Holography and renormalization in Lorentzian signature

De Boer et. al. have found an asymptotic equivalence between theHamilton-Jacobi equations for supergravity in (d+1)-dimensional asymptoticanti-de Sitter space, and the Callan-Symanzik equations for the duald-dimensional perturbed conformal field theory. We discuss this correspondencein Lorentzian signature. We construct a gravitational dual of the generatingfunction of correlation functions between initial and final states, inaccordance with the construction of Marolf, and find a class of states forwhich the result has a classical supergravity limit. We show how the dataspecifying the full set of solutions to the second-order supergravity equationsof motion are described in the field theory, despite the first-order nature ofthe renormalization group equations for the running couplings: one must specifyboth the couplings and the states, and the latter affects the solutions to theCallan-Symanzik equations.Comment: 45 pages, 2 figures. v2 -- reference added. v3 -- some typos corrected, one correction in section 5.