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

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 & 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.
[2]	S. Bernstein, Sur les liaisons entre les grandeurs aléatoires,, Verhand. Internat. Math. Kongr. Zürich, (1932).
[3]	A. Beurling, An automormhism of product measures,, Ann. Math., 72 (1960), 189. doi: 10.2307/1970151.
[4]	P. Billingsley, "Convergence of Probability Measures,", John Wiley & Sons, (1968).
[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.
[6]	P. Cattiaux and C. Léonard, Large deviations and Nelson's processes,, Forum Math., 7 (1995), 95. 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. 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, (2000), 41.
[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.
[11]	I. Csiszár, $I$-divergence geometry of probability distributions and minimization problems,, Annals of Probability, 3 (1975), 146. doi: 10.1214/aop/1176996454.
[12]	P. Dai Pra, A stochastic control approach to reciprocal diffusion processes,, Appl. Math. Optim., 23 (1991), 313. doi: 10.1007/BF01442404.
[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.
[14]	D. Dawson, L. Gorostiza and A. Wakolbinger, Schrödinger processes and large deviations,, J. Math. Phys., 31 (1990), 2385. 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. doi: 10.1080/17442508708833446.
[16]	A. Dembo and O. Zeitouni, "Large Deviations Techniques and Applications,", Second edition, (1998).
[17]	P. A. M. Dirac, The lagrangian in quantum mechanics,, Phys. Zeitsch. der Sowietunion, 3 (1933), 64.
[18]	J. L. Doob, "Stochastic Processes,", John Wiley & Sons, (1953).
[19]	J. L. Doob, Conditional Brownian motion and the boundary limits of harmonic functions,, Bull. Soc. Math. France, 85 (1957), 431.
[20]	J. L. Doob, "Classical Potential Theory and Its Probabilistic Counterpart,", 2nd ed., (2000).
[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., (2005). 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, (1362). doi: 10.1007/BFb0086180.
[25]	H. Föllmer and N. Gantert, Entropy minimization and Schrödinger processes in infinite dimensions,, Ann. Probab., 25 (1997), 901. doi: 10.1214/aop/1024404423.
[26]	R. Fortet, Résolution d'un système d'équations de M. Schrödinger,, J. Math. Pures Appl. 9 (1940), 9 (1940).
[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. doi: 10.1007/BF00532864.
[29]	B. Jamison, The Markov processes of Schrödinger,, Z. Wahrsch. verw. Geb., 32 (1975), 323. 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. doi: 10.1137/S0036141096303359.
[31]	M. Kac, On distributions of certain Wiener functionals,, Trans. Amer. Math. Soc., 65 (1949), 1. 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. doi: 10.1007/s00245-001-0019-5.
[38]	C. Léonard, Minimizers of energy functionals,, Acta Math. Hungar., 93 (2001), 281. doi: 10.1023/A:1017919422086.
[39]	C. Léonard, Entropic projections and dominating points,, ESAIM Probab. Stat., 14 (2010), 343. doi: 10.1051/ps/2009003.
[40]	C. Léonard, From the Schrödinger problem to the Monge-Kantorovich problem,, J. Funct. Anal., 262 (2012), 1879.
[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.
[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. doi: 10.4007/annals.2009.169.903.
[44]	R. McCann, "A Convexity Theory for Interacting Gases and Equilibrium Crystals,", PhD thesis, (1994).
[45]	R. McCann, A convexity principle for interacting gases,, Adv. Math., 128 (1997), 153. doi: 10.1006/aima.1997.1634.
[46]	T. Mikami, Variational processes from the weak forward equation,, Comm. Math. Phys., 135 (1990), 19. 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. 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. 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. 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. doi: 10.2996/kmj/1143122381.
[51]	T. Mikami, Marginal problem for semimartingales via duality,, International Conference for the 25th Anniversary of Viscosity Solutions, (2008), 133.
