Rényi's entropy on Lorentzian spaces. Timelike curvature-dimension conditions

For a Lorentzian space measured by $\mathfrak{m}$ in the sense of Kunzinger, Sämann, Cavalletti, and Mondino, we introduce and study synthetic notions of timelike lower Ricci curvature bounds by $K\in \mathrm{R}$ and upper dimension bounds by $N\in [1,\mathrm{\infty })$, namely the timelike curvature-dimension conditions ${\mathrm{T}\mathrm{C}\mathrm{D}}_p(K,N)$ and ${\mathrm{T}\mathrm{C}\mathrm{D}}_p^{\ast }(K,N)$ in weak and strong forms, where $p\in (0,1)$, and the timelike measure-contraction properties $\mathrm{T}\mathrm{M}\mathrm{C}\mathrm{P}(K,N)$ and ${\mathrm{T}\mathrm{M}\mathrm{C}\mathrm{P}}^{\ast }(K,N)$. These are formulated by convexity properties of the Rényi entropy with respect to $\mathfrak{m}$ along ${\mathrm{\ell }}_p$-geodesics of probability measures. We show many features of these notions, including their compatibility with the smooth setting, sharp geometric inequalities, stability, equivalence of the named weak and strong versions, local-to-global properties, and uniqueness of chronological ${\mathrm{\ell }}_p$-optimal couplings and chronological ${\mathrm{\ell }}_p$-geodesics. We also prove the equivalence of ${\mathrm{T}\mathrm{C}\mathrm{D}}_p^{\ast }(K,N)$ and ${\mathrm{T}\mathrm{M}\mathrm{C}\mathrm{P}}^{\ast }(K,N)$ to their respective entropic counterparts in the sense of Cavalletti and Mondino. Some of these results are obtained under timelike $p$-essential nonbranching, a concept which is a priori weaker than timelike nonbranching.