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.

Dimension doubling theorems

Let $(\Omega,\mathcal,P)$ be a probability space. 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.
:$W(A):=\\subset \R^d.$

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 surely, 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

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*{{cite book, title=Brownian Motion, first1=René L., last1=Schilling, first2=Lothar, last2=Partzsch, publisher=De Gruyter, pages=169, date=2014

References

Wiener process
Theorems in probability theory