  In this paper, we extend the paraxial conical refraction model to the case of
the partially coherent light using the unified optical coherence theory. We
demonstrate the decomposition of conical refraction correlation functions into
well-known conical refraction coherent modes for a Gaussian Schell-model
source. Assuming randomness of the electrical field phase of the input beam, we
reformulated and significantly simplified the rigorous conical refraction
theory. This approach allows us to consider the propagation of light through a
conical refraction crystal in exactly the same way as in the classical case of
coherent radiation. Having this in hand, we derive analytically the conical
refraction intensity both in the focal plane and in the far field, which allows
us to explain and rigorously justify earlier experimental findings and predict
new phenomena. The last include the counterintuitive effect of narrowing of the
conical refraction ring width, disappearance of the dark Poggendorff's ring in
the Lloyd's plane, and shift of Raman spots for the low-coherent conical
refraction light. We also demonstrate a universal power-law dependence of
conical refraction cones coherence degree on the input correlation length and
diffraction-free propagation of the low-coherent conical refraction light in
the far field.
