Heat Equations in $\mathbb{R}\times \mathbb{C}$

Let $p:\mathbb{C}\to \mathbb{R}$ be a subharmonic, nonharmonic polynomial and $\tau$ a real parameter. Define $\bar{Z}{\tau p} = \partial{\bar z} + \tau p_{\bar z}$,aclosed,densely-definedoperatoron$L^2($\mathbb{C}$)$\mathrm{.}If$\Box_{\tau p} = \bar{Z}{\tau p}\bar{Z}{\tau p}^*$and$\tau>0$, we solve the heat equation $({\mathrm{\partial }}_s+{\mathrm{\square }}_{\tau p})u=0$, $u(0,z)=f(z)$, on $(0,\mathrm{\infty })\times \mathbb{C}$. The solution comes via the heat semigroup $e^{-s{\mathrm{\square }}_{\tau p}}$, and we show that $u(s,z)$ is given as integration of the intial condition against a distributional kernel $H_{\tau p}(s,z,w)$. We prove that $H_{\tau p}$ is $C^{\mathrm{\infty }}$ off the diagonal ${(s,z,w):s=0 \text{and }z=w}$andthat$H_{\tau p}$ and its derivatives have exponential decay.