Is it possible to construct an inverse heat kernel in the following restricted sense: given a nice function $ h: \mathbb{R} \rightarrow \mathbb{R} $, can one find a function $ g: \mathbb{R} \rightarrow \mathbb{R} $ such that: $$ e^{-h(x)^2} = \int e^{-(x-y)^2} g(y) dy $$
I think this is equivalent to finding a nice way to express $ e^\Delta \; e^{-h(x)^2} $ but I am not having any luck with this...