Some of the noise is determined by quantum mechanics which is constant for all cameras (except Leica Ms which have a dispensation from the Creator... JUST JOKING). Contemporary high-quality sensors have essentially the same electronic noise levels and ADC noise. This means any CMOS sensor based camera sold in the past 18 months or so has similar electronic noise levels.
The signal, which is what we want and the only thing we can control, depends on how many photons are detected when the shutter is open. We can either let more light in (increase aperture) or increase the sensor area. Both increase the signal while the noise stays constant. This assumes the exposure is set such that the most intense light we care about completely fills some of the sensors. It is reasonable to assign a 1 stop advantage to a 24 x 36 mm sensor over an identical APS-C sensor. This means the XP1 lens has to let one stop more light hit the sensor to overcome the advantage of the larger sensor area.
I mis-stated the lens' aperture differences by 1/3 of a stop. But the point remains: a wide open fast lens with a small sensor has the same signal to noise ratio as a wide open slow lens with large sensor. Now lenses also have differences in transmission efficiency (T factor). But a first generation m4/3 camera with a fast lens will have the same signal to noise ratio (SNR) as a 24 x 36 mm camera with a much slower lens when both lenses are wide open. But a 25mm m4/3 lens has to have a f 0.7 lens compared to a less expensive 1.4 50mm lens for the 24 x 36 mm camera.
You asked what happens when the APS-C lens is stopped down. Both cameras have essentially the same noise. The D700 records more signal due to the sensor area. But note the D700 has less DOF (unless we move the camera). This does not affect the SNR, but it does affect the photograph. When the exposure, DOF and perspective are similar, both cameras have similar SNRs.
I was shocked when I learned well-purposed exposure can be more important than sensor size if SNR is your highest priority.
These ideas are presented much more completely and rigorously here:
http://www.falklumo.com/lumolabs/articles/equivalence/