This question was posted in <https://math.stackexchange.com> but I got hardly any view. If posting here is an objection please let me know I would delete it immediately.


This question has evolved from watching the video:
<https://www.youtube.com/watch?v=dh5hHpJ79jc>


In the video the speaker talked about white noise case and $\alpha$ related to the Holder condition, (wikipedia link to Holder condition <https://en.wikipedia.org/wiki/H%C3%B6lder_condition>)


The question: Is there a relation between SNR (<https://en.wikipedia.org/wiki/Signal-to-noise_ratio>) and Holder condition?


Let me be more specific, As I understand, Rough path theory (<https://en.wikipedia.org/wiki/Rough_path>) is related to the solution of differential equation driven by non-smooth signals and non-smoothness is defined in terms of Holder condition. In the video it clearly relates the white noise case to the value of $\alpha$. Consider our driver is simple sine wave, which is very smooth, but if we add a noise then we can define SNR of the driver based on the strength of the noise and at the same time can we define $\alpha$ based on the strength on the noise? 


In some sense I am trying to define rough path theory in terms of SNR, is it possible? Any comments would be highly appreciated 



Are Holder Condition and signal to noise ratio (SNR) related?
The Hölder exponent $h$ and the Hurst exponent $H$ (related by $h=H-1$ for a continuous signal) characterise correlations in the noise, not the size of the noise (as quantified by the signal-to-noise ratio). So these are different quantities. This plot illustrates two signals with the same signal-to-noise ratio but different $H$ (short-term correlations on the left, long-term correlations on the right).


![](https://ilorentz.org/beenakker/MO/hurst.png)
source: [On the relationship between the Hurst exponent, the ratio of the mean square successive difference to the variance, and the number of turning points](https://arxiv.org/abs/1512.02928)


