L 1 and L ∞ stability of transition densities of perturbed diffusions

In this paper, we derive a stability result for L 1 {L_{1}} and L ∞ {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 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 L 1 {L_{1}} - L 1 {L_{1}} metric between the transition densities of a given diffusion and the perturbed one using the McKean–Singer parametrix expansion. In the second part, we generalize the well-known result on the stability of diffusions with bounded coefficients to the case of at most linearly growing drift.

Partial regularity result for non-autonomous elliptic systems with general growth

<p style='text-indent:20px;'>In this paper we prove a partial Hölder regularity result for weak solutions <inline-formula><tex-math id="M1">\begin{document}$ u:\Omega\to \mathbb{R}^N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ N\geq 2 $\end{document}</tex-math></inline-formula>, to non-autonomous elliptic systems with general growth of the type:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} -{\rm{div}} a(x, u, Du) = b(x, u, Du) \quad \;{\rm{ in }}\; \Omega. \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>The crucial point is that the operator <inline-formula><tex-math id="M3">\begin{document}$ a $\end{document}</tex-math></inline-formula> satisfies very weak regularity properties and a general growth, while the inhomogeneity <inline-formula><tex-math id="M4">\begin{document}$ b $\end{document}</tex-math></inline-formula> has a controllable growth.</p>