Wasserstein space over the Wiener space

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Gradient flows of the entropy for jump processes (2012, background): “By now, similar interpretations of the heat flow have been established in a variety of settings ranging from Riemannian manifolds to abstract metric measure spaces, see [2,15,17,19,25].”
Ricci Curvature of Finite Markov Chains via Convexity of the Entropy (2011, background): “sense, W takes over the role of the Wasserstein metric, since it is known since the seminal work by Jordan, Kinderlehrer, and Otto that the heat ﬂow on Rn is the gradient ﬂow of the entropy [23] (see [2, 18, 19, 20, 32, 36] for variations and generalisations). Convexity along W-geodesics may thus be”
Convergence of pointed non‐compact metric measure spaces and stability of Ricci curvature bounds and heat flows (2013, background): “New problems arise when one considers sequences of spaces with non-uniformly bounded measures and possibly diverging dimensions or simply deals with measures whose locally doubling constants are not uniformly bounded (as, for example, in the infinite-dimensional setting of [12, 23, 35]).”
Heat Flow on Alexandrov Spaces (2010, background): “, it has been proved that in Rn, in Riemannian manifolds with Ricci curvature bounded below and in compact Finsler manifolds (for the latter see the work of Sturm and the third author [28]), the gradient flow of the Dirichlet energy with respect to the L2-distance coincides with the gradient flow of”