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 ${\displaystyle A}$ 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.^[1][2]

Dimension doubling theorems

Let ${\displaystyle (\Omega ,{\mathcal {F}},P)}$ be a probability space. For a ${\displaystyle d}$-dimensional Brownian motion ${\displaystyle W(t)}$ and a set ${\displaystyle A\subset [0,\infty )}$ we define the image of ${\displaystyle A}$ under ${\displaystyle W}$, i.e.

${\displaystyle W(A):=\{W(t):t\in A\}\subset \mathbb {R} ^{d}.}$

McKean's theorem

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

${\displaystyle \dim W(A)=2\dim A}$

${\displaystyle P}$-almost surely.

Kaufman's theorem

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

${\displaystyle \dim W(A)=2\dim A.}$

Difference of the theorems

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

Literature

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