Dynamics of the infinite discrete nonlinear Schr\"odinger equation

The discrete nonlinear Schr\"odinger equation on $\Z^d$, $d \geq 1$ is an example of a dispersive nonlinear wave system. Being a Hamiltonian system that conserves also the $\ell^2(\Z^d)$-norm, the well-posedness of the corresponding Cauchy problem follows for square-summable initial data. In this paper, we prove that the well-posedness continues to hold for much less regular initial data, namely anything that has at most a certain power law growth far away from the origin. The growth condition is loose enough to guarantee that, at least in dimension $d=1$, initial data sampled from any reasonable equilibrium distribution of the defocusing DNLS satisfies it almost surely.