The objective of this work is to extend the study of the Brownian motion to the case of many Brownian particles. It will be used functional Integration methods and Green functions, common in many body theory, to develop a theory of coupled fields to describe a system of interest made of many indistinguishable non-relativistic particles (fermions or bosons) immerse in a dissipative environment acting as thermal reservoir. The properties of the induced interaction between the sub-systems of interest will depend on the spectral function of excitations of the bath. The resulting non-linear theory will be used to study the phenomenon of transport in the system. We pretend to establish the criteria to propose semi-phenomenologic models describing the physical properties observed in several many-body systems.
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