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In the framework of the functional renormalization group and of theperturbative, algebraic approach to quantum field theory (pAQFT), in [DDPR23]it has been derived a Lorentian version of a flow equation \`a la Wetterich,which can be used to study non linear, quantum scalar field theories on aglobally hyperbolic spacetime. In this work we show that the realm of validityof this result can be extended to study interacting scalar field theories onglobally hyperbolic manifolds with a timelike boundary. By considering thespecific examples of half Minkowski spacetime and of the Poincar\'e patch ofAnti-de Sitter, we show that the form of the Lorentzian Wetterich equation isstrongly dependent on the boundary conditions assigned to the underlying fieldtheory. In addition, using a numerical approach, we are able to provide strongevidences that there is a qualitative and not only a quantitative difference inthe associated flow and we highlight this feature by considering Dirichlet andNeumann boundary conditions on half Minkowski spacetime.

On the interplay between boundary conditions and the Lorentzian Wetterich equation