L1 stability of transition densities of perturbed degenerate diffusions

In this paper, we derive a stability result for L1 perturbations of degenerate diffusions under weak regularity conditions on the coefficients. In particular, the drift terms we consider can be unbounded with at most linear growth, and the estimates reflect the transport of the initial condition by the unbounded drift through the corresponding flow. Our approach is based on the study of the distance in L1-L1 metric between the transition densities of a given diffusion and the perturbed one using the McKean – Singer parametrix expansion.