I want to know if there are general way to construct Green's function. I understand the Green's may not have explicit formula for a PDE consider in general. But what I want to know is as follow:
Suppose I have an operator $Lv=0$ on a Domain. If based on $L$, the backward Kolmogorov equation and forward Kolmogorov equation, denoted by $L\phi=0$ and $L^{*}\psi=0$ are solvable and have explicit formula, can I construct Green's function based on $\phi$ and $\psi$ for $L$ on the given Domain. For example, if $L=\partial_{t}+\partial_{xx}$ and $L^{*}=\partial_{t}+\partial_{xx}$ and $L\phi=0$ and $L^{*}\psi=0$ are solvable, can I recover the Green function from the knowledge of $\phi$ and $\psi$?