Dimension doubling theorem explained

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 , inf ty) we define the image of A under W , i.e.W(A): = \{W(t): t\in A\}\subset \R ^{d.}

McKean's theorem

Let

W(t) be a Brownian motion in dimension d\geq 2 . Let A\subset [ 0 , inf ty) , 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 surely, for any set A\subset [ 0 , inf ty) , 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

• Book: Peter. Mörters. Yuval. Peres. Brownian Motion. Cambridge University Press. Cambridge. 279. 2010.
• Book: Brownian Motion. René L.. Schilling. Lothar. Partzsch. De Gruyter. 169. 2014.

References

1. Robert. Kaufman. Une propriété métrique du mouvement brownien. C. R. Acad. Sci. Paris. 268. 727–728. 1969.
2. Book: Peter. Mörters. Yuval. Peres. Brownian Motion. Cambridge University Press. Cambridge. 279. 2010.