Anderson localization concerns the localization properties of the Schrödinger operator with a Hamiltonian of the form $$H=-\Delta+V(x),$$ where $V$ is a highly oscillatory random potential. A simple example in discrete space would be the case where $\big(V(x)\big)_{x\in\mathbb Z^d}$ is a field of i.i.d. random variables with a nice distribution. Although the results mentioned in the Wikipedia page are stated in terms of the decay of $|e^{-itH}(x,y)|^2$ (so-called dynamical localization), my understanding is that most mathematical results in this theory are formulated in terms of $H$'s spectrum, namely, discrete eigenvalues with exponentially-decaying eigenfunctions.
What I'm interested in is a time-dependent modification of the problem. Namely, consider the time-dependent Hamiltonian $$H_t=-\Delta+V(t,x),$$ where $V$ is a highly oscillatory random potential that depends on $t$. For instance, we could say that $\big(V(t,x)\big)_{x\in\mathbb Z^d}$ are i.i.d. random variables for all fixed $t$, and then for fixed $x$, the function $t\mapsto V(t,x)$ is some sort of stochastic process.
Is there anything that is known regarding the localization (or lack thereof) for these kinds of models in the mathematics literature? I expect that investigations of such models would require different techniques from what I perceive to be the standard in the time-independent case (i.e., spectral theory), since in this case the propagator of the Schrödinger equation will be a so-called ordered exponential, whose behavior bears no obvious connection (as far as I can tell) to the spectrum of the operators $H_t$.
