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 $A$ 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

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

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

McKean's theorem

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

\dim W(A)=2\dim A

$P$ -almost surely.

Kaufman's theorem

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

\dim W(A)=2\dim A.

Difference of the theorems

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

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.

References

Kaufman, Robert (1969). "Une propriété métrique du mouvement brownien". C. R. Acad. Sci. Paris. 268: 727–728.
Mörters, Peter; Peres, Yuval (2010). Brownian Motion. Cambridge: Cambridge University Press. p. 279.

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[1] Kaufman, Robert (1969). "Une propriété métrique du mouvement brownien". C. R. Acad. Sci. Paris. 268: 727–728.

[2] Mörters, Peter; Peres, Yuval (2010). Brownian Motion. Cambridge: Cambridge University Press. p. 279.