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 ${\displaystyle A}${\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]}

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

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

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

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

${\displaystyle P}${\displaystyle P}-almost surely.

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

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

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

• 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.

• Wiener process
• Probability theorems