Stochastic quantization and holographic Wilsonian renormalization group of scalar theory with generic mass, self-interaction and multiple trace deformation

We explore the mathematical relationship between holographic Wilsonian renormalization group(HWRG) and stochastic quantization(SQ) of scalar field theory with its generic mass, self-interaction and $n$-multiple-trace deformation on the $d$-dimensional conformal boundary defined in AdS${}_{d+1}$ spacetime. We understand that once we define our Euclidean action, $S_E$ as $S_E\equiv -2S_B$, then the stochastic process will reconstruct the holographic Wilsonian renormalization group data via solving Langevin equation and computing stochastic correlation functions. The $S_B$ is given by $S_B=S_{\mathrm{c}\mathrm{t}}+S_{\mathrm{d}\mathrm{e}\mathrm{f}}$, where $S_{\mathrm{c}\mathrm{t}}$ is the boundary counter term and $S_{\mathrm{d}\mathrm{e}\mathrm{f}}$ is the boundary deformation which gives a boundary condition. In our study, we choose the boundary condition adding (marginal)$n$-multiple trace deformation to the holographic dual field theory. In this theory, we establish maps bewteen ficticious time, $t$ evolution of stochastic $n$-point, ($2n-2$)-point correlation functions and the (AdS)radial, $r$ evolution of $n$-multiple-trace and ($2n-2$)-multiple-trace deformations respectively once we take identifications of $r=t$ and between some of constants appearing in both sides.