This thesis concerns the study of spin systems on an infinite lattice with the Langevin
diffusion dynamics. The projects are divided into two main topics.
In the first topic we construct Gibbs measures in infinite dimensions which are reversible
with respect to the Markov semigroup associated to the Langevin dynamics. We then find sufficient conditions for the Gibbs measures to satisfy spectral gap or logarithmic Sobolev inequalities, so that one can obtain ergodicity or contractivity of the semigroup. We pay particular attention to the cases where the single-state space is a Carnot group, and the
Gibbs measure is defined by homogeneous norms with vanishing or blowing up sub-gradient.
For the second topic, we study the quantum dynamics of the system by considering
the Schrödinger equation in infinite dimensions with a logarithmic nonlinearity. Namely,
we obtain estimates that hold in arbitrary dimensions, and prove the existence of infinite-dimensional weak solutions for potentials with bounded multi-particle interactions.