  The aim of this paper is to construct (explicit) heat kernels for some hybrid
evolution equations which arise in physics, conformal geometry and subelliptic
PDEs. Hybrid means that the relevant partial differential operator appears in
the form $\mathscr L_1 + \mathscr L_2 - \partial_t$, but the variables cannot
be decoupled. As a consequence, the relative heat kernel cannot be obtained as
the product of the heat kernels of the operators $\mathscr L_1 - \partial_t$
and $\mathscr L_2 - \partial_t$. Our approach is new and ultimately rests on
the generalised Ornstein-Uhlenbeck operators in the opening of H\"ormander's
1967 groundbreaking paper on hypoellipticity.
