  We examine under which conditions the canonical heat flow on glued manifolds
is ergodic and irreducible. Glued manifolds are spaces consisting of manifolds
of varying dimension connected by a weakly doubling measure. Moreover, we
construct a non-local perimeter functional, the heat excess, to raise the
question of its ${\Gamma}$-convergence to the standard perimeter functional. In
this context, we connect our work to the previous work on the convergence of
perimeter functionals, approximations, and existence of heat kernels, as well
as short-time expansions of Brownian motion.
