April  2014, 34(4): 1533-1574. doi: 10.3934/dcds.2014.34.1533

A survey of the Schrödinger problem and some of its connections with optimal transport

1. 

Modal-X. Université Paris Ouest, Bât. G, 200 av. de la République. 92001 Nanterre, France

Received  November 2012 Revised  March 2013 Published  October 2013

This article is aimed at presenting the Schrödinger problem and some of its connections with optimal transport. We hope that it can be used as a basic user's guide to Schrödinger problem. We also give a survey of the related literature. In addition, some new results are proved.
Citation: Christian Léonard. A survey of the Schrödinger problem and some of its connections with optimal transport. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1533-1574. doi: 10.3934/dcds.2014.34.1533
References:
[1]

J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393. doi: 10.1007/s002110050002.

[2]

S. Bernstein, Sur les liaisons entre les grandeurs aléatoires, Verhand. Internat. Math. Kongr. Zürich, (1932), no. Band I.

[3]

A. Beurling, An automormhism of product measures, Ann. Math., 72 (1960), 189-200. doi: 10.2307/1970151.

[4]

P. Billingsley, "Convergence of Probability Measures," John Wiley & Sons, Inc., New York-London-Sydney 1968 xii+253 pp.

[5]

J. M. Borwein and A. S. Lewis, Decomposition of multivariate functions, Can. J. Math., 44 (1992), 463-482. doi: 10.4153/CJM-1992-030-9.

[6]

P. Cattiaux and C. Léonard, Large deviations and Nelson's processes, Forum Math., 7 (1995), 95-115. doi: 10.1515/form.1995.7.95.

[7]

K. L. Chung and J.-C. Zambrini, "Introduction to Random Time and Quantum Randomness," World Scientific, 2008.

[8]

D. Cordero Erausquin, R. J. McCann and M. Schmuckenschläger, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb, Invent. Math., 146 (2001), 219-257. doi: 10.1007/s002220100160.

[9]

A. B. Cruzeiro, L. Wu and J.-C. Zambrini, Bernstein processes associated with a Markov process, Stochastic analysis and mathematical physics, ANESTOC'98. Proceedings of the Third International Workshop (Boston) (R. Rebolledo, ed.), Trends in Mathematics, Birkhäuser, 2000, pp. 41-71.

[10]

A. B. Cruzeiro and J.-C. Zambrini, Malliavin calculus and Euclidean quantum mechanics, I, Functional calculus. J. Funct. Anal., 96 (1991), 62-95. doi: 10.1016/0022-1236(91)90073-E.

[11]

I. Csiszár, $I$-divergence geometry of probability distributions and minimization problems, Annals of Probability, 3 (1975), 146-158. doi: 10.1214/aop/1176996454.

[12]

P. Dai Pra, A stochastic control approach to reciprocal diffusion processes, Appl. Math. Optim., 23 (1991), 313-329. doi: 10.1007/BF01442404.

[13]

G. Dal Maso, "An Introduction to $\gamma$-convergence," Progress in Nonlinear Differential Equations and Their Applications 8, Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0327-8.

[14]

D. Dawson, L. Gorostiza and A. Wakolbinger, Schrödinger processes and large deviations, J. Math. Phys., 31 (1990), 2385-2388. doi: 10.1063/1.528840.

[15]

D. A. Dawson and J. Gärtner, Large deviations from the McKean-Vlasov limit for weakly interacting diffusions, Stochastics, 20 (1987), 247-308. doi: 10.1080/17442508708833446.

[16]

A. Dembo and O. Zeitouni, "Large Deviations Techniques and Applications," Second edition, Applications of Mathematics 38, Springer Verlag, New York, 1998.

[17]

P. A. M. Dirac, The lagrangian in quantum mechanics, Phys. Zeitsch. der Sowietunion, 3 (1933), 64-72.

[18]

J. L. Doob, "Stochastic Processes," John Wiley & Sons, Inc., New York; Chapman & Hall, Limited, London, 1953. viii+654 pp.

[19]

J. L. Doob, Conditional Brownian motion and the boundary limits of harmonic functions, Bull. Soc. Math. France, 85 (1957), 431-458.

[20]

J. L. Doob, "Classical Potential Theory and Its Probabilistic Counterpart," 2nd ed., Classics in Mathematics, Springer, 2000, (reprint of the 1984 first edition).

