A survey of the Schr\"odinger problem and some of its connections with optimal transport

This article is aimed at presenting the Schr\"odinger problem and some of its connections with optimal transport. We hope that it can be used as a basic user's guide to Schr\"odinger problem. We also give a survey of the related literature. In addition, some new results are proved.

[1]  E. Schrödinger.  Sur la théorie relativiste de l'électron et l'interprétation de la mécanique quantique , 1932 .

[2]  P. Dirac,et al.  THE LAGRANGIAN IN QUANTUM MECHANICS. , 2022 .

[3]  A. Kolmogoroff.  Zur Theorie der Markoffschen Ketten , 1936 .

[4]  M. Kac.  On distributions of certain Wiener functionals , 1949 .

[5]  J. Doob.  Stochastic processes , 1953 .

[6]  J. L. Doob,et al.  Conditional brownian motion and the boundary limits of harmonic functions , 1957 .

[7]  J. Davenport.  Editor , 1960 .

[8]  A. Beurling,et al.  An Automorphism of Product Measures , 1960 .

[9]  R. Feynman,et al.  Quantum Mechanics and Path Integrals , 1965 .

[10]  Edward Nelson.  Dynamical Theories of Brownian Motion , 1967 .

[11]  P. Billingsley,et al.  Convergence of Probability Measures , 1969 .

[12]  B. Jamison.  Reciprocal processes , 1974 .

[13]  I. Csiszár.  $I$-Divergence Geometry of Probability Distributions and Minimization Problems , 1975 .

[14]  B. Jamison.  The Markov processes of Schrödinger , 1975 .

[15]  J. Doob.  Classical potential theory and its probabilistic counterpart , 1984 .

[16]  J. Zambrini.  Variational processes and stochastic versions of mechanics , 1986 .

[17]  QUANTUM FLUCTUATIONS (Princeton Series in Physics) , 1986 .

[18]  J. Gärtner,et al.  Large deviations from the mckean-vlasov limit for weakly interacting diffusions , 1987 .

[19]  Hans Föllmer,et al.  Random fields and diffusion processes , 1988 .

[20]  Persi Diaconis,et al.  Ecole d'Ete de Probabilites de Saint-Flour XV-XVII, 1985-87 , 1989 .

[21]  Masao Nagasawa,et al.  Transformations of diffusion and Schrödinger processes , 1989 .

[22]  M. Yor,et al.  Continuous martingales and Brownian motion , 1990 .

[23]  Toshio Mikami,et al.  Variational processes from the weak forward equation , 1990 .

[24]  Anton Wakolbinger,et al.  Schrödinger processes and large deviations , 1990 .

[25]  Paolo Dai Pra,et al.  A stochastic control approach to reciprocal diffusion processes , 1991 .

[26]  B. M. Fulk.  MATH , 1992 .

[27]  Jonathan M. Borwein,et al.  Decomposition of Multivariate Functions , 1992, Canadian Journal of Mathematics.

[28]  M. Thieullen.  Second order stochastic differential equations and non-Gaussian reciprocal diffusions , 1993 .

[29]  G. D. Maso,et al.  An Introduction to-convergence , 1993 .

[30]  L. Rüschendorf,et al.  Note on the Schrödinger equation and I-projections , 1993 .

[31]  Gianni Dal Maso,et al.  An Introduction to [gamma]-convergence , 1993 .

[32]  R. McCann.  A convexity theory for interacting gases and equilibrium crystals , 1994 .

[33]  Malliavin Calculus and Euclidean Quantum Mechanics II. Variational Principle for Infinite Dimensional Processes , 1995 .

[34]  P. Cattiaux,et al.  Large deviations and Nelson processes , 1995 .

[35]  D. Kinderlehrer,et al.  THE VARIATIONAL FORMULATION OF THE FOKKER-PLANCK EQUATION , 1996 .

[36]  R. McCann.  A Convexity Principle for Interacting Gases , 1997 .

[37]  Hans Föllmer,et al.  ENTROPY MINIMIZATION AND SCHRODINGER PROCESSES IN INFINITE DIMENSIONS , 1997 .

