  We study the statistical properties of the variation of the kinetic energy of
a spherical Brownian particle that freely moves in an incompressible fluid at
constant temperature. Based on the underdamped version of the generalized
Langevin equation that includes the inertia of both the particle and the
displaced fluid, we derive an analytical expression for the probability density
function of such a kinetic energy variation during an arbitrary time interval,
which exactly amounts to the energy exchanged with the fluid in absence of
external forces. We also determine all the moments of this probability
distribution, which can be fully expressed in terms of a function that is
proportional to the velocity autocorrelation function of the particle. The
derived expressions are verified by means of numerical simulations of the
stochastic motion of a particle in a viscous liquid with hydrodynamic backflow
for representative values of the time-scales of the system. Furthermore, we
also investigate the effect of viscoelasticity on the statistics of the kinetic
energy variation of the particle, which reveals the existence of three distinct
regimes of the energy exchange process depending on the values of the
viscoelastic parameters of the fluid.
