  It is well known that path probabilities of Brownian motion correspond to the
equilibrium configurational probabilities of flexible Gaussian polymers, while
those of active Brownian motion correspond to in-extensible semiflexible
polymers. Here we investigate the properties of the equilibrium polymer that
corresponds to the trajectories of particles acted on simultaneously by both
Brownian as well as active noise. Through this mapping we can see interesting
crossovers in mechanical properties of the polymer with changing contour
length. The polymer end-to-end distribution exhibits Gaussian behaviour for
short lengths, which changes to the form of semiflexible filaments at
intermediate lengths, to finally go back to a Gaussian form for long contour
lengths. By performing a Laplace transform of the governing Fokker-Planck
equation of the active Brownian particle, we discuss a direct method to derive
exact expressions for all the moments of the relevant dynamical variables, in
arbitrary dimensions. These are verified via numerical simulations and used to
describe interesting qualitative features such as, for example, dynamical
crossovers. Finally we discuss the kurtosis of the ABP's position which we
compute exactly and show that it can be used to differentiate between active
Brownian particles and active Ornstein-Uhlenbeck process.

Statistical Mechanics