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Benamou–Brenier and duality formulas for the entropic cost on \({\textsf {RCD}}^*(K,N)\) spaces

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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References

  1. Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich, 2nd edn. Birkhäuser Verlag, Basel (2008)

    MATH  Google Scholar 

  2. Ambrosio, L., Gigli, N., Savaré, G.: Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below. Invent. Math. 195, 289–391 (2014)

    Article  MathSciNet  Google Scholar 

  3. Ambrosio, L., Gigli, N., Savaré, G.: Metric measure spaces with Riemannian Ricci curvature bounded from below. Duke Math. J. 163, 1405–1490 (2014)

    Article  MathSciNet  Google Scholar 

  4. Ambrosio, L., Mondino, A., Savaré, G.: On the Bakry–Émery condition, the gradient estimates and the Local-to-Global property of \({RCD}^*({K}, {N})\) metric measure spaces. J. Geom. Anal. 26, 1–33 (2014)

    Google Scholar 

  5. Benamou, J.-D., Brenier, Y.: A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem. Numer. Math. 84, 375–393 (2000)

    Article  MathSciNet  Google Scholar 

  6. Bobkov, S.G., Gentil, I., Ledoux, M.: Hypercontractivity of Hamilton–Jacobi equations. J. Math. Pures Appl. 80, 669–696 (2001)

    Article  MathSciNet  Google Scholar 

  7. Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9, 428–517 (1999)

    Article  MathSciNet  Google Scholar 

  8. Cheeger, J., Colding, T.H.: On the structure of spaces with Ricci curvature bounded below. I. J. Differ. Geom. 46, 406–480 (1997)

    Article  MathSciNet  Google Scholar 

  9. Cheeger, J., Colding, T.H.: On the structure of spaces with Ricci curvature bounded below. II. J. Differ. Geom. 54, 13–35 (2000)

    Article  MathSciNet  Google Scholar 

  10. Cheeger, J., Colding, T.H.: On the structure of spaces with Ricci curvature bounded below. III. J. Differ. Geom. 54, 37–74 (2000)

    Article  MathSciNet  Google Scholar 

  11. Chen, Y., Georgiou, T.T., Pavon, M.: On the relation between optimal transport and Schrödinger bridges: a stochastic control viewpoint. J. Optim. Theory Appl. 169, 671–691 (2016)

    Article  MathSciNet  Google Scholar 

  12. Föllmer, H.: Random fields and diffusion processes, In: École d’Été de Probabilités de Saint-Flour XV–XVII, 1985–87. Lecture Notes in Mathematics, vol. 1362, pp. 101–203. Springer, Berlin (1988)

    Google Scholar 

  13. Gentil, I., Léonard, C., Ripani, L.: About the analogy between optimal transport and minimal entropy. Ann. Fac. Sci. Toulouse Math. Série 6(26), 569–600 (2017)

    Article  MathSciNet  Google Scholar 

  14. Gigli, N.: Nonsmooth differential geometry—an approach tailored for spaces with Ricci curvature bounded from below. Accepted at Memoirs of the American Mathematical Society. arXiv:1407.0809 (2014)

  15. Gigli, N.: On the differential structure of metric measure spaces and applications. Mem. Am. Math. Soc. 236, vi+91 (2015)

    MathSciNet  MATH  Google Scholar 

  16. Gigli, N., Han, B.: The continuity equation on metric measure spaces. Calc. Var. Partial Differ. Equ. 53, 149–177 (2013)

    Article  MathSciNet  Google Scholar 

  17. Gigli, N., Kuwada, K., Ohta, S.-I.: Heat flow on Alexandrov spaces. Commun. Pure Appl. Math. 66, 307–331 (2013)

    Article  MathSciNet  Google Scholar 

  18. Gigli, N., Mosconi, S.: The abstract Lewy–Stampacchia inequality and applications. Preprint. arXiv:1401.4911 (2014)

  19. Gigli, N., Tamanini, L.: Second order differentiation formula on \({RCD}^*({K},{N})\) spaces. Accepted at JEMS, arXiv:1802.02463 (2018)

