1,106
Views
4
CrossRef citations to date
0
Altmetric
Review Article

Path integrals and stochastic calculus

, , &

References

  • P. Lançon, G. Batrouni, L. Lobry, and N. Ostrowsky, Drift without flux: Brownian walker with a space-dependent diffusion coefficient, Europhys. Lett. 54 (2001) p. 28. Available at https://iopscience.iop.org/article/10.1209/epl/i2001-00103-6/meta.
  • P. Lançon, G. Batrouni, L. Lobry, and N. Ostrowsky, Brownian walker in a confined geometry leading to a space-dependent diffusion coefficient, Phys. A: Stat. Mech. Appl. 304 (2002) pp. 65–76. Available at https://www.sciencedirect.com/science/article/pii/S0378437101005106.
  • A.G. Cherstvy, A.V. Chechkin, and R. Metzler, Anomalous diffusion and ergodicity breaking in heterogeneous diffusion processes, New. J. Phys. 15 (2013) p. 083039. Available at https://dx.doi.org/10.1088/1367-2630/15/8/083039.
  • A.G. Cherstvy and R. Metzler, Population splitting, trapping, and non-ergodicity in heterogeneous diffusion processes, Phys. Chem. Chem. Phys. 15 (2013) pp. 20220–20235. Available at https://pubs.rsc.org/en/content/articlelanding/2013/cp/c3cp53056f.
  • D.T. Gillespie, The chemical Langevin equation, J. Chem. Phys. 113 (2000) pp. 297–306. Available at https://aip.scitation.org/doi/abs/10.1063/1.481811.
  • M. Kærn, T.C. Elston, W.J. Blake, and J.J. Collins, Stochasticity in gene expression: From theories to phenotypes, Nat. Rev. Genet. 6 (2005) pp. 451–464. Available at https://www.nature.com/articles/nrg1615.
  • E.D. Giuli and C. Scalliet, Dynamical mean-field theory: From ecosystems to reaction networks, J. Phys. A: Math. Theor. 55 (2022) p. 474002. Available at https://dx.doi.org/10.1088/1751-8121/aca3df.
  • Y. Hamao, R.W. Masulis, and V. Ng, Correlations in Price Changes and Volatility across International Stock Markets, Rev. Financ. Stud. 3 (1990) pp. 281–307. Available at https://doi.org/10.1093/rfs/3.2.281.
  • A. Matacz, A new theory of stochastic inflation, Phys. Rev. D 55 (1997) pp. 1860–1874. Available at https://link.aps.org/doi/10.1103/PhysRevD.55.1860.
  • B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, 6th ed., Universitext, Springer, Berlin, 2013.
  • P. Hänggi, Connection between deterministic and stochastic descriptions of nonlinear systems, Helv. Phys. Acta 53 (1980) p. 491. Available at https://dx.doi.org/10.5169/seals-115133.
  • P. Hänggi, Stochastic processes I: Asymptotic behaviour and symmetries, Helv. Phys. Acta 51 (1978) p. 183. Available at https://dx.doi.org/10.5169/seals-114941.
  • P. Hänggi and H. Thomas, Stochastic processes: Time-evolution, symmetries and linear response, Phys. Rep. 88 (1982) p. 207. Available at https://doi.org/10.1016/0370-1573(82)90045-X.
  • Y.L. Klimontovich, Nonlinear Brownian motion, Phys. Uspekhi 37 (1994) p. 737. Available at https://dx.doi.org/10.1070/PU1994v037n08ABEH000038.
  • C.W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences, 2nd ed., no. 13 in Springer series in synergetics, Springer-Verlag, Berlin, NY, 1994.
  • N.G. van Kampen, Stochastic Processes in Physics and Chemistry, 3rd ed., North-Holland personal library, Elsevier, Amsterdam; Boston, 2007.
  • R. Zwanzig, Nonequilibrium Statistical Mechanics, Oxford University Press, Oxford, NY, 2001.
