Let $\Delta f = f''$ on $\mathbb R$; we know the semigroup $(P_t)_{t \geq 0} = (e^{t\Delta})_{t \geq 0}$ acts on functions $f \in L^2(\mathbb R)$ via the heat kernel $$ u(x, t) = P_tf(x) = \int_{\mathbb R} \frac{1}{(4\pi t)^{1/2}}e^{-|x-y|^2/(4t)}f(y)dy. $$ This makes sense though for any function $f$ growing slower than any Gaussian, e.g. if $f(x) = e^x$ then $P_tf(x) = e^te^x$, and if $f(x) = x^2$ then $P_tf(x) = x^2 + 2t$. If I prescribe data at, say, $t = 1$, then I can solve backwards in time: if $u(x, 1) = P_1f(x) = e^x$ or $u(x, 1) = x^2$ then respectively the solution is $u(x, 0) = f(x) = e^{-t}e^x$ or $u(x, 0) = x^2 - 2$. Even if $u$ has very rapid growth at infinity, e.g. $u(x, 1) = e^{\alpha x^2}$ with $\alpha > 0$, we can solve to find $f(x) = \sqrt{1+4\alpha}e^{\alpha x^2/(1 + 4\alpha)}$. Formally, I know the solution should be $$e^{-\Delta}u(x, 1) = \sum_{k \geq 0} \frac{(-1)^k}{k!}\Delta^k u(x, 1)$$ which, e.g. for polynomials, is a finite sum and, e.g. for exponentials (and probably the Gaussian too), is a convergent infinite sum.
My question is: is there a class of functions (to which these non $L^2(\mathbb R)$ examples belong) for which this inversion/backwards heat equation makes sense and has nice estimates? For instance, if $u \in L^2(\gamma)$ where $\gamma$ is the standard Gaussian measure, we can write $u$ as an infinite sum of $u_n = \sum_{i=0}^\infty a_iH_i$ of Hermite polynomials (being an orthonormal base for $L^2(\gamma)$), and with enough decay on the $(a_i)$ the formal sum $f_n = e^{-\Delta}u_n$ converges to some $f \in L^2(\gamma)$ and it has the expected behaviour (that is, $P_1f = u$). But in doing so I lose (or do not know how to recover) statements about smoothness (which I expect is important since the heat kernel is smoothing) or even continuity.
I have an inkling this question is also related to the nature of the error term (based off a paper ["A well posed problem for the backward heat equation" by Miranker) in the expansion $$e^{-\Delta}u = u - \Delta u + \frac{1}{2}\Delta^2 u - \cdots$$ and I suspect the answer is that the question is "well-posed" if $u(x, 1) \leq Ce^{\alpha x}$ has at most exponential growth based on some simple examples for which the error $|e^{-\Delta u} - u| \leq C(1 + |\Delta u|)$ is roughly controlled by the first term. But I do not know if this problem is no longer well-posed if I introduce some pathological behaviour, e.g. by adding a fast oscillation like $u(x, 1) = e^x + x^2\sin^2(x^2)$ and if I need some extra conditions to prohibit this.