Additive proof of discrete log?
Chase et al.: Malleable Proof Systems and Applications. Eurocrypt 2012


This paper is the only one I know of that discuss a problem like this. They consider the problem of combining/modifying NIZK proofs -- for example, if I see a proof that $(A,B,C)=(g^a, g^b, g^c)$ is a DH-tuple ($c = ab$), then I can produce a proof that $(A^2, B^2, C^4)$ is a DH-tuple, even though I don't know their discrete logs.

Their focus is Groth-Sahai-style proofs in groups with bilinear pairings. I don't know how helpful the techniques will be for Schnorr-style proofs.