Nonhomogeneous semigroups of measures on compact Lie groups

A system of probability measures $\{P_{uv}\}$, $a\leq u\leq v\leq b$, on a group $G$ is called a non-homogeneous semi-group if
(i) $P_{uv}P_{uw}=P_{uw}$;
(ii) $P_{t,t+\Delta}\to\delta (e)$ weakly ($\delta (e)$ is the degenerate distribution concentrated at the identity element of $G$);
(iii) $P_{tt}=\delta (e)$, $a\leq t\leq b$. Semi-groups generate stochastically continuous processes with independent increments on $G$ and vice versa.
In the paper, it is proved that for the existence (at each point $t$, $a\leq t\leq b$) of the generator of $P_{t,t+\Delta}$ ($\Delta\to 0$) on $C_2$ it is necessary and sufficient that the Fourier coefficients of measures $P_{uv}$ be continuously differentiable in $u$ and $v$. The generator is then a Hant operator.