# In probability, what is exactly a noise and a white noise?

I'm reading a book on SDE, and the author often talk about Noise or white noise. But what is it exactly ? I saw on wikipedia that a Gaussian noise is a Normal r.v. but it doesn't really help to understand. For example, the author write : and SDE is an equation of the type : $$\frac{dS(t)}{dt}=S(t)+"Noise".$$ The he says that a Noise is a Brownian motion... but all this is confusing ! Why doesn't he write directly $$\frac{d S(t)}{dt}=S(t)+B(t) \ \ ?$$

And why a Noise is a Brownian motion ? Just call it "Brownian motion" instead of a Noise, no ?

Different types of noise are generated by different stochastic processes. The power spectrum of a noise signal is referred to using colors.

One type is white noise, which is when each component of the noise signal have a probability distribution with zero mean and finite variance, and are statistically independent. This results in a noise signal with spectral density that is even throughout all frequencies (flat power spectral density). Note that the name is drawn from the white light as it contains all colors.

Another type is red noise or Brownian noise, which refers to noise resulting from Brownian motion. The spectral density of this type is inversely proportional to the frequency squared. Meaning, its power drastically decreases as its frequency increases (has more energy at low frequencies). Note that it is called red noise as it is analogous to red light which has a low frequency.

I found the figure below here. It might help better convey the difference in the power spectrums. Update

One more thing I forgot to mention that could clear your confusion. First off, please note that red noise and Brownian noise are not synonyms. All Brownian noise are red noise but not vice versa. Brownian motion has a Gaussian probability distribution. In other words, $$B(t)$$ is a Gaussian random variable with mean $$0$$ and variance $$t$$. In this sense, white noise can be thought of as the derivative of a Brownian motion. This has more details to it but I'm just trying to highlight the relation between white noise and Brownian motion.

• Waou, very nice :) Even if you explain well what are different noise, you didn't explain what is a noise (sorry, I don't understand with your answer what's a noise in general) – user601023 Oct 6 '18 at 12:22
• @user601023 In communications, noise, as the name implies, is any form of unwanted signal that disturb the communication between to parties. – Lod Oct 6 '18 at 23:57

Depending on the application, one may refer to noise in different ways, but in general, Brownian motion is not "noise". White noise is a sequence of uncorrelated random variables with zero mean and finite variance. If I use the term noise, I am usually referring to some signal that is disturbed by some noise. In real life, signals have noise due to the device recording or transmitting the signal. This is probably why the author uses the term noise.