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 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

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For a  -dimensional Brownian motion   and a set   we define the image of   under  , i.e.

 

McKean's theorem

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Let   be a Brownian motion in dimension  . Let  , then

 

 -almost surely.

Kaufman's theorem

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Let   be a Brownian motion in dimension  . Then  -almost surely, for any set  , we have

 

Difference of the theorems

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

Literature

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  • 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

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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.