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Functional Hamilton-Jacobi (HJ) equation, the central equation of theholographic renormalization group (HRG), functional Schr\"{o}dinger equation,and generalized Wilson-Polchinski (WP) equation, the central equation of thefunctional renormalization group (FRG), are considered in $D$-dimensionalcoordinate and abstract (formal) spaces. Instead of extra coordinates or an FRGscale, a holographic scalar field $\varLambda$ is introduced. The extracoordinate (or scale) is obtained as the amplitude of delta-field or constantfield configurations of $\varLambda$. A rigorous derivation of correspondingintegro-differential equation hierarchies for Green functions (GFs) as well asthe integration formula for functionals are given. Using the integrationformula, the functional (arbitrary configuration of $\varLambda$) solution forthe translation-invariant two-particle GF is obtained. For the delta-field andthe constant field configurations of $\varLambda$ this solution is studied indetail. Separable solution for two-particle GF is briefly discussed. Then,rigorous derivation of the quantum HJ and the continuity functional equationsfrom the functional Schr\"{o}dinger equation as well as the semiclassicalapproximation are given. An iterative procedure for solving the functionalSchr\"{o}dinger equation is suggested. Translation-invariant solutions forvarious GFs (both hierarchies) on delta-field configuration of $\varLambda$ areobtained. In context of continuity equation and open quantum field systems anoptical potential is briefly discussed. Modes coarse graining growth functionalfor WP functional is analyzed. An approximation scheme is proposed for thegeneralized WP equation. With an optimized regulator translation-invariantsolutions for two-particle and four-particle amputated GFs from approximated WPhierarchy are found analytically.

On Functional Hamilton-Jacobi and Schr\"{o}dinger Equations and Functional Renormalization Group