The average thermal energy is $kT$ for each Harmonic oscillator, which is split equally between $kT/2$ kinetic and $kT/2$ potential. The average kinetic energy in a nonrelativistic system (or one with a quadratic kinetic energy) is always $kT/2$, for a quadratic potential, you get an equal potential contribution, while for a confining box-potential you get no potential energy contribution on average, because the potential acts in a negligible fraction of the total trajectory.
To find the Langevin dynamics appropriate to a given system coupled to a thermal bath, the coefficient of the thermal noise is determined by the condition that the Boltzmann distribution is stationary. Whether this is $1/2 kT$ or $kT$ depends on the Boltzmann distribution in question, but it's always known, since you know the energy as a function of the coordinates and velocities (or field values).
Once you find the coefficient of the Brownian noise, you make a Smulochowski approximation to find the pure Brownian limit of long times. See this answer for more detail on this limit: Cross-field diffusion from Smoluchowski approximation.