[21]

A. Dupuy and A. Galichon, Personality traits and the marriage market,, Preprint. Available at SSRN: , (). 

[22]

R. Feynman, "Feynman's Thesis. A New Approach to Quantum Theory," World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. xxii+121 pp. doi: 10.1142/9789812567635.

[23]

R. Feynman and A. Hibbs, "Quantum Mechanics and Path Integrals," McGraw-Hill, 1965.

[24]

H. Föllmer, "Random Fields and Diffusion Processes, in école D'été De Probabilités De Saint-Flour xv-xvii-1985-87," Lecture Notes in Mathematics, vol. 1362, Springer, Berlin, 1988. doi: 10.1007/BFb0086180.

[25]

H. Föllmer and N. Gantert, Entropy minimization and Schrödinger processes in infinite dimensions, Ann. Probab., 25 (1997), 901-926. doi: 10.1214/aop/1024404423.

[26]

R. Fortet, Résolution d'un système d'équations de M. Schrödinger, J. Math. Pures Appl. 9 (1940), 83.

[27]

A. Galichon and B. Salanie, Matching with trade-offs: Revealed preferences over competing characteristics,, Preprint. , (). 

[28]

B. Jamison, Reciprocal processes, Z. Wahrsch. verw. Geb., 30 (1974), 65-86. doi: 10.1007/BF00532864.

[29]

B. Jamison, The Markov processes of Schrödinger, Z. Wahrsch. verw. Geb., 32 (1975), 323-331. doi: 10.1007/BF00535844.

[30]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359.

[31]

M. Kac, On distributions of certain Wiener functionals, Trans. Amer. Math. Soc., 65 (1949), 1-13. doi: 10.1090/S0002-9947-1949-0027960-X.

[32]

A. Kolmogorov, Zur Theorie der Markoffschen Ketten, Mathematische Annalen, 112 (1936). doi: 10.1007/BF01565412.

[33]

C. Léonard, Lazy random walks and optimal transport on graphs,, Preprint., (). 

[34]

C. Léonard, On the convexity of the entropy along entropic interpolations,, Preprint., (). 

[35]

C. Léonard, Some properties of path measures,, Preprint., (). 

[36]

C. Léonard, Stochastic derivatives and $h$-transforms of Markov processes,, Preprint., (). 

[37]

C. Léonard, Minimization of energy functionals applied to some inverse problems, J. Appl. Math. Optim., 44 (2001), 273-297. doi: 10.1007/s00245-001-0019-5.

[38]

C. Léonard, Minimizers of energy functionals, Acta Math. Hungar., 93 (2001), 281-325. doi: 10.1023/A:1017919422086.

[39]

C. Léonard, Entropic projections and dominating points, ESAIM Probab. Stat., 14 (2010), 343-381. doi: 10.1051/ps/2009003.

[40]

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

[41]

C. Léonard, Girsanov theory under a finite entropy condition, Séminaire de probabilités XLIV., Lecture Notes in Mathematics 2046. Springer, 2012, pp. 429-465. doi: 10.1007/978-3-642-27461-9_20.

[42]

C. Léonard, S. Rœlly and J.-C. Zambrini, On the time symmetry of some stochastic processes,, Preprint., (). 

[43]

J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math., 169 (2009), 903-991. doi: 10.4007/annals.2009.169.903.

[44]

R. McCann, "A Convexity Theory for Interacting Gases and Equilibrium Crystals," PhD thesis, Princeton Univ., 1994.

[45]

R. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), 153-179. doi: 10.1006/aima.1997.1634.

[46]

T. Mikami, Variational processes from the weak forward equation, Comm. Math. Phys., 135 (1990), 19-40. doi: 10.1007/BF02097655.

[47]

T. Mikami, Dynamical systems in the variational formulation of the Fokker-Planck equation by the Wasserstein metric, Appl. Math. Optim., 42 (2000), 203-227. doi: 10.1007/s002450010008.

[48]

T. Mikami, Optimal control for absolutely continuous stochastic processes and the mass transportation problem, Electron. Comm. Probab., 7 (2002), 199-213. doi: 10.1214/ECP.v7-1061.

[49]

T. Mikami, Monge's problem with a quadratic cost by the zero-noise limit of $h$-path processes, Probab. Theory Relat. Fields, 129 (2004), 245-260. doi: 10.1007/s00440-004-0340-4.

