Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below
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
(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.
The paper is available on the
cvgmt preprint server.