# Autocorrelation and Power density spectrum : Continuous Markov Process

I've been reading through the paper from Gillespie on Brownian motion and Johnson Noise (DOI, PDF).

He considers $X_s(t)$, a zero-mean stochastic variable, that is stationary in the sense that all of its moments $\langle X_s(t)^k\rangle$ are time independent. He further defines the autocorrelation function (2.39)

$$\langle X_s(t)X_s(t+t')\rangle \equiv C_X(t')$$ which is independent of time since all the moments are time independent. The mean is over all possible values of X.

I cannot interpret this autocorrelation. I have been told that it measures how much the random variable fluctuates but I cannot convince myself of that.

I think this is the relevant question in order to answer my dilemma : on the same page of the definition of the autocorrelation, he goes on showing that its frequency fourrier transform

$$C_X(t) = \int \limits_{0}^\infty S_x(\nu) cos(2\pi \nu t)$$ can be related to the variance of $X_s(t)$ as $$\langle X_s(t)^2 \rangle = \int \limits_0^\infty S_x(\nu)d\nu$$

Now in the Ornstein Uhlenbeck process, $X_s(t)$ is to be interpreted as the speed of particle, subject to a drag force and a white-noise random force ($\Gamma$) $$\frac{dX_s(t)}{dt} = -\gamma X_s(t) + \sqrt{c}\Gamma(t)$$ One can then compute the dissipated power spectrum which originates from the drag force. Since power is force times speed, one gets

$$\langle P_{diss} \rangle = \gamma\langle X_s(t)^2\rangle \quad \to \quad P_{diss}(\nu) = \gamma S_X(\nu) = \gamma \frac{2c}{\gamma^2+(2\pi\nu)^2}$$ Where the last equality holds for the process at study. Gillespie simply says that this means that only the low frequency regime contributes to the dissipated power, and the high frequency regime doesn't. But frequency of what ?

My interpretation would be this : the frequency argument ammounts to saying that if the variable is auto-correlated also for long times (low frequency in fourier), then you will have significant dissipated power. Coming back to my original question: do long autocorrelation times mean that the particle fluctuates a lot, and if so, why ?

• Apr 23, 2022 at 14:54

The $C_X(t')= \langle X_s(t)X_s(t+t')\rangle$ should be compared with the statistical correlation (Wikipedia: http://en.wikipedia.org/wiki/Correlation_and_dependence). With the statistical correlation one measures the dependency between two veriables. If two variables are independent they will have a correlation equal to zero. If two variables tend to have the same value, the correlation is positive and if two varibles tend to have opposite values the correlation is negative.
For the case of Brownian motian, I guess that $X_s(t)$ is the velocity-distribution considering that the VACF (velocity autocorrelationfunction) is mostly used in the theory of Brownian motion. Then $C_X(t')$ tells you what the correlation is between the velocity at time $t$ and at time $t+t'$. For Continuous Markovchains this should drop exponentially, which is also the case for the Brownian motion.