Random process in a homogeneous Gaussian field

We consider a random process in a spatial-temporal homogeneous Gaussian field $V(\mathbf q,t)$ with the mean $\mathbf EV=0$ and the correlation function $W(|\mathbf q-\mathbf q'|,|t-t'|)\equiv\mathbf E[V(\mathbf q,t)V(\mathbf q',t')]$, where $\mathbf q\in\mathbb R^d$, $t\in\mathbb R^+$, and $d$ is the dimension of the Euclidean space $\mathbb R^d$. For a “density” $G(r,t)$ of the familiar model of a physical system averaged over all realizations of the random field $V$, we establish an integral equation which has the form of the Dyson equation. The invariance of the equation under the continuous renormalization group allows using the renormalization group method to find an asymptotic expression for $G(r,t)$ as $r\to\infty$ and $t\to\infty$.