= Positive diffusivity requirement for the forward heat kernel
{title2=$\varepsilon z>0$}
The real <Gaussian function> <heat kernel> $(4\pi\varepsilon z)^{-1/2}\exp[-(\phi-\theta)^2/(4\varepsilon z)]$ is integrable only when $\varepsilon z>0$. For forward time $z>0$, this requires positive diffusivity. A <Cole-Hopf transformation> is an algebraic substitution for any nonzero coefficient, but that does not make the real forward <convolution> valid for negative diffusivity. Constant initial heat data already produce a divergent integral with the wrong sign.
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