[52]	T. Mikami, Optimal transportation problem as stochastic mechanics,, Selected papers on probability and statistics, (2009), 75.
[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.
[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.
[55]	T. Mikami and M. Thieullen, Optimal transportation problem by stochastic optimal control,, SIAM J. Control Optim., 47 (2008), 1127. doi: 10.1137/050631264.
[56]	M. Nagasawa, Transformations of diffusion and Schrödinger processes,, Probab. Theory Related Fields, 82 (1989), 109. doi: 10.1007/BF00340014.
[57]	M. Nagasawa, "Stochastic Processes in Quantum Physics,", Monographs in Mathematics, (2000). doi: 10.1007/978-3-0348-8383-2.
[58]	E. Nelson, "Dynamical Theories of Brownian Motion,", Princeton University Press, (1967).
[59]	E. Nelson, "Quantum Fluctuations,", Princeton Series in Physics, (1985).
[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.
[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.
[62]	D. Revuz and M. Yor, "Continuous Martingales and Brownian Motion,", 3rd ed., (1999).
[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.
[64]	L. Rüschendorf and W. Thomsen, Closedness of sum spaces and the generalized Schrödinger problem,, Theory Probab. Appl., 42 (1998), 483.
[65]	E. Schrödinger, Über die umkehrung der naturgesetze,, Sitzungsberichte Preuss. Akad. Wiss. Berlin. Phys. Math. 144* (1931), 144* (1931), 144.
[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.
[67]	K-T. Sturm, On the geometry of metric measure spaces, I,, Acta Math., 196 (2006), 65. doi: 10.1007/s11511-006-0002-8.
[68]	K-T. Sturm, On the geometry of metric measure spaces, II,, Acta Math., 196 (2006), 133. 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. 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. 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. doi: 10.1007/s004400050091.
[72]	C. Villani, "Optimal Transport. Old and New,", Grundlehren der mathematischen Wissenschaften, (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. 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. doi: 10.1007/s002110050002.
[2]	S. Bernstein, Sur les liaisons entre les grandeurs aléatoires,, Verhand. Internat. Math. Kongr. Zürich, (1932).
[3]	A. Beurling, An automormhism of product measures,, Ann. Math., 72 (1960), 189. doi: 10.2307/1970151.
[4]	P. Billingsley, "Convergence of Probability Measures,", John Wiley & Sons, (1968).
[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.
[6]	P. Cattiaux and C. Léonard, Large deviations and Nelson's processes,, Forum Math., 7 (1995), 95. 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. 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, (2000), 41.
[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.
[11]	I. Csiszár, $I$-divergence geometry of probability distributions and minimization problems,, Annals of Probability, 3 (1975), 146. doi: 10.1214/aop/1176996454.
[12]	P. Dai Pra, A stochastic control approach to reciprocal diffusion processes,, Appl. Math. Optim., 23 (1991), 313. doi: 10.1007/BF01442404.
[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.
[14]	D. Dawson, L. Gorostiza and A. Wakolbinger, Schrödinger processes and large deviations,, J. Math. Phys., 31 (1990), 2385. 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. doi: 10.1080/17442508708833446.
[16]	A. Dembo and O. Zeitouni, "Large Deviations Techniques and Applications,", Second edition, (1998).
[17]	P. A. M. Dirac, The lagrangian in quantum mechanics,, Phys. Zeitsch. der Sowietunion, 3 (1933), 64.
[18]	J. L. Doob, "Stochastic Processes,", John Wiley & Sons, (1953).
[19]	J. L. Doob, Conditional Brownian motion and the boundary limits of harmonic functions,, Bull. Soc. Math. France, 85 (1957), 431.
[20]	J. L. Doob, "Classical Potential Theory and Its Probabilistic Counterpart,", 2nd ed., (2000).
[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., (2005). 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, (1362). doi: 10.1007/BFb0086180.
