Rooftop envelopes and residual plurisubharmonic functions

Given a negative plurisubharmonic function $\phi$ in a bounded pseudoconvex domain of ${\mathbb C}^n$, we introduce and study its residual function $g_\phi$ determined by the asymptotic behavior of $\phi$ near its singularity points, both inside the domain and on its boundary. For certain choices of $\phi$, the function $g_\phi$ coincides with different versions of pluricomplex Green functions. The considerations are motivated by a problem on when two given plurisubharmonic functions can be connected by a plurisubharmonic geodesic.