[38]  J. Zambrini,et al.  Symmetries in the stochastic calculus of variations , 1997 .

[39]  Amir Dembo,et al.  Large Deviations Techniques and Applications , 1998 .

[40]  L. Rüschendorf,et al.  Closedness of Sum Spaces andthe Generalized Schrödinger Problem , 1998 .

[41]  T. Mikami.  Dynamical Systems in the Variational Formulation of the Fokker—Planck Equation by the Wasserstein Metric , 1999 .

[42]  Yann Brenier,et al.  A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem , 2000, Numerische Mathematik.

[43]  Liming Wu,et al.  Bernstein Processes Associated with a Markov Process , 2000 .

[44]  C. Villani,et al.  Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality , 2000 .

[45]  Toshio Mikami,et al.  Optimal control for absolutely continuous stochastic processes and the mass transportation problem , 2000 .

[46]  Stochastic processes in quantum physics , 2000 .

[47]  F. Otto.  THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION , 2001 .

[48]  R. McCann,et al.  A Riemannian interpolation inequality à la Borell, Brascamp and Lieb , 2001 .

[49]  C. Léonard.  Minimization of Energy Functionals Applied to Some Inverse Problems , 2001 .

[50]  Christian Léonard,et al.  Minimizers of energy functionals , 2001 .

[51]  K. Chung,et al.  Introduction to Random Time and Quantum Randomness , 2003 .

[52]  Toshio Mikami,et al.  Duality Theorem for Stochastic Optimal Control Problem , 2004 .

[53]  T. Mikami.  Monge’s problem with a quadratic cost by the zero-noise limit of h-path processes , 2004 .

[54]  T. Mikami.  A Simple Proof of Duality Theorem for Monge-Kantorovich Problem , 2004 .

[55]  C. Villani,et al.  Ricci curvature for metric-measure spaces via optimal transport , 2004, math/0412127.

[56]  Feynman's thesis , 2005 .

[57]  Karl-Theodor Sturm,et al.  Transport inequalities, gradient estimates, entropy and Ricci curvature , 2005 .

[58]  L. Brown.  Feynman's Thesis - A New Approach To Quantum Theory , 2005 .

[59]  Karl-Theodor Sturm,et al.  On the geometry of metric measure spaces. II , 2006 .

[60]  Karl-Theodor Sturm,et al.  On the geometry of metric measure spaces , 2006 .

[61]  T. Nieuwenhuizen.  What are quantum fluctuations , 2007 .

[62]  T. Mikami.  Optimal Transportation Problem as Stochastic Mechanics , 2008 .

[63]  Toshio Mikami,et al.  Optimal Transportation Problem by Stochastic Optimal Control , 2005, SIAM J. Control. Optim..

[64]  C. Villani.  Optimal Transport: Old and New , 2008 .

[65]  A. Galichon,et al.  Matching with Trade-Offs: Revealed Preferences Over Competing Characteristics , 2009, 2102.12811.

[66]  Christian L'eonard.  Entropic projections and dominating points , 2010 .

[67]  Christian L'eonard,et al.  O C ] 1 1 N ov 2 01 0 FROM THE SCHRÖDINGER PROBLEM TO THE MONGE-KANTOROVICH , 2010 .

[68]  C. Léonard.  Girsanov theory under a finite entropy condition , 2011, 1101.3958.

[69]  Toshio Mikami.  A Characterization of the Knothe-Rosenblatt Processes by a Convergence Result , 2012, SIAM J. Control. Optim..

[70]  Christian L'eonard.  On the convexity of the entropy along entropic interpolations , 2013, 1310.1274.

[71]  Christian L'eonard.  Lazy random walks and optimal transport on graphs , 2013, 1308.0226.

[72]  On the time symmetry of some stochastic processes , 2013 .

[73]  A. Dupuy,et al.  Personality Traits and the Marriage Market , 2014, Journal of Political Economy.

[74]  Christian L'eonard.  Some properties of path measures , 2013, 1308.0217.