  Motivated from the classical expressions of the mean squared displacement and
the velocity autocorrelation function of Brownian particles suspended either in
a Newtonian viscous fluid or trapped in a harmonic potential, we show that for
all time-scales the mean squared displacement of Brownian microspheres with
mass $m$ and radius $R$ suspended in any linear, isotropic viscoelastic
material is identical to the creep compliance of a linear mechanical network
that is a parallel connection of the linear viscoelastic material with an
inerter with distributed inertance, $m_R=\frac{m}{\text{6}\pi R}$. The
synthesis of this mechanical network leads to the statement of a
viscous-viscoelastic correspondence principle for Brownian motion which
simplifies appreciably the calculations of the mean squared displacement and
the velocity autocorrelation function of Brownian particles suspended in
viscoelastic materials where inertia effects are non-negligible at longer
time-scales. The viscous-viscoelastic correspondence principle established in
this paper by introducing the concept of the inerter is equivalent to the
viscous-viscoelastic analogy adopted by Mason and Weitz (1995).
