Variable Exponent Wasserstein Spaces: Stability of Entropy Convexity and Modified Rényi Entropy

We study the Wasserstein space $\mathcal{P}(M)$ equipped with a distance $\text{Wp}$ constructed from the Lagrangian $L(x,v)=\mathrm{\mid }v{\mathrm{\mid }}^{p(x)}$ where $p(x)=2+\epsilon (x)$ with $\epsilon$ small. Building on the fundamental work of Lott and Villani on the $K$-geodesic convexity of the Boltzmann entropy in $(\mathcal{P}(M),\text{Wb})$, we establish a generalized inequality showing that the entropy remains $(K-C\mathrm{\parallel }\epsilon {\mathrm{\parallel }}_{\mathrm{\infty }})$-convex along $\text{Wp}$-geodesics. We then introduce a modified Rényi entropy that exactly compensates the logarithmic divergence that appears in the expansions of ${\text{Wp}}^2$, obtaining thus a sharp equivalence that reaveals the Bakry-Émery tensor as the effective curvature in the variable exponent setting. As applications, we derive perturbed versions of the Log-Sobolev and Talagrand inequalities in variable exponent Wasserstein spaces, showing that these fundamental functional inequalities are robust under small perturbations of the transport exponent. This work generalizes the Lott-Villani theorem and its consequences (\emph{J. Lott and C. Villani, Ann. of Math. $\mathbf{\text{169}}$ (2009), 903-991}) to situations where the transport metric varies spatially.