Brownian motion is almost surely not differentiable everywhere

Question

Could anyone point out the difference between the statement of the following theorems:

1) For any $t\ge0$, Brownian motion is almost surely not differentiable at $t$.

2) Almost surely, the sample paths of Brownian motion are not Lipschitz continuous at any point. In particular, the sample of Brownian motion are almost surely not differentiable at any point.

The author of the book argues that to show 2) a more refined argument is needed. I'm a bit confused about the difference between the above statements. Could someone explain this?

Answer 1

Since there are uncountably many $t$'s, it is not automatic that something that is almost surely true for any given $t$ is almost surely true for all $t$. For example, consider a Poisson process $N(t)$. For each $t$, $N(t)$ almost surely doesn't have a jump at $t$. On the other hand, with probability $1$ the process does have a jump somewhere.

Answer 2

Let me restate:

1. At each time $t$ the probability that Brownian motion is differentiable at $t$ is $0$.

2. The probability that there exists even a single point $t$ at which Brownian motion is differentiable is $0$.

It's kind of a probabilistic version of $\forall \exists$ vs. $\exists\forall$.