  We introduce a Brownian $p$-state clock model in two dimensions and
investigate the nature of phase transitions numerically. As a nonequilibrium
extension of the equilibrium lattice model, the Brownian $p$-state clock model
allows spins to diffuse randomly in the two-dimensional space of area $L^2$
under periodic boundary conditions. We find three distinct phases for $p>4$: a
disordered paramagnetic phase, a quasi-long-range-ordered critical phase, and
an ordered ferromagnetic phase. In the intermediate critical phase, the
magnetization order parameter follows a power law scaling $m \sim
L^{-\tilde{\beta}}$, where the finite-size scaling exponent $\tilde{\beta}$
varies continuously. These critical behaviors are reminiscent of the double
Berezinskii-Kosterlitz-Thouless~(BKT) transition picture of the equilibrium
system. At the transition to the disordered phase, the exponent takes the
universal value $\tilde\beta = 1/8$ which coincides with that of the
equilibrium system. This result indicates that the BKT transition driven by the
unbinding of topological excitations is robust against the particle diffusion.
On the contrary, the exponent at the symmetry-breaking transition to the
ordered phase deviates from the universal value $\tilde{\beta} = 2/p^2$ of the
equilibrium system. The deviation is attributed to a nonequilibrium effect from
the particle diffusion.