[50]

T. Mikami, A simple proof of duality theorem for Monge-Kantorovich problem, Kodai Math. J., 29 (2006), 1-4. doi: 10.2996/kmj/1143122381.

[51]

T. Mikami, Marginal problem for semimartingales via duality, International Conference for the 25th Anniversary of Viscosity Solutions, Gakuto International Series. Mathematical Sciences and Applications, vol. 30, 2008, pp. 133-152.

[52]

T. Mikami, Optimal transportation problem as stochastic mechanics, Selected papers on probability and statistics, 75-94, Amer. Math. Soc. Transl. Ser. 2, 227, Amer. Math. Soc., Providence, RI, 2009.

[53]

T. Mikami, A characterization of the Knothe-Rosenblatt processes by a convergence result, SIAM J. Control and Optim., 50 (2012), 1903-1920. doi: 10.1137/100791129.

[54]

T. Mikami and M. Thieullen, Duality theorem for the stochastic optimal control problem, Stoch. Proc. Appl., 116 (2006), 1815-1835. doi: 10.1016/j.spa.2006.04.014.

[55]

T. Mikami and M. Thieullen, Optimal transportation problem by stochastic optimal control, SIAM J. Control Optim., 47 (2008), 1127-1139. doi: 10.1137/050631264.

[56]

M. Nagasawa, Transformations of diffusion and Schrödinger processes, Probab. Theory Related Fields, 82 (1989), 109-136. doi: 10.1007/BF00340014.

[57]

M. Nagasawa, "Stochastic Processes in Quantum Physics," Monographs in Mathematics, vol. 94, Birkhäuser Verlag, Basel, 2000. doi: 10.1007/978-3-0348-8383-2.

[58]

E. Nelson, "Dynamical Theories of Brownian Motion," Princeton University Press, Princeton, N.J. 1967 iii+142 pp.

[59]

E. Nelson, "Quantum Fluctuations," Princeton Series in Physics, Princeton University Press, Princeton, NJ, 1985.

[60]

F. Otto, The geometry of dissipative evolution equations: The porous medieum equation., Comm. Partial Differential Equations, 26 (2001), 101-174. doi: 10.1081/PDE-100002243.

[61]

F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173 (2000), 361-400. doi: 10.1006/jfan.1999.3557.

[62]

D. Revuz and M. Yor, "Continuous Martingales and Brownian Motion," 3rd ed., Grundlehren der Mathematischen Wissenschaften, [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1999..

[63]

L. Rüschendorf and W. Thomsen, Note on the Schrödinger equation and $I$-projections, Statist. Probab. Lett., 17 (1993), 369-375. doi: 10.1016/0167-7152(93)90257-J.

[64]

L. Rüschendorf and W. Thomsen, Closedness of sum spaces and the generalized Schrödinger problem, Theory Probab. Appl., 42 (1998), 483-494.

[65]

E. Schrödinger, Über die umkehrung der naturgesetze, Sitzungsberichte Preuss. Akad. Wiss. Berlin. Phys. Math. 144 (1931), 144-153.

[66]

E. Schrödinger, Sur la théorie relativiste de l'électron et l'interprétation de la mécanique quantique, (French) Ann. Inst. H. Poincaré, 2 (1932), 269-310.

[67]

K-T. Sturm, On the geometry of metric measure spaces, I, Acta Math., 196 (2006), 65-131. doi: 10.1007/s11511-006-0002-8.

[68]

K-T. Sturm, On the geometry of metric measure spaces, II, Acta Math., 196 (2006), 133-177. doi: 10.1007/s11511-006-0003-7.

[69]

K-T. Sturm and M-K. von Renesse, Transport inequalities, gradient estimates, entropy, and Ricci curvature, Comm. Pure Appl. Math., 58 (2005), 923-940. doi: 10.1002/cpa.20060.

[70]

M. Thieullen, Second order stochastic differential equations and non Gaussian reciprocal diffusions, Probab. Theory Related Fields, 97 (1993), 231-257. doi: 10.1007/BF01199322.

[71]

M. Thieullen and J.-C. Zambrini, Symmetries in the stochastic calculus of variations, Probab. Theory Related Fields, 107 (1997), 401-427. doi: 10.1007/s004400050091.

