  The fluctuation-dissipation theorem, in the Kubo original formulation, is
based on the decomposition of the thermal agitation forces into a dissipative
contribution and a stochastically fluctuating term. This decomposition can be
avoided by introducing a stochastic velocity field, with correlation properties
deriving from linear response theory. Here, we adopt this field as the
comprehensive hydrodynamic/fluctuational driver of the kinematic equations of
motion. With this description, we show that the Langevin equations for a
Brownian particle interacting with a solvent fluid become particularly simple
and can be applied even in those cases in which the classical approach, based
on the concept of a stochastic thermal force, displays intrinsic difficulties
e.g., in the presence of the Basset force. We show that a convenient way for
describing hydrodynamic/thermal fluctuations is by expressing them in the form
of Extended Poisson-Kac Processes possessing prescribed correlation properties
and a continuous velocity density function. We further highlight the importance
of higher-order correlation functions in the description of the stochastic
hydrodynamic velocity field with special reference to short-time properties of
Brownian motion. We conclude by outlining some practical implications in
connection with the statistical description of particle motion in confined
geometries.
