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

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.