Does renormalization flow work in curved spacetime?

Renormalization group equations describe how quantum fields evolve under changes in scale, but applying them to curved spacetime has been mathematically unclear. This work derives the Wetterich RG flow equations for interacting scalar and Dirac fields on curved backgrounds using algebraic quantum field theory, computing the beta functions that govern coupling evolution and proving the solutions actually exist. The results hint at a deeper link between renormalization and stochastic field theory.