  We define a class of not necessarily linear $C_0$-semigroups $(P_t)_{t\geq0}$
on $C_b(E)$ (more generally, on $C_\kappa(E):=\frac1\kappa C_b(E)$, for some
bounded function $\kappa$, which is the pointwise limit of a decreasing
sequence of continuous functions) equipped with the mixed topology
$\tau_1^{\mathscr M}$ for a large class of topological state spaces $E$. If
these semigroups are linear, classical theory of operator semigroups on locally
convex spaces as well as the theory of bicontinuous semigroups apply to them.
In particular, they are infinitesimally generated by their generator $(L,D(L))$
and thus reconstructable through an Euler formula from their strong derivative
at zero in $(C_b(E),\tau_1^{\mathscr M})$. In the linear case, we characterize
such $(P_t)_{t\geq0}$ as integral operators given by measure kernels satisfying
certain tightness properties. As a consequence, transition semigroups of Markov
processes are $C_0$-semigroups on $(C_b(E),\tau_1^{\mathscr M})$, if they leave
$C_b(E)$ invariant and they are jointly weakly continuous in space and time.
Hence, they can be reconstructed from their strong derivative at zero and thus
have a fully infinitesimal description. Furthermore, we introduce the notion of
a Markov core operator $(L_0,D(L_0))$ for the above generators $(L,D(L))$ and
prove that uniqueness of the Fokker-Planck-Kolmogorov equations corresponding
to $(L_0,D(L_0))$ for all Dirac initial conditions implies that $(L_0,D(L_0))$
is a Markov core operator for $(L,D(L))$. If each $P_t$ is merely convex, we
prove that $(P_t)_{t\geq0}$ gives rise to viscosity solutions to the Cauchy
problem given by its associated (nonlinear) infinitesimal generator. We also
show that value functions of optimal control problems, both, in finite and
infinite dimensions are particular instances of convex $C_0$-semigroups on
$(C_\kappa(E),\tau_\kappa^{\mathscr M})$.
