Today I Learned

Some of the things I've learned every day since Oct 10, 2016

36: Scale-Invariance of Brownian Motion

Brownian motion (also known as the Wiener process (after Norbert Wiener)) is scale-invariant, a feature which distinguishes it from other stochastic processes like standard random walks. This means that if W_t is a Wiener process, then for any non-zero constant \alpha,

\alpha ^{-1} W_{\alpha ^2 t}

is also a Wiener process.

[Cool GIF of a Wiener process under re-scaling]

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