  The peculiar velocity encodes rich information about the formation, dynamics,
evolution, and merging history of binary black holes. In this work, we employ a
hierarchical Bayesian model to infer the peculiar velocity distribution of
binary black holes for the first time using GWTC-3 by assuming a
Maxwell-Boltzmann distribution for the peculiar velocities. The constraint on
the peculiar velocity distribution parameter is rather weak and uninformative
with the current GWTC-3 data release. However, the measurement of the peculiar
velocity distribution can be significantly improved with the next-generation
ground-based gravitational wave detectors. For instance, the uncertainty on the
peculiar velocity distribution parameter will be measured within $\sim$ 10\%
with $10^3$ golden binary black hole events for the Einstein Telescope. We,
therefore, conclude that our statistical approach provides a robust inference
for the peculiar velocity distribution.

Cosmology and Nongalactic Astrophysics;General Relativity and Quantum Cosmology