Let $D>0$ be a constant. Imagine we have the following forward heat conduction problem: \begin{align*} \begin{cases} \partial_t u = D \partial_x^2u &, \quad (x,t) \in \mathbb{R} \times (0, \infty). \\ u = \phi(x) &, \quad (x,t) \in \mathbb{R} \times \{0\}. \end{cases} \end{align*} We define the Heat Kernel: \begin{align*} \Phi(x,t) = \frac{1}{\sqrt{4 \pi D t}} \exp{\left( \frac{-x^2}{4Dt}\right)} \ . \end{align*} A well know solutions to this problem is the one expressed in terms of a convolution \begin{align*} u(x,t) = (\Phi * \phi)(x,t) \ . \end{align*} Where $*$ is the convolution operator with respect to the spatial variable $x$.
Now we define the backward heat conduction problem: \begin{align*} \begin{cases} \partial_t v = -D \partial_x^2v &, \quad (x,t) \in \mathbb{R} \times (0, T). \\ v = \psi(x) &, \quad (x,t) \in \mathbb{R} \times \{0\}. \end{cases} \end{align*} This problem is not as well behaving but if we have Gaussian initial data: \begin{align*} \psi(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} \exp{\left( \frac{-x^2}{2\sigma^2}\right)} \ . \end{align*} Then a possible solution to the Backward problem is \begin{align*} v(x,t) = \frac{1}{\sqrt{2 \pi ( \sigma^2-2 D t)}} \exp{\left( \frac{-x^2}{2( \sigma^2-2 D t)}\right)} \ . \end{align*} I define $T = \sigma^2/2D$. An interesting observation is that we can write the initial data as a convolution \begin{align*} (v * \Phi)(x) = \psi(x) \ . \end{align*} Now to my question is it possible to define a Backward Heat Kernel $\Phi^{-1}(x,t)$ in the following way \begin{align*} \Phi^{-1}(x,t) = \frac{1}{\sqrt{4 \pi (-D) t}} \exp{\left( \frac{-x^2}{4(-D)t}\right)} \ . \end{align*} Such that we have \begin{align*} (\Phi^{-1} * v * \Phi)(x,t) &= (\Phi^{-1} *\psi)(x,t) \\ v &= (\Phi^{-1} *\psi)(x,t) \ . \end{align*} This integral can ofcourse not make sense in the classical sense, since $\Phi^{-1}$ is not integrable, but is it possible to make sense of it in terms of distributions? And if so can we solve if for other initial data than Gaussian?
Edit
I didn't mention that I am interested in the case where $u,v,\phi,\psi$ go to zero for $x$ goint to $\pm \infty$. One can see them as probability distributions. But all comments about how this will work, if possible, is welcome.