Is there a known mathematical foundation to the concept of emergence?

There is no method (known) in full generality. The does exist a huge amount of literature, mainly in physics, on deriving the equations of continuous or "large" systems as a statistical mechanics limit of N-particle systems.

Emergence usually means something broader than reduction of macroscopic equations to microscopic ones. The idea more is different is that phenomena and quantities that describe a large system can be qualitatively different than those useful for the description of its microscopic building blocks, and knowing the lower level completely is not always sufficient to understand the higher level.

There is no known way of deriving the Navier-Stokes equation from the Bolzmann equations.

There are attempts at putting emergence on a firm mathematical foundation in very wide generality. While the following introduces it only in the context of cellular automata, it generalises well to other domains:

Robert S. MacKay.Space-time phases, page 387–426. London Mathematical Society Lecture Note Series. Cambridge University Press, 2013.

See also this paper for another method for quantifying emergence using shannon entropy:

ROBIN C. BALL, MARINA DIAKONOVA, and ROBERT S. MACKAY.Quantifying emergence in terms of persistent mutual information.Advancesin Complex Systems, 13(03):327–338, 2010.