In this theoretical study, the vibration suppression, energy transfer, and bifurcation characteristics, as a function of dimensionless parameters, are investigated for an inerter-based pendulum vibration absorber (IPVA) attached to a linear single-degree-of-freedom spring-mass-damper system (primary structure), subject to white noise excitation. A perturbation method is introduced to detect and track the bifurcation points of the system. It is shown that the marginal probability density function of the pendulum angular displacement undergoes a P-bifurcation at critical parameter values, transitioning from monomodality to bi-modality. A cumulant-neglect technique is used to predict the mean squares of the system, which are compared to the response of a linear system without the IPVA to quantify the vibration suppression. It is shown that the IPVA leads to effective vibration mitigation of the structure in the neighborhood of the P-bifurcation, which is achieved by transferring the kinetic energy of the structure to the pendulum. The results are validated by a Monte Carlo simulation that is used to numerically approximate the marginal probability density function for the pendulum angle as well as the mean squares.