The research program of Stochastic Deformation (with a view toward Geometric Mechanics)

We give an overview of a program of Stochastic Deformation of Classical Mechanics and the Calculus of Variations, strongly inspired by the quantization method.

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

[2]  Christian L'eonard,et al.  Reciprocal processes. A measure-theoretical point of view , 2013, 1308.0576.

[3]  J. Marsden,et al.  Introduction to mechanics and symmetry , 1994 .

[4]  Local Invertibility of Adapted Shifts on Wiener Space and Related Topics , 2013 .

[5]  Duke Benadom Forward , 1996, Nursing standard (Royal College of Nursing (Great Britain) : 1987).

[6]  J. Zambrini,et al.  Probability and quantum symmetries. I. The theorem of Noether in Schrödinger's euclidean quantum mechanics , 1997 .

[7]  P. Kazinski Fluctuations as stochastic deformation. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[9]  Yaozhong Hu,et al.  Malliavin calculus and stochastic analysis : a festschrift in honor of David Nualart , 2013 .

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

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

[12]  U. Haussmann On the Adjoint Process for Optimal Control of Diffusion Processes , 1981 .

[13]  J. V. Casteren Markov Processes, Feller Semigroups And Evolution Equations , 2010 .

[14]  池田 信行,et al.  Stochastic differential equations and diffusion processes , 1981 .

[15]  V. Rohatgi A Course in the Theory of Stochastic Processes , 1983 .

[16]  Stochastic variational derivations of the Navier-Stokes equation , 1981 .

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

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

[19]  R. L. Dobrushin,et al.  A. N. Kolmogorov - the founder of the theory of reversible Markov processes , 1988 .

[20]  G. G. Stokes "J." , 1890, The New Yale Book of Quotations.

[21]  Andrea Raimondo,et al.  A Deformation of the Method of Characteristics and the Cauchy Problem for Hamiltonian PDEs in the Small Dispersion Limit , 2012, 1211.2676.

[22]  L. Rogers Stochastic differential equations and diffusion processes: Nobuyuki Ikeda and Shinzo Watanabe North-Holland, Amsterdam, 1981, xiv + 464 pages, Dfl.175.00 , 1982 .

[23]  Francesco Russo,et al.  Forward, backward and symmetric stochastic integration , 1993 .

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

[25]  Nicolas Privault,et al.  Markovian bridges and reversible diffusion processes with jumps , 2004 .

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

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

[28]  P. Lescot On affine interest rate models , 2009, 0911.2757.

[29]  On the geometry of the Hamilton-Jacobi-Bellman equation , 2009 .

[30]  Klaus Reinhardt,et al.  Introduction To Random Time And Quantum Randomness , 2016 .

[31]  Jan A. van Casteren,et al.  Non-autonomous Kato classes and Feynman-Kac propagators , 2006 .

[32]  Radoslaw K. Wojciechowski Heat Kernel and Essential Spectrum of Infinite Graphs , 2008, 0802.2745.

[33]  E. Villaseñor Introduction to Quantum Mechanics , 2008, Nature.

[34]  Jean-Claude Zambrini,et al.  Probabilistic Deformation of Contact Geometry, Diffusion Processes and Their Quadratures , 2007 .

[35]  Edward Nelson Feynman Integrals and the Schrödinger Equation , 1964 .

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

[37]  B. Jamison Reciprocal processes , 1974 .

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

[39]  Nicolas Privault,et al.  Stochastic deformation of integrable dynamical systems and random time symmetry , 2010 .

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

[41]  F. Estabrook,et al.  Geometric Approach to Invariance Groups and Solution of Partial Differential Systems , 1971 .

[42]  Phillip A. Griffiths,et al.  Exterior Differential Systems and Euler-Lagrange Partial Differential Equations , 2003 .

[43]  Schrödinger problem, Lévy processes, and noise in relativistic quantum mechanics. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[44]  J. Ortega,et al.  Stochastic hamiltonian dynamical systems , 2007, math/0702787.

[45]  P. Lee,et al.  A Course in the Theory of Stochastic Processes , 1983, The Mathematical Gazette.

[46]  A characterization of reciprocal processes via an integration by parts formula on the path space , 2002 .

[47]  Tihomir Car Lectures on Physics , 2013 .

[48]  R. Leighton,et al.  Feynman Lectures on Physics , 1971 .

[49]  L. Alili,et al.  Boundary-Crossing Identities for Diffusions Having the Time-Inversion Property , 2009, 0904.2680.

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

[51]  Anton Wakolbinger A simplified variational characterization of Schrödinger processes , 1989 .

[52]  Nicolas Privault,et al.  Feynman-Kac formula for Levy processesand semiclassical (Euclidean) momentum representation , 2013, 1308.2363.

[53]  J. Janssen,et al.  Deterministic and Stochastic Optimal Control , 2013 .

[54]  R. Cameron A Family of Integrals Serving to Connect the Wiener and Feynman Integrals , 1960 .

[55]  S. Albeverio,et al.  Probability and Quantum Symmetries. II. The Theorem of Nœther in quantum mechanics , 2006 .

[56]  E. Bohl,et al.  Discrete and continuous dynamical systems , 1996 .

[57]  Christian L'eonard A survey of the Schr\"odinger problem and some of its connections with optimal transport , 2013, 1308.0215.

[58]  Kiyosi Itô,et al.  The Brownian Motion and Tensor Fields on Riemannian Manifold , 1987 .

[59]  D. Griffiths,et al.  Introduction To Quantum Mechanics Second Edition , 2005 .

[60]  Adam Bowditch Stochastic Analysis , 2013 .

[61]  Nicolas Privault,et al.  Euclidean quantum mechanics in the momentum representation , 2005 .

[62]  J. Zambrini,et al.  Bernstein Diffusions for a Class of Linear Parabolic Partial Differential Equations , 2013, 1305.4392.

[63]  Yehoshua Y. Zeevi,et al.  Image enhancement and denoising by complex diffusion processes , 2004, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[64]  P. Kramer,et al.  Local kinetic interpretation of entropy production through reversed diffusion. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[66]  P. Olver Applications of Lie Groups to Differential Equations , 1986 .

[67]  Jarek Duda,et al.  From Maximal Entropy Random Walk to quantum thermodynamics , 2011, 1111.2253.

[68]  C. Léonard,et al.  A Probabilistic Deformation of Calculus of Variations with Constraints , 2011 .

[69]  Kiyosi Itô Extension of stochastic integrals , 1978 .