[72]

C. Villani, "Optimal Transport. Old and New," Grundlehren der mathematischen Wissenschaften, [Fundamental Principles of Mathematical Sciences], 338. Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.

[73]

J.-C. Zambrini, Variational processes and stochastic versions of mechanics, J. Math. Phys., 27 (1986), 2307-2330. doi: 10.1063/1.527002.

show all references

References:
[1]

J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393. doi: 10.1007/s002110050002.

[2]

S. Bernstein, Sur les liaisons entre les grandeurs aléatoires, Verhand. Internat. Math. Kongr. Zürich, (1932), no. Band I.

[3]

A. Beurling, An automormhism of product measures, Ann. Math., 72 (1960), 189-200. doi: 10.2307/1970151.

[4]

P. Billingsley, "Convergence of Probability Measures," John Wiley & Sons, Inc., New York-London-Sydney 1968 xii+253 pp.

[5]

J. M. Borwein and A. S. Lewis, Decomposition of multivariate functions, Can. J. Math., 44 (1992), 463-482. doi: 10.4153/CJM-1992-030-9.

[6]

P. Cattiaux and C. Léonard, Large deviations and Nelson's processes, Forum Math., 7 (1995), 95-115. doi: 10.1515/form.1995.7.95.

[7]

K. L. Chung and J.-C. Zambrini, "Introduction to Random Time and Quantum Randomness," World Scientific, 2008.

[8]

D. Cordero Erausquin, R. J. McCann and M. Schmuckenschläger, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb, Invent. Math., 146 (2001), 219-257. doi: 10.1007/s002220100160.

[9]

A. B. Cruzeiro, L. Wu and J.-C. Zambrini, Bernstein processes associated with a Markov process, Stochastic analysis and mathematical physics, ANESTOC'98. Proceedings of the Third International Workshop (Boston) (R. Rebolledo, ed.), Trends in Mathematics, Birkhäuser, 2000, pp. 41-71.

[10]

A. B. Cruzeiro and J.-C. Zambrini, Malliavin calculus and Euclidean quantum mechanics, I, Functional calculus. J. Funct. Anal., 96 (1991), 62-95. doi: 10.1016/0022-1236(91)90073-E.

[11]

I. Csiszár, $I$-divergence geometry of probability distributions and minimization problems, Annals of Probability, 3 (1975), 146-158. doi: 10.1214/aop/1176996454.

[12]

P. Dai Pra, A stochastic control approach to reciprocal diffusion processes, Appl. Math. Optim., 23 (1991), 313-329. doi: 10.1007/BF01442404.

[13]

G. Dal Maso, "An Introduction to $\gamma$-convergence," Progress in Nonlinear Differential Equations and Their Applications 8, Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0327-8.

[14]

D. Dawson, L. Gorostiza and A. Wakolbinger, Schrödinger processes and large deviations, J. Math. Phys., 31 (1990), 2385-2388. doi: 10.1063/1.528840.

[15]

D. A. Dawson and J. Gärtner, Large deviations from the McKean-Vlasov limit for weakly interacting diffusions, Stochastics, 20 (1987), 247-308. doi: 10.1080/17442508708833446.

[16]

A. Dembo and O. Zeitouni, "Large Deviations Techniques and Applications," Second edition, Applications of Mathematics 38, Springer Verlag, New York, 1998.

[17]

P. A. M. Dirac, The lagrangian in quantum mechanics, Phys. Zeitsch. der Sowietunion, 3 (1933), 64-72.

[18]

J. L. Doob, "Stochastic Processes," John Wiley & Sons, Inc., New York; Chapman & Hall, Limited, London, 1953. viii+654 pp.

[19]

J. L. Doob, Conditional Brownian motion and the boundary limits of harmonic functions, Bull. Soc. Math. France, 85 (1957), 431-458.

[20]

J. L. Doob, "Classical Potential Theory and Its Probabilistic Counterpart," 2nd ed., Classics in Mathematics, Springer, 2000, (reprint of the 1984 first edition).

[21]

A. Dupuy and A. Galichon, Personality traits and the marriage market,, Preprint. Available at SSRN: , (). 

[22]

R. Feynman, "Feynman's Thesis. A New Approach to Quantum Theory," World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. xxii+121 pp. doi: 10.1142/9789812567635.

