Consider a general parabolic partial differential equation with its spatial dimensions on $R^n$, such as a heat equation with the diffusion coefficient possibly dependent on the spacial variables. Does its Green's function $g(t,x)\rightarrow 0$ as $x\rightarrow\infty$ for any fixed time $t$? If it is not in general:
What would be a counterexample?
What are reasonable but general conditions on the diffusion coefficients for the Green's function to possess that limiting property?