I am trying to understand why Brownian Motion is not differentiable.
Here we are tossing a coin n-times.
So we form a sample space $ \Omega $.
Each of the paths are represented by ω = ω1 ω2 ω3…. ωn
In all likely situation, sample space will include the following two paths
- outcomes of the tosses are either purely Heads-H or purely Tails Then
these two paths can be written as o (omega) ω =HHHHHHHHHHHHHHHHHHHHHH…n times o (omega) ω = TTTTTTTTTTTTTTTTTTTTTTTTTT..ntimes
In my understanding both the above cases the paths/Brownian motion is a straight line. In other words, not a zig-zag
In my view, the above two omegas are differentiable as both are straight lines. So two of the random walks/paths in the Omega $ \omega $ are differentiable.
I have come across in several texts and videos that Brownian motion is not differentiable. I agree that except for the above two paths remaining are not differentiable.
The clarification I seek is whether Brownian Motion is differentiable at each time period • For the above two paths at every/any point • for the remaining paths in the omega (sample space other than the above two) This might sound naïve, why is it important to be differentiable at every step.
Coming from a non-math background, I have built my skills to this stage with the help of PhD student. But there is always the possibility my fundamentals may not be perfect or complete.
Kindly guide/help me.
Thank you
The proof of a.s. nondifferentiability of Brownian motion is explained in Theorem 1.30, page 21 of the book [1]. The theorem was first proved by Paley, Wiener and Zygmund in [PWZ33], but the proof in the book is due to Dvoretzky, Erdos and Kakutani [DEK61].
[1] Mörters, Peter, and Yuval Peres. Brownian motion. Vol. 30. Cambridge University Press, 2010. https://yuvalperes.com/brownian-motion/