Evolution in a Gaussian Random Field

We consider an evolution process in a Gaussian random field $V(q)$ with the mean $\bigl\langle V(q)\bigr\rangle=0$ and the correlation function $W\bigl(|\mathbf{q}-\mathbf{q}^{\prime}|\bigr)\equiv \bigl\langle V(q)V(q^{\prime})\bigr\rangle$ where $\mathbf{q}\in \mathbb{R}^{d}$, $d$ is the dimension of the Euclidean space $\mathbb{R}^{d}$. For the value $\bigl\langle G(\mathbf{q},t;\mathbf{q}_{0})\bigr\rangle$, $t>0$, of the Green's function of the evolution equation averaged over all realizations of the random field, we use the Feynman–Kac formula to establish an integral equation that is invariant with respect to a continuous renormalization group. This invariance property allows using the renormalization group method to find an asymptotic expression for $\bigl\langle G(\mathbf{q},t;\mathbf{q}_{0})\bigr\rangle$, $|\mathbf{q}-\mathbf{q}_{0}|\rightarrow\infty$ and $t\rightarrow\infty$.