  20. Gozlan, N., Roberto, C., Samson, P.-M.: Hamilton Jacobi equations on metric spaces and transport entropy inequalities. Revis. Mat. Iberoam. 30, 133–163 (2014)

    Article  MathSciNet  Google Scholar 

  21. Gozlan, N., Roberto, C., Samson, P.-M., Tetali, P.: Kantorovich duality for general transport costs and applications. J. Funct. Anal. 273, 3327–3405 (2017)

    Article  MathSciNet  Google Scholar 

  22. Kantorovich, L.V.: On the translocation of masses. Dokl. Akad. Nauk. USSR (NS) 37, 199–201 (1942)

    MathSciNet  MATH  Google Scholar 

  23. Léger, F.: A geometric perspective on regularized optimal transport. Preprint, arXiv:1703.10243 (2017)

  24. Léonard, C.: From the Schrödinger problem to the Monge–Kantorovich problem. J. Funct. Anal. 262, 1879–1920 (2012)

    Article  MathSciNet  Google Scholar 

  25. Léonard, C.: Some properties of path measures. Sémin. Probab. XLVI, 207–230 (2014)

    MathSciNet  MATH  Google Scholar 

  26. Léonard, C.: A survey of the Schrödinger problem and some of its connections with optimal transport. Discrete Contin. Dyn. Syst. 34, 1533–1574 (2014)

    Article  MathSciNet  Google Scholar 

  27. Mikami, T.: Monge’s problem with a quadratic cost by the zero-noise limit of \(h\)-path processes. Probab. Theory Relat, Fields 129, 245–260 (2004)

    Article  MathSciNet  Google Scholar 

  28. Mikami, T., Thieullen, M.: Duality theorem for the stochastic optimal control problem. Stoch. Process. Appl. 116, 1815–1835 (2006)

    Article  MathSciNet  Google Scholar 

  29. Mikami, T., Thieullen, M.: Optimal transportation problem by stochastic optimal control. SIAM J. Control Optim. 47, 1127–1139 (2008)

    Article  MathSciNet  Google Scholar 

  30. Mondino, A., Naber, A.: Structure theory of metric-measure spaces with lower Ricci curvature bounds I. Preprint. arXiv:1405.2222 (2014)

  31. Nagasawa, M.: Schrödinger Equations and Diffusion Theory. Monographs in Mathematics, vol. 86. Birkhäuser Verlag, Basel (1993)

    Book  Google Scholar 

  32. Nelson, E.: Dynamical Theories of Brownian Motion. Princeton University Press, Princeton, NJ (1967)

    MATH  Google Scholar 

  33. Otto, F., Villani, C.: Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173, 361–400 (2000)

    Article  MathSciNet  Google Scholar 

  34. Shanmugalingam, N.: Newtonian spaces: an extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoam. 16, 243–279 (2000)

    Article  MathSciNet  Google Scholar 

  35. Tamanini, L.: Analysis and geometry of RCD spaces via the Schrödinger problem. Ph.D. thesis, Université Paris Nanterre and SISSA (2017)

Download references

Acknowledgements

Funding was provided by Ministero dell’Istruzione, dell’Università e della Ricerca (Grant SIR No. RBSI147UG4).

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Correspondence to Nicola Gigli.

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Dedicated to the memory of Prof. Kazumasa Kuwada.

