American Institute of Mathematical Sciences

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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

Christian Léonard 1,

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.
Keywords: relative entropy, displacement interpolations, Markov measures, large deviations., optimal transport, Schrödinger problem.
Mathematics Subject Classification: 46N10, 60J25, 60F1.
Citation: Christian Léonard. A survey of the Schrödinger problem and some of its connections with optimal transport. Discrete & Continuous Dynamical Systems - A, 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. doi: 10.1007/s002110050002. Google Scholar

[2]

S. Bernstein, Sur les liaisons entre les grandeurs aléatoires,, Verhand. Internat. Math. Kongr. Zürich, (1932). Google Scholar

[3]

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

[4]

P. Billingsley, "Convergence of Probability Measures,", John Wiley & Sons, (1968). Google Scholar

[5]

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

[6]

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

[7]

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

[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. doi: 10.1007/s002220100160. Google Scholar

[9]

A. B. Cruzeiro, L. Wu and J.-C. Zambrini, Bernstein processes associated with a Markov process,, Stochastic analysis and mathematical physics, (2000), 41. Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

A. Dembo and O. Zeitouni, "Large Deviations Techniques and Applications,", Second edition, (1998). Google Scholar

[17]

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

[18]

J. L. Doob, "Stochastic Processes,", John Wiley & Sons, (1953). Google Scholar

[19]

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

[20]

J. L. Doob, "Classical Potential Theory and Its Probabilistic Counterpart,", 2nd ed., (2000). Google Scholar

[21]

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

[22]

R. Feynman, "Feynman's Thesis. A New Approach to Quantum Theory,", World Scientific Publishing Co. Pte. Ltd., (2005). doi: 10.1142/9789812567635. Google Scholar

[23]

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

[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, (1362). doi: 10.1007/BFb0086180. Google Scholar

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

[34]

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

[35]

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

[36]

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

[37]

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

[38]

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

[39]

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

[40]

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

[41]

C. Léonard, Girsanov theory under a finite entropy condition,, Séminaire de probabilités XLIV., (2046), 429. doi: 10.1007/978-3-642-27461-9_20. Google Scholar

[42]

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

[43]

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

[44]

R. McCann, "A Convexity Theory for Interacting Gases and Equilibrium Crystals,", PhD thesis, (1994). Google Scholar

[45]

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

[46]

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

[47]

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

[48]

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

[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. doi: 10.1007/s00440-004-0340-4. Google Scholar

[50]

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

[51]

T. Mikami, Marginal problem for semimartingales via duality,, International Conference for the 25th Anniversary of Viscosity Solutions, (2008), 133. Google Scholar

[52]

T. Mikami, Optimal transportation problem as stochastic mechanics,, Selected papers on probability and statistics, (2009), 75. Google Scholar

[53]

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

[54]

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

[55]

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

[56]

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

[57]

M. Nagasawa, "Stochastic Processes in Quantum Physics,", Monographs in Mathematics, (2000). doi: 10.1007/978-3-0348-8383-2. Google Scholar

[58]

E. Nelson, "Dynamical Theories of Brownian Motion,", Princeton University Press, (1967). Google Scholar

[59]

E. Nelson, "Quantum Fluctuations,", Princeton Series in Physics, (1985). Google Scholar

[60]

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

[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. doi: 10.1006/jfan.1999.3557. Google Scholar

[62]

D. Revuz and M. Yor, "Continuous Martingales and Brownian Motion,", 3rd ed., (1999). Google Scholar

[63]

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

[64]

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

[65]

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

[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. Google Scholar

[67]

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

[68]

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

[69]

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

[70]

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

[71]

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

[72]

C. Villani, "Optimal Transport. Old and New,", Grundlehren der mathematischen Wissenschaften, (2009). doi: 10.1007/978-3-540-71050-9. Google Scholar

[73]

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

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. doi: 10.1007/s002110050002. Google Scholar

[2]

S. Bernstein, Sur les liaisons entre les grandeurs aléatoires,, Verhand. Internat. Math. Kongr. Zürich, (1932). Google Scholar

[3]

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

[4]

P. Billingsley, "Convergence of Probability Measures,", John Wiley & Sons, (1968). Google Scholar

[5]

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

[6]

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

[7]

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

[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. doi: 10.1007/s002220100160. Google Scholar

[9]

A. B. Cruzeiro, L. Wu and J.-C. Zambrini, Bernstein processes associated with a Markov process,, Stochastic analysis and mathematical physics, (2000), 41. Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

A. Dembo and O. Zeitouni, "Large Deviations Techniques and Applications,", Second edition, (1998). Google Scholar

[17]

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

[18]

J. L. Doob, "Stochastic Processes,", John Wiley & Sons, (1953). Google Scholar

[19]

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

[20]

J. L. Doob, "Classical Potential Theory and Its Probabilistic Counterpart,", 2nd ed., (2000). Google Scholar

[21]

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

[22]

R. Feynman, "Feynman's Thesis. A New Approach to Quantum Theory,", World Scientific Publishing Co. Pte. Ltd., (2005). doi: 10.1142/9789812567635. Google Scholar

[23]

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

[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, (1362). doi: 10.1007/BFb0086180. Google Scholar

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

[34]

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

[35]

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

[36]

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

[37]

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

[38]

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

[39]

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

[40]

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

[41]

C. Léonard, Girsanov theory under a finite entropy condition,, Séminaire de probabilités XLIV., (2046), 429. doi: 10.1007/978-3-642-27461-9_20. Google Scholar

[42]

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

[43]

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

[44]

R. McCann, "A Convexity Theory for Interacting Gases and Equilibrium Crystals,", PhD thesis, (1994). Google Scholar

[45]

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

[46]

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

[47]

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

[48]

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

[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. doi: 10.1007/s00440-004-0340-4. Google Scholar

[50]

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

[51]

T. Mikami, Marginal problem for semimartingales via duality,, International Conference for the 25th Anniversary of Viscosity Solutions, (2008), 133. Google Scholar

[52]

T. Mikami, Optimal transportation problem as stochastic mechanics,, Selected papers on probability and statistics, (2009), 75. Google Scholar

[53]

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

[54]

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

[55]

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

[56]

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

[57]

M. Nagasawa, "Stochastic Processes in Quantum Physics,", Monographs in Mathematics, (2000). doi: 10.1007/978-3-0348-8383-2. Google Scholar

[58]

E. Nelson, "Dynamical Theories of Brownian Motion,", Princeton University Press, (1967). Google Scholar

[59]

E. Nelson, "Quantum Fluctuations,", Princeton Series in Physics, (1985). Google Scholar

[60]

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

[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. doi: 10.1006/jfan.1999.3557. Google Scholar

[62]

D. Revuz and M. Yor, "Continuous Martingales and Brownian Motion,", 3rd ed., (1999). Google Scholar

[63]

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

[64]

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

[65]

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

[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. Google Scholar

[67]

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

[68]

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

[69]

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

[70]

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

[71]

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

[72]

C. Villani, "Optimal Transport. Old and New,", Grundlehren der mathematischen Wissenschaften, (2009). doi: 10.1007/978-3-540-71050-9. Google Scholar

[73]

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

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