Conditions for unique (not exact) recovery from noisy compressed measurements
"Recovery of Exact Sparse Representations in the Presence of Bounded Noise", by J. Fuchs, deals with the case you ask. From the abstract:


  The purpose of this contribution is to extend some recent results on sparse representations of signals in redundant bases developed in the noise-free case to the case of noisy observations. [..] We consider the case $b = Ax_0 + e$ [...] and seek conditions under which $x_0$ can be recovered from $b$ ...


I think theorem 2 and 3 are what you're looking for (at least in part).