Regularity preservation in Kolmogorov equations for non-Lipschitz coefficients under Lyapunov conditions

Given global Lipschitz continuity and differentiability of high enough order on the coefficients in Itô’s equation, differentiability of associated semigroups, existence of twice differentiable solutions to Kolmogorov equations and weak convergence rates of order one for numerical approximations are known. In this work and against the counterexamples of Hairer et al. (Ann Probab 43(2):468–527, https://doi.org/10.1214/13-AOP838, 2015), the drift and diffusion coefficients having Lipschitz constants that are  \(o(\log V)\) $o(logV)$ and \(o(\sqrt{\log V})\) $o(\sqrt{logV})$ respectively for a function V satisfying \((\partial _t + L)V\le CV\) $({\partial }_t+L)V\le CV$ is shown to be a generalizing condition in place of global Lipschitz continuity for the above.