Why isn’t Brownian motion differentiable?

Question

Intuitively, if increments become infinitesimally small, why doesn’t Brownian motion become a differentiable function?

Answer 1

Imagine a particle moving around on some trajectory.

Its trajectory being continuous means that as you slow time down, the particle stays closer and closer to where it was: no big jumps.

Its trajectory being differentiable means that as you slow time down, the particle doesn't just stay near where it was, it moves more and more in a straight line.

Differentiability is a much, much stronger condition than mere continuity. As you take a limit in Brownian motion, you get a continuous function -- but you have no guarantees on its direction, which is what you need for differentiability.

Answer 2

Here's another intuitive explanation using self-similarity. We know that $a W(\frac{t}{a^2})$ is also a Brownian motion. If W is a wiggly curve, when we zoom in billions of times, it is still very wiggly. This means there's no tangent line approximating the curvature of the local curve.

Consider $$ Z = \frac{W(t+\Delta) - W(t)}{\Delta} $$ $Var(Z) = 1/\Delta$, which becomes larger when $\Delta\to0$. So the limitation does not exist.