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

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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

Abstract
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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 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: http://ssrn.com/abstract=2162812.
[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. http://hal.archives-ouvertes.fr/hal-00473173/en/.
[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: http://ssrn.com/abstract=2162812.
[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. http://hal.archives-ouvertes.fr/hal-00473173/en/.
[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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