[23]

R. Feynman and A. Hibbs, "Quantum Mechanics and Path Integrals," McGraw-Hill, 1965.

[24]

H. Föllmer, "Random Fields and Diffusion Processes, in école D'été De Probabilités De Saint-Flour xv-xvii-1985-87," Lecture Notes in Mathematics, vol. 1362, Springer, Berlin, 1988. doi: 10.1007/BFb0086180.

[25]

H. Föllmer and N. Gantert, Entropy minimization and Schrödinger processes in infinite dimensions, Ann. Probab., 25 (1997), 901-926. doi: 10.1214/aop/1024404423.

[26]

R. Fortet, Résolution d'un système d'équations de M. Schrödinger, J. Math. Pures Appl. 9 (1940), 83.

[27]

A. Galichon and B. Salanie, Matching with trade-offs: Revealed preferences over competing characteristics,, Preprint. , (). 

[28]

B. Jamison, Reciprocal processes, Z. Wahrsch. verw. Geb., 30 (1974), 65-86. doi: 10.1007/BF00532864.

[29]

B. Jamison, The Markov processes of Schrödinger, Z. Wahrsch. verw. Geb., 32 (1975), 323-331. doi: 10.1007/BF00535844.

[30]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359.

[31]

M. Kac, On distributions of certain Wiener functionals, Trans. Amer. Math. Soc., 65 (1949), 1-13. doi: 10.1090/S0002-9947-1949-0027960-X.

[32]

A. Kolmogorov, Zur Theorie der Markoffschen Ketten, Mathematische Annalen, 112 (1936). doi: 10.1007/BF01565412.

[33]

C. Léonard, Lazy random walks and optimal transport on graphs,, Preprint., (). 

[34]

C. Léonard, On the convexity of the entropy along entropic interpolations,, Preprint., (). 

[35]

C. Léonard, Some properties of path measures,, Preprint., (). 

[36]

C. Léonard, Stochastic derivatives and $h$-transforms of Markov processes,, Preprint., (). 

[37]

C. Léonard, Minimization of energy functionals applied to some inverse problems, J. Appl. Math. Optim., 44 (2001), 273-297. doi: 10.1007/s00245-001-0019-5.

[38]

C. Léonard, Minimizers of energy functionals, Acta Math. Hungar., 93 (2001), 281-325. doi: 10.1023/A:1017919422086.

[39]

C. Léonard, Entropic projections and dominating points, ESAIM Probab. Stat., 14 (2010), 343-381. doi: 10.1051/ps/2009003.

[40]

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

[41]

C. Léonard, Girsanov theory under a finite entropy condition, Séminaire de probabilités XLIV., Lecture Notes in Mathematics 2046. Springer, 2012, pp. 429-465. doi: 10.1007/978-3-642-27461-9_20.

[42]

C. Léonard, S. Rœlly and J.-C. Zambrini, On the time symmetry of some stochastic processes,, Preprint., (). 

[43]

J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math., 169 (2009), 903-991. doi: 10.4007/annals.2009.169.903.

[44]

R. McCann, "A Convexity Theory for Interacting Gases and Equilibrium Crystals," PhD thesis, Princeton Univ., 1994.

[45]

R. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), 153-179. doi: 10.1006/aima.1997.1634.

[46]

T. Mikami, Variational processes from the weak forward equation, Comm. Math. Phys., 135 (1990), 19-40. doi: 10.1007/BF02097655.

[47]

T. Mikami, Dynamical systems in the variational formulation of the Fokker-Planck equation by the Wasserstein metric, Appl. Math. Optim., 42 (2000), 203-227. doi: 10.1007/s002450010008.

[48]

T. Mikami, Optimal control for absolutely continuous stochastic processes and the mass transportation problem, Electron. Comm. Probab., 7 (2002), 199-213. doi: 10.1214/ECP.v7-1061.

[49]

T. Mikami, Monge's problem with a quadratic cost by the zero-noise limit of $h$-path processes, Probab. Theory Relat. Fields, 129 (2004), 245-260. doi: 10.1007/s00440-004-0340-4.

[50]

T. Mikami, A simple proof of duality theorem for Monge-Kantorovich problem, Kodai Math. J., 29 (2006), 1-4. doi: 10.2996/kmj/1143122381.

