Initial dirac distribution for heat kernel

[pabryan]

Does it follow from the set equation in defn 4.1 that $\rho_t \to \delta$ as $t \to 0$ distributionally, or must that be imposed as an additional condition in the definition?

[lashputin]

It probably follows, but one would have to prove that any such L^2 solution is given by the semigroup. That takes extra work, and it's just adding technicalities for no reason. I think it's okay just to ask for this because it's a definiton.