  In this paper, we derive a stability result for $L_1$ and $L_{\infty}$
perturbations of diffusions under weak regularity conditions on the
coefficients. In particular, the drift terms we consider can be unbounded with
at most linear growth, and we do not require uniform convergence of perturbed
diffusions. Instead, we require a weaker convergence condition in a special
metric introduced in this paper, related to the Holder norm of the diffusion
matrix differences. Our approach is based on a special version of the
McKean-Singer parametrix expansion.
