It is well-known that Brownian motion attains infinitely many maxima in each time interval $[0,T]$ a.s..

From a physics perspective it seems reasonable that when the disorder of the path of a particle decreases and the motion becomes more deterministic, then the number of maxima should decrease.

But I could not find anything on that. Now, there were two natural things to look at:

Is there a way to quantify that a Brownian motion with large variance (large disorder) has more maxima than one with little disorder?

Or is there a way to say that a diffusion process

$dX_t = \mu (X_t) \ dt + \alpha dB_t $

has "less" maxima when $\alpha $ is small compared to $\alpha$ large?

I guess it is hard to make this question more precise, since this is not a question of cardinality of maxima but more about finding a suitably chosen measure that could capture such an effect.

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