Brownian Motion - Self-Affinity

Recall a fractal is self-affine if different directions must be scaled by different factors.

In these pictures, t is scaled by t → 4⋅t. Only one of the three Y scaling preserve the approximate appearance of the

Y → (4/3)⋅Y Y → 2⋅Y Y → 4⋅Y Note the right picture looks steeper than the left.

The graph of Brownian motion is self-affine with the rescaling t → 4⋅t and Y → 2⋅Y.

That is, the Y rescaling factor is the square root of the t rescaling factor. These pictures give a visual interpretation of the rescaling formula for the probabilities.

Here is a more analytic, but still visual, approach.

Here is a more mathematical approach.

Return to Mathematical Properties of Brownian Motion.