  The properties of the thermal force driving micron particles in
incompressible fluids are studied within the hydrodynamic theory of the
Brownian motion. It is shown that the assumption used for the hydrodynamic
Langevin equation in its usual form, according to which the random force at a
time t and the velocity of the particle at the initial time equal to zero are
uncorrelated, leads to super-diffusion of the particle. To obtain the correct
Einstein diffusion at long times, the mentioned hypothesis must be abandoned,
which however does not contradict causality. The corresponding correlations are
explicitly evaluated. We consider also the "color" of thermal noise, recently
measured experimentally (Th. Franosch et al., Nature 478, 85 (2011)), and
correct the interpretation of these experiments. The time correlation functions
for the thermal random force are obtained using the exact solution of the
Langevin equation, and on the basis of the theorem that in the linear response
theory connects the mobility of the particle and its velocity autocorrelation
function.
