Momentum-space Langevin dynamics of holographic Wilsonian RG flow: self-interacting massive scalar field with marginal deformation

Abstract We explore the mathematical relationship between the holographic Wilsonian renormalization group (HWRG) and stochastic quantization(SQ) motivated by the similarity of the monotonicity in RG flow with Langevin dynamics of non-equilibrium thermodynamics. We look at scalar field theory in AdS space with its generic mass, self-interaction, and marginal boundary deformation in the momentum space. Identifying the stochastic time t with radial coordinate r in AdS, we establish maps between the fictitious time evolution of stochastic multi-point correlation function and the radial evolution of multi-trace deformation, which respectively, express the relaxation process of Langevin dynamics and holographic RG flow. We show that the multi-trace deformations in the HWRG are successfully captured by the Langevin dynamics of SQ.