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We build a regular version of the field $Z_{\beta}(t,x|s,y)$ which describesthe Green's function, or fundamental solution, of the parabolic Anderson model(PAM) with white noise forcing on $\mathbb{R}^{1+1}$: $\partial_t Z_{\beta}(t,x| s,y) =$ $\frac{1}{2}\partial_{xx} Z_{\beta}(t,x|s,y) + \beta Z_{\beta}(t,x |s,y)W(t,x)$, $Z_{\beta}(s,x | s,y) = \delta(x-y)$ for all $-\infty < s \leq t <\infty$, all $x,y \in \mathbb{R}$, and all $\beta \in \mathbb{R}$simultaneously. Through the superposition principle, our construction gives apointwise coupling of all solutions to the PAM with initial or terminalconditions satisfying sharp growth assumptions, for all initial and terminaltimes. Using this coupling, we show that the PAM with a (sub-)exponentiallygrowing initial condition admits conserved quantities given by the limits$\displaystyle \lim_{x\to \pm\infty} x^{-1}\log Z_{\beta}(t,x)$, in addition toproving many new basic properties of solutions to the PAM with general initialconditions. These properties are then connected to the existence, regularity,and continuity of the quenched continuum polymer measures. Through the polymerconnection, we also show that the kernel $(x,y) \mapsto Z_{\beta}(t,x | s,y)$is strictly totally positive for all $t>s$ and $\beta\in \mathbb{R}$.

The Green's function of the parabolic Anderson model and the continuum directed polymer