Abstract
In this paper we prove that, within the framework of \(\textsf {RCD}^*(K,N)\) spaces with \(N<\infty \), the entropic cost (i.e. the minimal value of the Schrödinger problem) admits:
A threefold dynamical variational representation, in the spirit of the Benamou–Brenier formula for the Wasserstein distance;
A Hamilton–Jacobi–Bellman dual representation, in line with Bobkov–Gentil–Ledoux and Otto–Villani results on the duality between Hamilton–Jacobi and continuity equation for optimal transport;
A Kantorovich-type duality formula, where the Hopf–Lax semigroup is replaced by a suitable ‘entropic’ counterpart.
We thus provide a complete and unifying picture of the equivalent variational representations of the Schrödinger problem as well as a perfect parallelism with the analogous formulas for the Wasserstein distance. Riemannian manifolds with Ricci curvature bounded from below are a relevant class of \(\textsf {RCD}^*(K,N)\) spaces and our results are new even in this setting.
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Funding was provided by Ministero dell’Istruzione, dell’Università e della Ricerca (Grant SIR No. RBSI147UG4).
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Proofs of Lemmas 3.3, 3.4, 3.5
Proofs of Lemmas 3.3, 3.4, 3.5
This appendix is devoted to the proofs of the auxiliary lemmas contained in Sect. 3.
Proof of Lemma 3.3
Since \((\rho _t^\varepsilon ) \in C([0,1],L^2(\mathrm{X}))\) and \((h_t^\varepsilon ),(|\nabla h_t^\varepsilon ) \in C({\mathcal {I}},L^2(\mathrm{X},e^{-V}{\mathfrak {m}}))\), all the functions appearing in the statement are continuous from \({\mathcal {I}}\) to \(L^0(\mathrm{X})\) equipped with the topology of convergence in measure on bounded sets. Therefore the continuity of \({\mathcal {I}} \ni t \mapsto \int H_t^\varepsilon \,\mathrm {d}{\mathfrak {m}}\) for these maps will follow as soon as we show that they are, locally in \(t \in {\mathcal {I}}\), uniformly dominated by an \(L^1(\mathrm{X})\) function. This will also imply all the other statements. Furthermore, it is sufficient to consider the case \(h_t^\varepsilon = \varphi _t^\varepsilon \), as the estimates for \(\psi _t^\varepsilon \) can be obtained by symmetric arguments and the ones for \(\vartheta _t^\varepsilon ,\log \rho _t^\varepsilon \) follow from the identities \(\vartheta _t^\varepsilon = \frac{\psi _t^\varepsilon - \varphi _t^\varepsilon }{2}\) and \(\varepsilon \log \rho _t^\varepsilon = \varphi _t^\varepsilon + \psi _t^\varepsilon \).
From (2.14a) and (2.14b) we immediately see that for any \({\bar{x}} \in \mathrm{X}\) and \(\delta > 0\) there exist constants \(c_1,c_2 > 0\) depending on \(K,N,\delta ,{\bar{x}},\rho _0,\rho _1\) only such that
for every \(t \in [\delta ,1]\) and \(\varepsilon \in (0,1)\). The volume growth (2.8) then implies that the right-hand side is integrable and thus the conclusion.
For \(\rho _t^\varepsilon \varphi _t^\varepsilon \) and \(\rho _t^\varepsilon |h_t^\varepsilon |^2\), observe that from (2.14b) and the fact that \(\mathrm{X}\) is a geodesic space it follows that
which means that \(\varphi _t^\varepsilon \) has quadratic growth. We then argue as before, coupling this information with (2.14a) and (2.8). \(\square \)
Proof of Lemma 3.4
Fix \(x \in \mathrm{X}\), \(M>0\) and let V be defined as in the statement. By the maximum principle for the heat flow \(\log \delta \le \phi _t^\delta \le \log (\Vert u\Vert _{L^\infty (\mathrm{X})} + \delta )\) for all \(t \ge 0\), so that
Since \(\log \) is smooth with bounded derivatives on \([\delta ,+\infty )\), the chain and Leibniz rules entail that
and, by the a priori estimates (2.3) and the Lipschitz regularization (2.5), (iii) follows. Furthermore, notice that (2.2) and the chain rule grant that \({\mathfrak {m}}\)-a.e. (3.1) holds for a.e. t; since (iii) implies in particular that \((|\nabla \phi _t^\delta |),(\Delta \phi _t^\delta ) \in L^2_{loc}((0,\infty ),L^2(\mathrm{X},e^{-V}{\mathfrak {m}}))\), this means that \((\phi _t^\delta ) \in AC_{loc}((0,\infty ),L^2(\mathrm{X},e^{-V}{\mathfrak {m}}))\) and (3.1) actually holds when the left-hand side is intended as limit of the difference quotients in \(L^2(\mathrm{X},e^{-V}{\mathfrak {m}})\).
The continuity in \(t=0\) follows by dominated convergence from (i) and the fact that for any sequence \(t_n \downarrow 0\) there exists a subsequence \(t_{n_k} \downarrow 0\) such that \({\textsf {h}}_{t_{n_k}}u \rightarrow u\)\({\mathfrak {m}}\)-a.e.
Finally, given \({\mathcal {C}} \subset \subset (0,\infty )\), observe that from (i), (iii) and the fact that \(\mu _t \le C{\mathfrak {m}}\) for all \(t \ge 0\) we get
whence integrability on \({\mathcal {C}} \times \mathrm{X}\) by Fubini’s theorem. For \(|\nabla \phi _t^\delta |\eta _t\) it is sufficient to notice that \(|\nabla \phi _t^\delta |\eta _t \le \frac{1}{2}|\nabla \phi _t^\delta |^2\eta _t + \frac{1}{2}\eta _t\) and then argue as above. \(\square \)
Proof of Lemma 3.5
The absolute continuity of \(t \mapsto \int f\,\mathrm {d}\mu _t\) and the bound (3.2) are trivial consequences of Definition 3.1. The fact that the exceptional set can be chosen independently of f follows from the separability of \(W^{1,2}(\mathrm{Y})\) and standard approximation procedures, carried out, for instance, in [14].
This implies that the second derivative in the right hand side of (3.3) exists for a.e. t, so that the claim makes sense. The absolute continuity of \(t\mapsto \int f_t\,\mathrm {d}\mu _t\) follows from the fact that for any \(t_0,t_1 \in [0,1]\), \(t_0 < t_1\) it holds
and our assumptions on \((f_t)\). Now fix a point t of differentiability for \((f_t)\) and observe that the fact that \(\frac{f_{t+h}-f_t}{h}\) strongly converges in \(L^2(\mathrm{Y})\) to \({\frac{\mathrm {d}}{{\mathrm {d}t}}}f_t\) and \(\mu _{t+h}\) weakly converges to \(\mu _t\) as \(h \rightarrow 0\) and the densities are equibounded is sufficient to get
Hence the conclusion comes dividing by h the trivial identity
and letting \(h\rightarrow 0\).
The last statement is straightforward. \(\square \)
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Gigli, N., Tamanini, L. Benamou–Brenier and duality formulas for the entropic cost on \({\textsf {RCD}}^*(K,N)\) spaces. Probab. Theory Relat. Fields 176, 1–34 (2020). https://doi.org/10.1007/s00440-019-00909-1
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DOI: https://doi.org/10.1007/s00440-019-00909-1
Mathematics Subject Classification
- 49-XX
- 60J25
- 60J60