Stochastic quantization and holographic Wilsonian renormalization group

We study relation between stochastic quantization and holographic Wilsonian renormalization group flow. Considering stochastic quantization of the boundary on-shell actions with the Dirichlet boundary condition for certain $AdS$ bulk gravity theories, we find that the radial flows of double trace deformations in the boundary effective actions are completely captured by stochastic time evolution with identification of the $AdS$ radial coordinate `$r$' with the stochastic time '$t$' as $r=t$. More precisely, we investigate Langevin dynamics and find an exact relation between radial flow of the double trace couplings and 2-point correlation functions in stochastic quantization. We also show that the radial evolution of double trace deformations in the boundary effective action and the stochastic time evolution of the Fokker-Planck action are the same. We demonstrate this relation with a couple of examples: (minimally coupled)massless scalar fields in $AdS_2$ and U(1) vector fields in $AdS_4$.