Consider the heat equation $$ \partial_tu=H(t)u, $$ where $H(t)=H_0+\epsilon H_1(t)$ is some generic parabolic differential operator in one dimension. For $\epsilon=0$ the unperturbed system is associated with the $C_0$ semigroup $$ T=\{T(t)=e^{H_0t}\,|\,t\in[0,\infty)\}. $$ Now, in analogy with quantum mechanics, I would like to do time-dependent perturbation theory and I would need the inverse operator $$ T^{-1}(t)=T(-t)=e^{-H_0t}. $$ This would naturally be interpreted as propagation backwards in time. But strictly speaking this object does not exist because its kernel blows up $$ u(x,t-\tau)=\int dy\frac{1}{\sqrt{4\pi(-\tau)}}\exp\left(+\frac{(x-y)^2}{4\tau}\right)u(y,t),\quad(\tau>0) $$
My question:
Is there a proper way to define this inverse, and as such embed the semigroup into a larger group?
A naive try would be to regularize the integrals by adding a small imaginary part to the time and use analytic continuation where necessary (to avoid the square-root branch cuts for negative time arguments): $$ u(x,t-\tau)=\int dy\frac{1}{\sqrt{4\pi(-\tau\pm i0)}}\exp\left(-\frac{(x-y)^2}{4(-\tau\pm i0)}\right)u(y,t),\quad(\tau>0) $$
Edit:
I did some research on the subject and I tend to feel that the proper theory of extending the semigroup to complex times is that of the analytical semigroups which deals with sectorial operators/generators. For example if $H_0$ is the Laplacian then one can show it is sectorial and the time evolution operator $e^{H_0t}$ can be extended to complex times, that is to the whole complex plane excluding the negative reals.