  • H. Grabert, P. Hänggi, and P. Talkner, Is quantum mechanics equivalent to a classical stochastic process?, Phys. Rev. A. 19 (1979) p. 2440. Available at https://doi.org/10.1103/PhysRevA.19.2440.
  • E. Nelson, Quantum Fluctuations, Princeton series in physics, Princeton University Press, Princeton, NJ, 1985.
  • I.M. Sokolov, Ito, stratonovich, hänggi and all the rest: The thermodynamics of interpretation, Chem. Phys. 375 (2010) pp. 359–363. Available at https://doi.org/10.1016/j.chemphys.2010.07.024.
  • P. Hanggi, Nonlinear fluctuations: The problem of deterministic limit and reconstruction of stochastic dynamics, Phys. Rev. A. 25 (1982) pp. 1130–1136. Available at https://dx.doi.org/10.1103/PhysRevA.25.1130.
  • K. Itō, Stochastic integral, Proc. Imper. Acad. 20 (1944) pp. 519–524. Available at https://projecteuclid.org/euclid.pja/1195572786.
  • N. Wiener, Differential-Space, J. Math. Phys. 2 (1923) pp. 131–174. Available at http://doi.wiley.com/10.1002/sapm192321131.
  • N. Wiener, The Average value of a Functional, Proc. Lond. Math. Soc. s2–22 (1924) pp. 454–467. Available at http://doi.wiley.com/10.1112/plms/s2-22.1.454.
  • R.P. Feynman, Space-Time Approach to Non-Relativistic Quantum Mechanics, Rev. Mod. Phys. 20 (1948) p. 367. Available at http://link.aps.org/doi/10.1103/RevModPhys.20.367.
  • L. Onsager and S. Machlup, Fluctuations and Irreversible Processes, Phys. Rev. 91 (1953) pp. 1505–1512. Available at http://link.aps.org/doi/10.1103/PhysRev.91.1505.
  • S. Machlup and L. Onsager, Fluctuations and Irreversible Process. II. Systems with Kinetic Energy, Phys. Rev. 91 (1953) pp. 1512–1515. Available at http://link.aps.org/doi/10.1103/PhysRev.91.1512.
  • B. Caroli, C. Caroli, and B. Roulet, Diffusion in a bistable potential: The functional integral approach, J. Stat. Phys. 26 (1981) p. 83.
  • C. Aron, G. Biroli, and L.F. Cugliandolo, Symmetries of generating functionals of Langevin processes with colored multiplicative noise, J. Stat. Mech. 2010 (2010) p. P11018. Available at https://dx.doi.org/10.1088/1742-5468/2010/11/P11018.
  • U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep. Prog. Phys. 75 (2012) p. 126001. Available at https://dx.doi.org/10.1088/0034-4885/75/12/126001.
  • L.F. Cugliandolo, Dynamics of glassy systems, in Slow Relaxations and Nonequilibrium Dynamics in Condensed Matter: Les Houches Session LXXVII, Springer, Berlin, 2002.
  • S.F. Edwards and Y.V. Gulyaev, Path integrals in polar co-ordinates, Proc. R. Soc. Lond. A 279 (1964) pp. 229–235. Available at http://rspa.royalsocietypublishing.org/content/279/1377/229.
  • J.L. Gervais and A. Jevicki, Point canonical transformations in the path integral, Nucl. Phys. B 110 (1976) pp. 93–112. Available at http://www.sciencedirect.com/science/article/pii/0550321376904223.
  • P. Salomonson, When does a non-linear point transformation generate an extra o(ℏ2) potential in the effective Lagrangian?, Nucl. Phys. B 121 (1977) pp. 433–444. Available at http://www.sciencedirect.com/science/article/pii/0550321377901651.
  • U. Deininghaus and R. Graham, Nonlinear point transformations and covariant interpretation of path integrals, Z. Phys. B Con. Matt. 34 (1979) pp. 211–219. Available at https://link.springer.com/article/10.1007/BF01322143.
  • J. Alfaro and P.H. Damgaard, Field transformations, collective coordinates and BRST invariance, Ann. Phys. (N. Y) 202 (1990) pp. 398–435. Available at http://www.sciencedirect.com/science/article/pii/000349169090230L.