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Proofs of Lemmas 3.3, 3.4, 3.5

Proofs of Lemmas 3.33.43.5

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

$$\begin{aligned} \rho _t^\varepsilon |\nabla \varphi _t^\varepsilon |^2 \le c_1\big (1+\textsf {d}^2(\cdot ,{\bar{x}})\big )\exp \big (-c_2\textsf {d}^2(\cdot ,{\bar{x}})\big ) \qquad {\mathfrak {m}}\text {-a.e.}\end{aligned}$$

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

$$\begin{aligned} |\varphi _t^\varepsilon (x) - \varphi _t^\varepsilon ({\bar{x}})| \le C_{\delta }\textsf {d}(x,{\bar{x}})(1+\textsf {d}(x,{\bar{x}})) \le C_{\delta }(1+\textsf {d}^2(x,{\bar{x}})) \quad \forall t \in [\delta ,1], \end{aligned}$$

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

$$\begin{aligned} \sup _{t \in [0,\infty )}\Vert \phi _t^\delta \Vert _{L^\infty (\mathrm{X})} < \infty \qquad \text {and} \qquad (\phi _t^\delta ) \in L^\infty ((0,\infty ),L^2(\mathrm{X},e^{-V}{\mathfrak {m}})). \end{aligned}$$

Since \(\log \) is smooth with bounded derivatives on \([\delta ,+\infty )\), the chain and Leibniz rules entail that

$$\begin{aligned} |\nabla \phi _t^\delta | \le \frac{|\nabla {\textsf {h}}_t u|}{\delta } \quad |\Delta \phi _t^\delta | \le \frac{|\Delta {\textsf {h}}_t u|}{\delta } + \frac{|\nabla {\textsf {h}}_t u|^2}{\delta ^2} \end{aligned}$$

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

$$\begin{aligned} \sup _{t \in {\mathcal {C}}}\int |\phi _t^\delta |\eta _t\,\mathrm {d}{\mathfrak {m}}+ \int |\phi _t^\delta |^2\eta _t\,\mathrm {d}{\mathfrak {m}}+ \int |\nabla \phi _t^\delta |^2\eta _t\,\mathrm {d}{\mathfrak {m}}< \infty , \end{aligned}$$

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

$$\begin{aligned} \begin{aligned} \Big |\int f_{t_1}\,\mathrm {d}\mu _{t_1} - \int f_{t_0}\,\mathrm {d}\mu _{t_0}\Big |&\le \Big |\int f_{t_1}\,\mathrm {d}\mu _{t_1} - \int f_{t_1}\,\mathrm {d}\mu _{t_0}\Big | + \Big |\int \big (f_{t_1} - f_{t_0}\big )\mathrm {d}\mu _{t_0}\Big | \\&\le \int _{t_0}^{t_1}\int \big (|\mathrm {d}f_{t_1}||X_t| + c|\Delta f_{t_1}|\big )\mathrm {d}\mu _t{\mathrm {d}t}+ \iint _{t_0}^{t_1}\Big |{\frac{\mathrm {d}}{{\mathrm {d}t}}}f_t\Big |\,\mathrm {d}t\,\mathrm {d}\mu _{t_0} \end{aligned} \end{aligned}$$

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

$$\begin{aligned} \lim _{h\rightarrow 0}\int \frac{f_{t+h} - f_t}{h}\,\mathrm {d}\mu _{t+h} = \int {\frac{\mathrm {d}}{{\mathrm {d}t}}}f_t\,\mathrm {d}\mu _t = \lim _{h \rightarrow 0} \int \frac{f_{t+h}-f_t}{h}\,\mathrm {d}\mu _t. \end{aligned}$$

Hence the conclusion comes dividing by h the trivial identity

$$\begin{aligned} \begin{aligned} \int f_{t+h}\,\mathrm {d}\mu _{t+h} - \int f_t\,\mathrm {d}\mu _t =&\int f_t\,\mathrm {d}\mu _{t+h} - \int f_t\,\mathrm {d}\mu _t + \int (f_{t+h} - f_t)\mathrm {d}\mu _t + \\&+ \int (f_{t+h}-f_t)\mathrm {d}\mu _{t+h} - \int (f_{t+h}-f_t)\mathrm {d}\mu _t \end{aligned} \end{aligned}$$

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