A Bismut-Elworthy inequality for a Wasserstein diffusion on the circle

We introduce in this paper a strategy to prove gradient estimates for some infinite-dimensional diffusions on $L_2$-Wasserstein spaces. For a specific example of a diffusion on the $L_2$-Wasserstein space of the torus, we get a Bismut-Elworthy-Li formula up to a remainder term and deduce a gradient estimate with a rate of blow-up of order $\mathcal{O}(t^{-(2+\epsilon)})$.