V. Arnold, Sur la g??om??trie diff??rentielle des groupes de Lie de dimension infinie et ses applications ?? l'hydrodynamique des fluides parfaits, Annales de l???institut Fourier, vol.16, issue.1, pp.319-361, 1966.
DOI : 10.5802/aif.233
URL : http://archive.numdam.org/article/AIF_1966__16_1_319_0.pdf

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

A. Beurling, An Automorphism of Product Measures, The Annals of Mathematics, vol.72, issue.1, pp.189-200, 1960.
DOI : 10.2307/1970151

Y. Brenier, The least action principle and the related concept of generalized flows for incompressible perfect fluids, Journal of the American Mathematical Society, vol.2, issue.2, pp.225-255, 1989.
DOI : 10.1090/S0894-0347-1989-0969419-8

G. Conforti, P. Dai-pra, and S. Roelly, Reciprocal classes of jump processes. Preprint. http://opus.kobv On quasi-Markov random fields, J. Multivariate Anal, vol.7077, issue.2, pp.14-76, 1972.

K. L. Chung, A course in probability theory. Harcourt, Brace & World, Conforti and C. Léonard. Reciprocal classes of random walks on graphs, 1968.

J. M. Clark, A local characterization of reciprocal diffusions, Applied Stoch. Analysis, vol.5, pp.45-59, 1991.

G. Conforti, C. Léonard, R. Murr, and S. Roelly, Bridges of Markov counting processes. Reciprocal classes and duality formulas, Electronic Communications in Probability, vol.20, issue.0
DOI : 10.1214/ECP.v20-3697
URL : https://hal.archives-ouvertes.fr/hal-01053875

. Ph, P. Courrège, and . Renouard, Oscillateurs harmoniques mesures quasi-invariantes sur c(R, R) et théorie quantique des champs en dimension 1, Astérisque, pp.22-23, 1975.

L. Chaumont and G. Uribe-bravo, Markovian bridges: Weak continuity and pathwise constructions, The Annals of Probability, vol.39, issue.2, pp.609-647, 2011.
DOI : 10.1214/10-AOP562
URL : https://hal.archives-ouvertes.fr/hal-00384359

K. L. Chung and J. B. Walsh, Markov processes, Brownian motion and time symmetry, Lecture Notes in Mathematics, vol.249, 2005.
DOI : 10.1007/0-387-28696-9

A. B. Cruzeiro, L. Wu, and J. Zambrini, Bernstein processes associated with a Markov process Stochastic analysis and mathematical physics, ANESTOC'98 Malliavin calculus and Euclidean quantum mechanics, I, Proceedings of the Third International WorkshopCZ08] K.L. Chung and J.-C. Zambrini. Introduction to Random Time and Quantum Randomness, pp.41-7162, 1991.
DOI : 10.1007/978-1-4612-1372-7_4
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.512.7666

N. , D. Ngoc, and M. Yor, Champs markoviens et mesures de Gibbs sur R, Ann. Sci. ´ Ec

J. L. Doob, Conditional brownian motion and the boundary limits of harmonic functions, Bulletin de la Société mathématique de France, vol.79, pp.431-458, 1957.
DOI : 10.24033/bsmf.1494

E. B. Dynkin, Theory of Markov processes, 1961.

H. Föllmer and N. Gantert, Entropy minimization and Schrödinger processes in infinite dimensions, Ann. Probab, vol.25, issue.2, pp.901-926, 1997.

H. Föllmer, Random fields and diffusion processes, inÉcolein´inÉcole d'´ eté de Probabilités de Saint-Flour XV-XVII-1985-87, Lecture Notes in Mathematics, vol.1362, 1988.

P. Fitzsimmons, J. Pitman, and M. Yor, Markovian Bridges: Construction, Palm Interpretation, and Splicing, Progr. Probab, vol.33, pp.101-134, 1992.
DOI : 10.1007/978-1-4612-0339-1_5
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.159.239

H. Georgii, Gibbs measures and phase transitions, In Studies in Mathematics, vol.9, 2011.
DOI : 10.1515/9783110850147

B. Jamison, Reciprocal Processes: The Stationary Gaussian Case, The Annals of Mathematical Statistics, vol.41, issue.5, pp.1624-1630, 1970.
DOI : 10.1214/aoms/1177696805

