Non-stationary Lattice Anderson Model with Non-local Laplacian and Correlated White Noise

We study the non-stationary Anderson parabolic problem on the lattice $Z^d$, i.e., the equation \begin{equation}\label{andersonmodel} \begin{aligned} \frac{\partial u}{\partial t} &=\varkappa \mathcal{A}u(t,x)+\xi_{t}(x)u(t,x) u(0,x) &\equiv 1, \, (t,x) \in [0,\infty)\times Z^d. \end{aligned} \end{equation} Here $\mathcal{A}$ is non-local Laplacian, $\xi_t (x), \ t \geq 0, \ x \in Z^d$ is the family of the correlated white noises and $\varkappa >0$ is the diffusion coefficient. The changes of $\varkappa$ (large versus small) are responsible for the qualitative phase transition in the model. At the first step the analysis of the model is reduced to the solution of the stochastic differential equation(SDE) (in the standard It\^{o}'s form) on the weighted Hilbert space $l^2(Z^d,\mu)$ with appropriate measure $\mu$. The equations of first two moments of the solution $u(t,x)$ are derived and studied using the spectral analysis of the corresponding Schr\"{o}dinger operators with special class of the positive definite potentials. The analysis reveals several bifurcations depending on the properties of the kernel of $\mathcal{A}$ and the correlation function in the potential.