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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    $\begingroup$ This doesn't really match my intuition. If we drop the drift term (so $\mu = 0$), then increasing the variance really just stretches the Brownian motion vertically, which in my book has no effect on the "number" of maxima. $\endgroup$ – Nate Eldredge May 24 '19 at 22:10
  • $\begingroup$ @NateEldredge mhmm, but you would agree that if a particle that performs a deterministic walk has a more "straight motion" whereas a disordered motion has more change of direction and therefore more maxima? Perhaps there is a better to capture this behaviour?-I just wanted to present two starting ideas to model such an effect (which may be wrong?) $\endgroup$ – Sascha May 24 '19 at 22:12
  • $\begingroup$ Sure, but in this case I think that's just the difference between $\alpha = 0$ and $\alpha \ne 0$. It's more of an abrupt "phase transition", to abuse physics terminology, than a continuous change. $\endgroup$ – Nate Eldredge May 24 '19 at 22:14
  • $\begingroup$ @NateEldredge I understand, but maybe if you look at fractal dimensions, one could see something like that?-Sorry, I am just trying to bounce ideas. I am sorry that my question is not purely technical. $\endgroup$ – Sascha May 24 '19 at 22:18
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    $\begingroup$ Another possible way to interpret the question is to fix the volatility and increase the drift. Does increasing the drift “reduce the size of the set of local maxima”? Here’s where my BM knowledge is weak: if $(X_t)$ is a driftless BM, does $(X_t+at)$ have the same set of local maxima? $\endgroup$ – Anthony Quas May 25 '19 at 1:41

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.

  • $\begingroup$ thank you, I appreciate it very much that you tried to make sense out of my question. $\endgroup$ – Sascha May 24 '19 at 22:31

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