Low Order Perturbation Theory

We are analyzing what happens as a particle moves from $x$ to $x^{\prime }$ with some interactions. The free propagator describes how the particle moves with no interactions. There is different notation that seems to get used to convey the same idea. For example

$G_0(x),G_0(x-x^{\prime })$

Which both convey a particle starting at some point then moving just changing where the origin is.

Now say we wanted to include some interaction term like a ${\varphi }^4$.