[25]	H. Föllmer and N. Gantert, Entropy minimization and Schrödinger processes in infinite dimensions,, Ann. Probab., 25 (1997), 901. doi: 10.1214/aop/1024404423.
[26]	R. Fortet, Résolution d'un système d'équations de M. Schrödinger,, J. Math. Pures Appl. 9 (1940), 9 (1940).
[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. doi: 10.1007/BF00532864.
[29]	B. Jamison, The Markov processes of Schrödinger,, Z. Wahrsch. verw. Geb., 32 (1975), 323. 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. doi: 10.1137/S0036141096303359.
[31]	M. Kac, On distributions of certain Wiener functionals,, Trans. Amer. Math. Soc., 65 (1949), 1. 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. doi: 10.1007/s00245-001-0019-5.
[38]	C. Léonard, Minimizers of energy functionals,, Acta Math. Hungar., 93 (2001), 281. doi: 10.1023/A:1017919422086.
[39]	C. Léonard, Entropic projections and dominating points,, ESAIM Probab. Stat., 14 (2010), 343. doi: 10.1051/ps/2009003.
[40]	C. Léonard, From the Schrödinger problem to the Monge-Kantorovich problem,, J. Funct. Anal., 262 (2012), 1879.
[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.
[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. doi: 10.4007/annals.2009.169.903.
[44]	R. McCann, "A Convexity Theory for Interacting Gases and Equilibrium Crystals,", PhD thesis, (1994).
[45]	R. McCann, A convexity principle for interacting gases,, Adv. Math., 128 (1997), 153. doi: 10.1006/aima.1997.1634.
[46]	T. Mikami, Variational processes from the weak forward equation,, Comm. Math. Phys., 135 (1990), 19. 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. 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. 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. 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. doi: 10.2996/kmj/1143122381.
[51]	T. Mikami, Marginal problem for semimartingales via duality,, International Conference for the 25th Anniversary of Viscosity Solutions, (2008), 133.
[52]	T. Mikami, Optimal transportation problem as stochastic mechanics,, Selected papers on probability and statistics, (2009), 75.
[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.
[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.
[55]	T. Mikami and M. Thieullen, Optimal transportation problem by stochastic optimal control,, SIAM J. Control Optim., 47 (2008), 1127. doi: 10.1137/050631264.
[56]	M. Nagasawa, Transformations of diffusion and Schrödinger processes,, Probab. Theory Related Fields, 82 (1989), 109. doi: 10.1007/BF00340014.
[57]	M. Nagasawa, "Stochastic Processes in Quantum Physics,", Monographs in Mathematics, (2000). doi: 10.1007/978-3-0348-8383-2.
[58]	E. Nelson, "Dynamical Theories of Brownian Motion,", Princeton University Press, (1967).
[59]	E. Nelson, "Quantum Fluctuations,", Princeton Series in Physics, (1985).
[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.
[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.
[62]	D. Revuz and M. Yor, "Continuous Martingales and Brownian Motion,", 3rd ed., (1999).
[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.
[64]	L. Rüschendorf and W. Thomsen, Closedness of sum spaces and the generalized Schrödinger problem,, Theory Probab. Appl., 42 (1998), 483.
[65]	E. Schrödinger, Über die umkehrung der naturgesetze,, Sitzungsberichte Preuss. Akad. Wiss. Berlin. Phys. Math. 144* (1931), 144* (1931), 144.
[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.
[67]	K-T. Sturm, On the geometry of metric measure spaces, I,, Acta Math., 196 (2006), 65. doi: 10.1007/s11511-006-0002-8.
[68]	K-T. Sturm, On the geometry of metric measure spaces, II,, Acta Math., 196 (2006), 133. 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. 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. 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. doi: 10.1007/s004400050091.
[72]	C. Villani, "Optimal Transport. Old and New,", Grundlehren der mathematischen Wissenschaften, (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. doi: 10.1063/1.527002.

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