Kalman Filter Process Noise - Model Where It Vanished
The process noise $w(t)$, which is typically assumed to be a zero mean, white Gaussian noise with (power) variance $\sigma^2$, is used to account for any mismatches between the assumed dynamic model of the states and the actual truth. 

If your assumed constant state (DC) model perfectly matches with the underlying nature of the observed signal, then you should set the process noise to zero; i.e., your model is exact.

Then you only have measurement noise due to the sensor's intrinsic noise. 

If you do not set your $\sigma^2$ to zero (adding non-zero process noise to a constant dynamic model) then your state estimation will never reach steady-state, but wander around it; as you are telling the Kalman filter that the true signal is not a constant...