Let $u(t,x)$ be Green's function of the heat equation and let $v(x)$ be Greens function of the Laplacian. Over what domains do we have $\int_0^{\infty}u(t,x) \,dt= v(x)$?
I'm trying to find a standard reference for this. There's two things I'm looking at. One is bounded domain vs unbounded domains and one is the dimension of $\mathbb{R}^n$
In dimensions $n<3$ the integral $$ v(x,y)=\int_0^\infty \frac{1}{(4\pi t)^{n/2}}\exp\Big(-\frac{|x-y|^2}{4t}\Big)\,dt $$ is divergent. The only reference I can offer quickly is [1] p. 111. For $n\ge 3$ this integral is Green's function for the whole domain $\mathbb R^n\,,$ that is $v(x,y)$ vanishes at infinity and is a fundamental solution for the Poisson equation $\Delta f=-\rho\,$ that is: $$ \Delta_x v(x,y)=-\varepsilon_y(x)\,. $$ For $n=1,2$ the situation is not totally hopeless when one considers instead of the Green function the Resolvent [2]: $$ R(x,y,\lambda)=\int_0^\infty e^{-\lambda t}\frac{1}{(4\pi t)^{n/2}}\exp\Big(-\frac{|x-y|^2}{4t}\Big)\,dt\,. $$ This is finite for all $\lambda>0$ and in all dimensions. It yields the fundamental solution for the equation $$ \Delta f-\lambda f=-\rho\,. $$ and for the whole domain $\mathbb R^n.$
For other domains I only remember that Green functions may exist but are sometimes hard to find. Let's first see what you think about the above.
[1] Ch. Berg, G. Forst, Potential Theory on Locally Compact Abelian Groups. Springer-Verlag Berlin Heidelberg New York 1975.
[2] R.L. Schilling, L. Partzsch, Brownian Motion. An Introduction to Stochastic Processes. de Gruyter Graduate, Berlin 2012.
