*Submitted Paper*

**Inserted:** 10 jun 2011

**Last Updated:** 17 nov 2012

**Year:** 2011

**Abstract:**

This and a companion forthcoming paper are devoted to a deeper understanding of the heat flow in metric measure spaces $(X,d,m)$. Our results apply in particular to spaces satisfying Ricci curvature bounds in the sense of Lott & Villani and Sturm. Indeed, the development of a ``calculus'' in this class of spaces is one of our motivations. In this paper the main goals are:

(i) The proof of equivalence of the heat flow in $L^2$ generated by a suitable Dirichlet energy and the Wasserstein gradient flow in the space of probability measuress of the relative entropy functional w.r.t. $m$.

(ii) The equivalence of two weak notions of modulus of the
gradient: the first one (inspired by Cheeger), that we call *relaxed
gradient*, is defined by $L^2(X,\mm)$-relaxation of the pointwise
Lipschitz constant in the class of Lipschitz functions; the second
one (inspired by Shanmugalingam), that we
call *weak upper gradient*, is based on the validity of the
fundamental theorem of calculus along almost all curves. These two
notions of gradient will be compared and identified under very mild
assumptions on $(X,d,m)$ which include all finite measures.
Under additional assumptions, fulfilled in $LSV$ spaces,
these derivatives will be identified
with a third object, namely the energy density appearing in the
so-called Fisher information functional, representing
the energy dissipation rate of entropy w.r.t. the Wasserstein
distance.

(iii) A fine and very general analysis of the differentiability properties of a large class of Kantorovich potentials, in connection with the optimal transport problem.

**Tags:**
GeMeThNES

**Keywords:**
Optimal transport, entropy, Ricci curvature, Heat Flow

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