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 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.
Dimension doubling theorems Let ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},P)} be a probability space. For a d {\displaystyle d} -dimensional Brownian motion W ( t ) {\displaystyle W(t)} and a set A ⊂ [ 0 , ∞ ) {\displaystyle A\subset [0,\infty )} we define the image of A {\displaystyle A} under W {\displaystyle W} , i.e.
W ( A ) := { W ( t ) : t ∈ A } ⊂ R d . {\displaystyle W(A):=\{W(t):t\in A\}\subset \mathbb {R} ^{d}.}
McKean's theorem Let W ( t ) {\displaystyle W(t)} be a Brownian motion in dimension d ≥ 2 {\displaystyle d\geq 2} . Let A ⊂ [ 0 , ∞ ) {\displaystyle A\subset [0,\infty )} , then
dim W ( A ) = 2 dim A {\displaystyle \dim W(A)=2\dim A}
P {\displaystyle P} -almost surely.
Kaufman's theorem Let W ( t ) {\displaystyle W(t)} be a Brownian motion in dimension d ≥ 2 {\displaystyle d\geq 2} . Then P {\displaystyle P} -almost surely, for any set A ⊂ [ 0 , ∞ ) {\displaystyle A\subset [0,\infty )} , we have
dim W ( A ) = 2 dim A . {\displaystyle \dim W(A)=2\dim A.}
Difference of the theorems The difference of the theorems is the following: in McKean's result the P {\displaystyle P} -null sets, where the statement is not true, depends on the choice of A {\displaystyle A} . Kaufman's result on the other hand is true for all choices of A {\displaystyle A} simultaneously. This means Kaufman's theorem can also be applied to random sets A {\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.
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