  Given global Lipschitz continuity and differentiability of high enough order
on the coefficients in It\^{o}'s equation, differentiability of associated
semigroups, existence of twice differentiable solutions to Kolmogorov equations
and weak convergence rates of numerical approximations are known results. In
this work and against the counterexamples of Hairer et al.(2015), the drift and
diffusion coefficients having Lipschitz constants that are $o(\log V)$ and
$o(\sqrt{\log V})$ respectively for a function $V$ satisfying $(\partial_t +
L)V\leq CV$ is shown to be a generalizing condition in place of global
Lipschitz continuity for the above.
