx. \end{aligned}\]Insert the hitting-time tail: \(P_0(T_x>t) = \int_{t}^{\infty} \frac{x}{\sqrt{2\pi u^3}} e^{-x^2/(2u)}\, du.\)
Substitute this into the integral and evaluate. After transformation $x = \sqrt{s/(1-s)}\, y$, one obtains the classical arcsine law:
Arcsine Law
\(P_0(L\le s)=\frac{2}{\pi}\arcsin(\sqrt{s}), \qquad 0\le s\le 1.\)
This is one of Lévy’s arcsine laws.
6. Structure of the Zero Set
Let \(A(\omega)=\{t\ge 0 : B_t(\omega)=0\}.\)
Claim:
Almost surely there are no isolated zeroes.
That is, with probability 1,
If $t\in A(\omega)$, then there exist sequences $t_n\downarrow t$ and
$s_n\uparrow t$ such that $B_{t_n}=B_{s_n}=0$.
The handwritten sketches on page 2 (lower right) show BM oscillating infinitely often around each zero—this follows from the law of the iterated logarithm and infinite variation.
:contentReference[oaicite:2]{index=2}
Thus the zero set is perfect (closed and has no isolated points) and uncountable.
Comments