  Motivated by the simple models of molecular motor obeying a linear
force-velocity relation, we have studied the stochastic dynamics of a Brownian
particle in the presence of a linear velocity dependent force,
$f_s(1-\frac{v}{v_0})$ where $f_{s}$ is a constant. The position and velocity
auto correlation functions in different situations of the dynamics are
calculated exactly. We observed that the velocity auto correlation function
shows an exponentially decaying behaviour with time and saturates to a constant
value in the time asymptotic limit, for a fixed $f_s$. It attains saturation
faster with increase in the $f_{s}$ value. When the particle is confined in a
harmonic well, the spectral density exhibits a symmetric behaviour and the
corresponding velocity auto correlation function shows a damped oscillatory
behaviour before decaying to zero in the long time limit. With viscous
coefficient, a non-systematic variation of the velocity auto correlation
function is observed. Further, in the presence of a sinusoidal driving force,
the correlation in velocities increases with increase in the amplitude of
driving in the transient regime. For the particle confined in a harmonic well,
the correlation corresponding to the shift relative to the average position is
basically the thermal contribution to the total position correlation. Moreover,
in the athermal regime, the total correlation is entirely due to the velocity
dependent force.
