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

$L_1$ and $L_{\infty}$ stability of transition densities of perturbed diffusions