Diffusion processes play an important role in describing systems in many fields of science, as in physics, biology, finance and social science. One of the most famous examples of the diffusion process is the Brownian motion. 

At mesoscopic scale, the Brownian theory describes the very irregular and animated motion of a particle suspended in a fluid. In this thesis, the dynamics and thermodynamics of diffusion processes driven out of equilibrium, at mesoscopic scale, are investigated. 

For dynamics, the theory of Brownian motion for a particle which is able to rotate and translate in three dimensions is presented. 

Moreover, it is presented how to treat diffusion process on n-dimensional Riemann manifolds defining the Kolmogorov forward equation on such manifold.

For thermodynamics, this thesis describes how to define thermodynamics quantities at mesoscopic scale using the tools of Brownian theory. The theory

of stochastic energetics and how to compute entropy production along a trajectory are presented introducing the new field of stochastic thermodynamics.

Moreover, the "anomalous entropy production" is introduced. This anomaly in the entropy production arises when diffusion processes are driven out of equilibrium by space dependent temperature field. The presence of this term expresses the fallacy of the overdamped approximation in computing thermodynamic quantities. 

In the first part of the thesis the translational and rotational motion of an ellipsoidal particle in a heterogeneous thermal environment, with a space-dependent temperature field, is analyzed from the point of view of stochastic thermodynamics. 

In the final part of the thesis, the motion of a Brownian rigid body three-dimensional space in a homogeneous thermal environment under the presence of an external force field is analyzed, using multiscale method and homogenization.