Geometry and probability on the noncommutative $2$-torus in a magnetic field

We describe the geometric and probabilistic properties of a noncommutative $2$-torus in a magnetic field. We study the volume invariance, integrated scalar curvature, and the volume form by using the operator method of perturbation by an inner derivation of the magnetic Laplacian operator on the noncommutative $2$-torus. We then analyze the magnetic stochastic process describing the motion of a particle subject to a uniform magnetic field on the noncommutative $2$-torus, and discuss the related main properties.