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.



Maxima of Brownian motion
A good way to measure the set of maxima is the Hausdorff dimension of the set of records, which for BM is a.s. 1/2. Because of time/scale invariance, the dimension is the same for $\alpha B\_\cdot$, and if the drift is nice enough to have absolute continuity, the same holds for your drifted diffusion.


This however changes when you move from BM to fractional BM, which I would suggest is the right frame for your question. There, the change in regularity is reflected in the Hausdorff dimension of the set of zeros. See
<https://projecteuclid.org/download/pdfview_1/euclid.ecp/1522375381> for details.


