How can fluctuations be summarized by giving its first and second moments?

Question

In the book of Zwanzig, Nonequilibrium statistical mechanics, at page 5, while explaning Langevin equation for Brownian motion, he states that

$$m \frac{d v}{d t}=-\zeta v+\delta F(t)$$ [...] The force during an impact is supposed to vary with extreme rapidity over the time of any observation, in fact, in any infinitesimal time interval. This clearly cannot be strictly true in any real system. Then the effects of the fluctuating force can be summarized by giving its first and second moments, as time averages over an infinitesimal time interval, $$ \langle\delta F(t)\rangle=0, \quad\left\langle\delta F(t) \delta F\left(t^{\prime}\right)\right\rangle=2 B \delta\left(t-t^{\prime}\right) $$ $B$ is a measure of the strength of the fluctuating force.

However, I cannot understand what does he mean by "the effects of the fluctuating force can be summarized by giving its first and second moments...". I mean, if we assume that the distribution of random force is normal distribution, then the first and the second moments of $\delta F$ specifies the distribution uniquely, but I don't see any such assumption and quite frankly, that would not be an assumption that I would like to make.

Question: So, what does the author mean exactly with that statement? and what is the domain of validity of that statement?

Answer 1

The force is assumed Gaussian/Normal (functional) - it is fully characterized by its mean and the correlation function (the fact that it is delta-correlated greatly simplifies things, but this is not necessary for the noise being Gaussian). The solution of this Langevin equation is a linear function of the force, i.e. linear combination of Gaussian variables, which therefore is also a Gaussian variable.