B. Jamison, Reciprocal processes, Zeitschrift f???r Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol.32, issue.1, pp.65-86, 1974.
DOI : 10.1007/BF00532864

B. Jamison, The Markov processes of Schr???dinger, Zeitschrift f???r Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol.22, issue.4, pp.323-331, 1975.
DOI : 10.1007/BF00535844

O. Kallenberg, Splitting at Backward Times in Regenerative Sets, The Annals of Probability, vol.9, issue.5, pp.781-799, 1981.
DOI : 10.1214/aop/1176994308

S. Kullback and R. A. Leibler, On Information and Sufficiency, The Annals of Mathematical Statistics, vol.22, issue.1, pp.79-86, 1951.
DOI : 10.1214/aoms/1177729694

A. J. Kolmogorovkre88-]-a and . Krener, Zur Theorie der Markoffschen Ketten Reciprocal diffusions and stochastic differential equations of second order Some properties of path measures, Mathematische Annalen Stochastics, vol.112, issue.46, pp.393-422, 1936.

C. Léonard, A survey of the Schrödinger problem and some of its connections with optimal transport. Discrete Contin, Dyn. Syst. A, vol.34, issue.4, pp.1533-1574, 2014.

R. Murr, Reciprocal classes of Markov processes. An approach with duality formulae, 2012.
URL : https://hal.archives-ouvertes.fr/hal-01053875

M. Nagasawa, Schrödinger equations and diffusions theory, Monographs in Mathematics . Birkhäuser, vol.86, 1993.

E. Nelson, Dynamical theories of Brownian motion at: www.math.princeton.edu/?nelson/books.html. [OS99] H. Osada and H. Spohn. Gibbs measures relative to Brownian motion Markovian bridges and reversible diffusions with jumps, Annals Probab, vol.27, issue.40, pp.1183-1207599, 1967.

S. Roelly, Reciprocal Processes: A Stochastic Analysis Approach, Modern Stochastics with Applications, volume 90 of Optimization and Its Applications, pp.53-67, 2014.
DOI : 10.1007/978-3-319-03512-3_4

S. Roelly and M. Thieullen, A characterization of reciprocal processes via an integration by parts formula on the path space. Probab. Theory Related Fields, pp.97-120, 2004.

S. Roelly and M. Thieullen, Duality formula for the bridges of a Brownian diffusion: Application to gradient drifts, Stochastic Processes and their Applications, pp.1677-1700, 2005.
DOI : 10.1016/j.spa.2005.04.010
URL : https://hal.archives-ouvertes.fr/hal-00101964

G. Royer and M. Yor, Représentation intégrale de certaines mesures quasi-invariantes sur C(R)
DOI : 10.24033/asens.1290
URL : http://archive.numdam.org/article/ASENS_1975_4_8_3_319_0.pdf

E. Schrödinger, Sur la théorie relativiste de l'´ electron et l'interprétation de la mécanique quantique, Ann. Inst. H. Poincaré, vol.2, pp.269-310, 1932.

M. Thieullen, Second order stochastic differential equations and non-gaussian reciprocal diffusions . Probab. Theory Related Fields, pp.231-257, 1993.
DOI : 10.1007/bf01199322

M. Thieullen, Reciprocal diffusions and symmetries of parabolic PDE: The nonflat case, Potential Analysis, vol.16, issue.1, pp.1-28, 2002.
DOI : 10.1023/A:1024813508835

M. Thieullen and J. Zambrini, Probability and quantum symmetries. I. The theorem of Noether in Schrödinger's Euclidean quantum mechanics, Ann. Inst. H. Poincaré Phys. Théor, vol.67, pp.297-338, 1997.

M. Thieullen and J. Zambrini, Symmetries in the stochastic calculus of variations. Probab. Theory Related Fields, pp.401-427, 1997.

C. Van-putten and J. H. Van-schuppen, Invariance Properties of the Conditional Independence Relation, The Annals of Probability, vol.13, issue.3, pp.934-945, 1985.
DOI : 10.1214/aop/1176992915

P. Vuillermot and J. Zambrini, Bernstein diffusions for a class of linear parabolic PDEs, Journal of Theor. Probab, pp.1-44, 2012.

J. Zambrini, Variational processes and stochastic versions of mechanics, Journal of Mathematical Physics, vol.27, issue.9, pp.2307-2330, 1986.
DOI : 10.1063/1.527002

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