Dimension doubling theorem

In probability theory, the dimension doubling theorems are two results about the Hausdorff dimension of an image of a Brownian motion. In their core both statements say, that the dimension of a set [math]\displaystyle{ A }[/math] under a Brownian motion doubles almost surely. The first result is due to Henry P. McKean jr and hence called McKean's theorem (1955). The second theorem is a refinement of McKean's result and called Kaufman's theorem (1969) since it was proven by Robert Kaufman.

Dimension doubling theorems

For a [math]\displaystyle{ d }[/math]-dimensional Brownian motion [math]\displaystyle{ W(t) }[/math] and a set [math]\displaystyle{ A\subset [0,\infty) }[/math] we define the image of [math]\displaystyle{ A }[/math] under [math]\displaystyle{ W }[/math], i.e.

[math]\displaystyle{ W(A):=\{W(t): t\in A\}\subset \R^d. }[/math]

McKean's theorem

Let [math]\displaystyle{ W(t) }[/math] be a Brownian motion in dimension [math]\displaystyle{ d\geq 2 }[/math]. Let [math]\displaystyle{ A\subset [0,\infty) }[/math], then

[math]\displaystyle{ \dim W(A)=2\dim A }[/math]

[math]\displaystyle{ P }[/math]-almost surely.

Kaufman's theorem

Let [math]\displaystyle{ W(t) }[/math] be a Brownian motion in dimension [math]\displaystyle{ d\geq 2 }[/math]. Then [math]\displaystyle{ P }[/math]-almost surley, for any set [math]\displaystyle{ A\subset [0,\infty) }[/math], we have

[math]\displaystyle{ \dim W(A)=2\dim A. }[/math]

Difference of the theorems

The difference of the theorems is the following: in McKean's result the [math]\displaystyle{ P }[/math]-null sets, where the statement is not true, depends on the choice of [math]\displaystyle{ A }[/math]. Kaufman's result on the other hand is true for all choices of [math]\displaystyle{ A }[/math] simultaneously. This means Kaufman's theorem can also be applied to random sets [math]\displaystyle{ A }[/math].

Literature

• Mörters, Peter; Peres, Yuval (2010). Brownian Motion. Cambridge: Cambridge University Press. pp. 279. 
• Schilling, René L.; Partzsch, Lothar (2014). Brownian Motion. De Gruyter. pp. 169.