Where do we stand now? Since Chap. 5 , we have been following up the choice to adopt a Markovian approach, which led us to a particle state vector made up by the discrete particle positions, their velocities and the velocity of the fluid seen \({\mathbf {Z}}_{\mathrm {p}}=({\mathbf {X}}_{\mathrm {p}},{\mathbf {U}}_{\mathrm {p}},{\mathbf {U}}_{\mathrm {s}})\) . In Chaps. 6 and 8 , we brought forward the physical reasons that justify to represent the velocity of the fluid seen by a Langevin model and proposed new research directions. We can wrap up our progress by saying that \({\mathbf {Z}}_{\mathrm {p}}\) is modeled as a stochastic diffusion process. This implies that each stochastic particle is characterized by the knowledge of a 9-component vector while the PDF \(p(t;{\mathbf {y}}_{\mathrm {p}},{\mathbf {V}}_{\mathrm {p}},{\mathbf {V}}_{\mathrm {s}})\) is the solution of a Fokker-Planck equation in a 9-dimensional space. We are thus evolving in a high-dimensional sample space, whose dimension is likely to increase if we consider additional phenomena requiring to add extra variables. The different ways to account for Brownian motion described in the preceding chapter suggest something enticing: can we eliminate some variables from the above description and derive a reduced description? What are the physical arguments allowing such projections into a reduced space to be made and, when justified, what is the form of SDEs characterizing this reduced stochastic model?