The Symmetric exclusion process (SEP) is the simplest (IPS) in which particles jump randomly to an unoccupied neighbour site with symmetric rate. In this case, the rate depends on a distance-based kernel. The hydrodynamic limit is obtained on random neighbourhood graphs approximating a sub-Riemannian manifold via duality. 

At macroscopic time-scale, this system is described by a diffusion process whose drift is induced by the initial location of the sites drawn according to a Boltzmann-type probability measure. This process can also be lifted from a base Riemannian manifold to its principal bundle to describe horizontal diffusions.

Geodesic random walks (GRW) are a type of random walk designed on a Riemannian manifold in order to approximate the Brownian motion with the Laplace-Beltrami operator as generator. Eels-Elworthy-Malliavin use a fundamental relation between this operator and the horizontal Laplacian on the orthonormal frambe bundle O(M) to define the Brownian motion by the rolling without slipping method.

The (GRW) can be lifted to O(M) through this identity defining a Sub-Riemannian random walk, and more generally random walks in submersions, yielding different types of invariance principles.

The Vlasov–Poisson system describes the evolution of electrons in an idealised plasma without collision. Under some physical assumptions on the energy and initial datum, the well-posedness of the system is guaranteed.

Relaxing the conditions on the initial datum can still lead to existence and or uniqueness of the solution. The stability estimates are optained in Wasserstein or in kinetic Wasserstein distances taking into additional consideration the anisotropy between the position and velocity of the transport flow.