[51]

T. Mikami, Marginal problem for semimartingales via duality, International Conference for the 25th Anniversary of Viscosity Solutions, Gakuto International Series. Mathematical Sciences and Applications, vol. 30, 2008, pp. 133-152.

[52]

T. Mikami, Optimal transportation problem as stochastic mechanics, Selected papers on probability and statistics, 75-94, Amer. Math. Soc. Transl. Ser. 2, 227, Amer. Math. Soc., Providence, RI, 2009.

[53]

T. Mikami, A characterization of the Knothe-Rosenblatt processes by a convergence result, SIAM J. Control and Optim., 50 (2012), 1903-1920. doi: 10.1137/100791129.

[54]

T. Mikami and M. Thieullen, Duality theorem for the stochastic optimal control problem, Stoch. Proc. Appl., 116 (2006), 1815-1835. doi: 10.1016/j.spa.2006.04.014.

[55]

T. Mikami and M. Thieullen, Optimal transportation problem by stochastic optimal control, SIAM J. Control Optim., 47 (2008), 1127-1139. doi: 10.1137/050631264.

[56]

M. Nagasawa, Transformations of diffusion and Schrödinger processes, Probab. Theory Related Fields, 82 (1989), 109-136. doi: 10.1007/BF00340014.

[57]

M. Nagasawa, "Stochastic Processes in Quantum Physics," Monographs in Mathematics, vol. 94, Birkhäuser Verlag, Basel, 2000. doi: 10.1007/978-3-0348-8383-2.

[58]

E. Nelson, "Dynamical Theories of Brownian Motion," Princeton University Press, Princeton, N.J. 1967 iii+142 pp.

[59]

E. Nelson, "Quantum Fluctuations," Princeton Series in Physics, Princeton University Press, Princeton, NJ, 1985.

[60]

F. Otto, The geometry of dissipative evolution equations: The porous medieum equation., Comm. Partial Differential Equations, 26 (2001), 101-174. doi: 10.1081/PDE-100002243.

[61]

F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173 (2000), 361-400. doi: 10.1006/jfan.1999.3557.

[62]

D. Revuz and M. Yor, "Continuous Martingales and Brownian Motion," 3rd ed., Grundlehren der Mathematischen Wissenschaften, [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1999..

[63]

L. Rüschendorf and W. Thomsen, Note on the Schrödinger equation and $I$-projections, Statist. Probab. Lett., 17 (1993), 369-375. doi: 10.1016/0167-7152(93)90257-J.

[64]

L. Rüschendorf and W. Thomsen, Closedness of sum spaces and the generalized Schrödinger problem, Theory Probab. Appl., 42 (1998), 483-494.

[65]

E. Schrödinger, Über die umkehrung der naturgesetze, Sitzungsberichte Preuss. Akad. Wiss. Berlin. Phys. Math. 144 (1931), 144-153.

[66]

E. Schrödinger, Sur la théorie relativiste de l'électron et l'interprétation de la mécanique quantique, (French) Ann. Inst. H. Poincaré, 2 (1932), 269-310.

[67]

K-T. Sturm, On the geometry of metric measure spaces, I, Acta Math., 196 (2006), 65-131. doi: 10.1007/s11511-006-0002-8.

[68]

K-T. Sturm, On the geometry of metric measure spaces, II, Acta Math., 196 (2006), 133-177. doi: 10.1007/s11511-006-0003-7.

[69]

K-T. Sturm and M-K. von Renesse, Transport inequalities, gradient estimates, entropy, and Ricci curvature, Comm. Pure Appl. Math., 58 (2005), 923-940. doi: 10.1002/cpa.20060.

[70]

M. Thieullen, Second order stochastic differential equations and non Gaussian reciprocal diffusions, Probab. Theory Related Fields, 97 (1993), 231-257. doi: 10.1007/BF01199322.

[71]

M. Thieullen and J.-C. Zambrini, Symmetries in the stochastic calculus of variations, Probab. Theory Related Fields, 107 (1997), 401-427. doi: 10.1007/s004400050091.

[72]

C. Villani, "Optimal Transport. Old and New," Grundlehren der mathematischen Wissenschaften, [Fundamental Principles of Mathematical Sciences], 338. Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.

[73]

J.-C. Zambrini, Variational processes and stochastic versions of mechanics, J. Math. Phys., 27 (1986), 2307-2330. doi: 10.1063/1.527002.

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