Here is a heuristic reasoning.
Suppose that the function $u(x, t)$ solves $$\partial_t u = \Delta u.$$ Integrating in $t$ we can define a new function $v$: $$v(x)=\int_0^\infty u(x, t)\, dt.$$ Applying the operator $-\Delta$ to $v$ we get $$-\Delta v(x)=\int_0^\infty -\partial_t u (x, t)\, dt = u(x, 0).$$ In particular, if $u_0=\delta$, that is if $u(x, t)$ is a fundamental solution for the heat equation, then $v$ is a fundamental solution for the Laplace equation.
Question Is there some truth in the above reasoning? Can it be formalized somehow?
EDIT: I asked the owner of the local course in PDE. He replied that there is some truth in this and suggested to look for the keywords "subordination principle".