the Brownian zero set and its Hausdorff dimension
/ Hausdorff: HOWS-dorf /
Where does Brownian motion equal zero? The set Z = { t >= 0 : B_t = 0 } of zero times is one of the most striking fractals in probability: it is closed, has no isolated points (it is perfect), contains no interval (it has Lebesgue measure zero), and yet is uncountable. It looks like a random Cantor set on the time axis, and its size is captured not by length but by Hausdorff dimension.
The precise facts: Z is almost surely a closed, perfect, nowhere-dense set of Lebesgue measure zero, and its Hausdorff dimension is exactly 1/2. Hausdorff dimension generalises 'dimension' to fractals by asking, for a gauge h(r) = r^s, when the s-dimensional Hausdorff measure (the infimum over covers by sets of diameter r_i of the sum of h(r_i)) is finite or zero; the critical exponent s where it jumps from infinity to zero is the dimension. For Z this critical s is 1/2, reflecting the sqrt-scaling B_(c t) =d sqrt(c) B_t: doubling the time-resolution only quadruples the count of zeros, the signature of a half-dimensional set. The zeros accumulate immediately: 0 is a limit of zeros from the right, and around every zero there are infinitely many others. The geometry is organised by excursion theory — the complement of Z is a countable union of open excursion intervals — and local time at 0 is the natural 'measure' on Z, growing only on Z and flat off it.
Why it matters: the half-dimensional zero set is the prototype for fractal sample-path geometry, and the same ideas give the dimensions of the Brownian graph (3/2), the range/image in d dimensions, the set of double points, and the multifractal spectrum of fast points. It explains the arcsine law for the last zero (the zero set clusters near the endpoints) and is the stage on which excursion theory and the Ray-Knight theorems play out. The honest caveat: 'dimension 1/2' does NOT mean 'half the points' or positive length — the set has length zero; dimension is a finer fractal gauge, and a measure-zero set can still be uncountable and richly structured.
If you simulate Brownian motion and zoom in on any one zero, you see not an isolated crossing but an infinite cluster of further zeros at every scale — the same picture at every magnification, the hallmark of a self-similar set of dimension 1/2.
The zero set is a measure-zero, uncountable random Cantor set of Hausdorff dimension exactly 1/2.
Dimension 1/2 does not mean positive length — the zero set has Lebesgue measure zero yet is uncountable and perfect; Hausdorff dimension is a finer gauge than length.