In my field, stochastic differential equations are written like
$$ \frac{dr_i}{dt} = \sum_j F(r_i - r_j) + \eta_i(t) $$ Then, the authors state that having the conditions on $\eta$ s.t. $\langle\eta_i(t)\rangle = 0$ and $\langle\eta_i(t)\eta_j(t')\rangle = T \delta_{ij}\delta(t-t')$ is equivalent to the noise, $\eta$, being uncorrelated Gaussian noise.
How does this follow? Namely, how does $$ \int \langle\eta_i(t)\eta_i(t')\rangle dt' = T $$ relate to a Gaussian?