In this highly interesting answer by Ron Maimon here, under the section of fractional order infinitesimals, he explains fractional order infinitesimals by usage of an example from brownian motion. After defining the velocity as,

$$ \frac{dx}{dt} = \frac{ \Delta x}{\Delta t}$$

He writes,

In time δt it goes an amount proportional $\sqrt{dt}$. So the velocity of a Brownian motion is divergent at every time, it doesn't have a limit. But it does have a well-defined limit as a distribution because its integral is continuous.

I don't understand this paragraph due to the following reasons.

  1. Is Ron Maimon talking about the velocity when he writes "in time $\delta t$ it goes an amount proportional to $\sqrt{dt}$"?
  2. How does the position (the integral) being continuous suggest that the distribution has a well-defined limit?

How I'm understanding it currently: He goes around taking the limit for multiple particles set forth in similar starting taking the same random walk and statistically it comes out that there is a well-defined velocity (?). However, I'm not sure how integral being continuous would suggest this? What would happen if integral wasn't continuous?


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