Brownian transformations and invariance

Flip, stretch, restart or invert time on a Brownian path: is the result still a Brownian path?

See that four natural transformations of W again give Brownian motion: the variance of each is t.

One path, four transformations

Reflection: −W(t)

Scaling: W(2·t)/√2

Time inversion: t·W(1/t)

Restart: W(t + 0.5) − W(0.5)

Each coloured path looks like a fresh Brownian path. Time inversion is drawn from t = 0.02 on.

Empirical variance over 400 paths

Var = tReflection: −W(t)Scaling: W(2·t)/√2Time inversion: t·W(1/t)Restart: W(t + 0.5) − W(0.5)

All four empirical variances lie on the dashed line Var = t: the transforms have the same variance as W itself.