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The real Gaussian function heat kernel $(4\pi \epsilon z{)}^{-1/2}\exp [-(\varphi -\theta {)}^2/(4\epsilon z)]$ is integrable only when $\epsilon z>0$. 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.

The remnant profile is the time integral of a decaying heat kernel. Its length scale is $\sqrt{D/\mu }$ and its total mass is $N\lambda /\mu$, the initial population times the mean deposition over an exponential lifetime.