Brownian Self-Affinity: Mathematical Appendix

The self-affine scaling of Brownian motion is expressed in the statement for all t ≥ 0, for all h > 0, and for all s > 0,

The proof of this is straightforward, using Wiener's definition of Brownian increments:

For with this definition,

Change varaibles to

Then note

and

so when we have

Thus we obtain

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