In K.T. Sturm's "On the geometry of metric measure spaces. I", Definition 4.5, he introduces a "lax" version of the usual CD(K,$\infty$) lower bound. Namely, one allows for $\epsilon$-approximate length-minimizing curves, so that a metric measure space satisfies "lax CD(K,$\infty$)" (in Sturm's notation, $\underline{Curv}_{lax} (M,d,m)\geq K$) provided that, for any $\epsilon > 0$, any two measures $\rho_0 $ and $\rho_1$ absolutely continuous with respect to $m$, and any absolutely continuous curve $\rho_t$ connecting these measures with length $\leq W_2 (\rho_0, \rho_1)+\varepsilon$, then $$ Ent(\rho_t \mid m) \leq (1-t) Ent( \rho_0 \mid m) + t Ent (\rho_1 \mid m) - 0.5 K t (1-t) W_2^2(\rho_0 , \rho_1 ) + \epsilon.$$

Sturm goes on to show this coincides with the usual CD(K,$\infty$) lower bound if the underlying space is compact.

My question is: has anyone else used this lax CD(K,$\infty$) notion, in the literature?

In K.T. Sturm's "On the geometry of metric measure spaces. I", Definition 4.5, he introduces a "lax" version of the usual CD(K,$\infty$) lower bound. Namely, one allows for $\epsilon$-approximate length-minimizing curves, so that a metric measure space satisfies "lax CD(K,$\infty$)" (in Sturm's notation, $\underline{Curv}_{lax} (M,d,m)\geq K$) provided that, for any $\epsilon > 0$, any two measures $\rho_0 $ and $\rho_1$ absolutely continuous with respect to $m$, there exists an absolutely continuous curve $\rho_t$ connecting these measures with length $\leq W_2 (\rho_0, \rho_1)+\varepsilon$, with $$ Ent(\rho_t \mid m) \leq (1-t) Ent( \rho_0 \mid m) + t Ent (\rho_1 \mid m) - 0.5 K t (1-t) W_2^2(\rho_0 , \rho_1 ) + \epsilon.$$

Sturm goes on to show this coincides with the usual CD(K,$\infty$) lower bound if the underlying space is compact.

My question is: has anyone else used this lax CD(K,$\infty$) notion, in the literature?