We consider homogeneous open quantum walks on a lattice with finite dimensional local Hilbert space and we study in particular the position process of the quantum trajectories of the walk. We prove that the properly rescaled position process asymptotically approaches a mixture of Gaussian measures. We can generalize the existing central limit type results and give more explicit expressions for the involved asymptotic quantities, dropping any additional condition on the walk. We use deformation and spectral techniques, together with reducibility properties of the local channel associated with the open quantum walk. Further, we can provide a large deviation principle in the case of a fast recurrent local channel and at least lower and upper bounds in the general case.

This paper is concerned about random walks on random environments in the lattice Z d . This model is analyzed through ergodicity in the form of the logarithmic Sobolev inequality. We assume that the environments are random variables being independent and identically distributed. Here, we give heat kernel estimates for non-diagonal random matrices leading in dimension d ≥ 3 a Berry-Esseen upper bound with a rate of convergence t − 1 10 .

The present work aims to provide Meyers estimates throughout a finer inner regularity theory in perforated domains. We also provide a hypercontractivity property on correctors whenever the perforations are controlled with a uniformly bounded random variable and the underlying probability space admits a weaker form of ergodicity which we called coarsened logarithmic Sobolev inequality. Contents 1 Preliminaries 42 Proof of the main results 10 3 C 1,1 regularity of harmonic maps 16