  • K.M. Apfeldorf and C. Ordóñez, Coordinate redefinition invariance and “extra” terms, Nucl. Phys. B 479 (1996) pp. 515–526. Available at http://www.sciencedirect.com/science/article/pii/0550321396004518.
  • C. Aron, D.G. Barci, L.F. Cugliandolo, Z. González Arenas, and G.S. Lozano, Dynamical symmetries of Markov processes with multiplicative white noise, J. Stat. Mech. 2016 (2016) p. 053207. Available at http://stacks.iop.org/1742-5468/2016/i=5/a=053207.
  • L.F. Cugliandolo and V. Lecomte, Rules of calculus in the path integral representation of white noise Langevin equations: The Onsager–Machlup approach, J. Phys. A: Math. Theor. 50 (2017) pp. 345001. Available at https://dx.doi.org/10.1088/1751-8121/aa7dd6.
  • P. Cartier and C. DeWitte-Morette, Functional Integration: Action and Symmetries, Cambridge University Press, Cambridge, 2006.
  • J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 4th ed., no. 113 in International series of monographs on physics, Clarendon Press; Oxford University Press, Oxford, NY, 2002.
  • M. Chaichian and A.P. Demičev, Stochastic Processes and Quantum Mechanics, Path integrals in physics, Vol. 1, Inst. of Physics Publ, Bristol, 2001.
  • H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, World Scientific, Singapore; Hackensack, NJ, 2009. Available at http://ebooks.worldscinet.com/ISBN/9789814273572/9789814273572.shtml.
  • M. Itami and S.I. Sasa, Universal Form of Stochastic Evolution for Slow Variables in Equilibrium Systems, J. Stat. Phys. 167 (2017) pp. 46–63. Available at https://link.springer.com/article/10.1007/s10955-017-1738-6.
  • V. Vennin and A.A. Starobinsky, Correlation functions in stochastic inflation, Eur. Phys. J. C 75 (2015) pp. 413. Available at https://link.springer.com/article/10.1140/epjc/s10052-015-3643-y.
  • L. Pinol, S. Renaux-Petel, and Y. Tada, Inflationary stochastic anomalies, Class. Quantum Grav. 36 (2019) pp. 07LT01. Available at https://dx.doi.org/10.1088/1475-7516/2021/04/048.
  • G.A. Gottwald, D.T. Crommelin, and C.L.E. Franzke, Stochastic Climate Theory, in Nonlinear and Stochastic Climate Dynamics, C.L.E. Franzke and T.J. O'Kane, eds., Cambridge University Press, 2017, pp. 209–240.
  • A. Einstein, Über die von der molekularkinetischen theorie der wärme geforderte bewegung von in ruhenden flüssigkeiten suspendierten teilchen, Ann. Phys. 4 (1905) p.549. Available at https://doi.org/10.1002/andp.2005517S112.
  • R.A. Duine and H.T.C. Stoof, Stochastic dynamics of a trapped Bose-Einstein condensate, Phys. Rev. A 65 (2001) p. 013603. Available at https://link.aps.org/doi/10.1103/PhysRevA.65.013603.
  • I. Karatzas and S.E. Shreve, Methods of Mathematical Finance, corrected 4th printing ed., no. volume 39 in Probability theory and stochastic modelling, Springer, New York Heidelberg Dordrecht London, 2016.
  • C. Bustamante, D. Keller, and G. Oster, The Physics of Molecular Motors, Acc. Chem. Res. 34 (2001) pp. 412–420. Available at https://doi.org/10.1021/ar0001719.
  • M. Ding and X. Xing, Time-Slicing Path-integral in Curved Space, Quantum 6 (2022) pp. 694. Available at https://quantum-journal.org/papers/q-2022-04-21-694/.
  • B.S. DeWitt, Dynamical Theory in Curved Spaces. I. A Review of the Classical and Quantum Action Principles, Rev. Mod. Phys. 29 (1957) pp. 377–397. Available at https://link.aps.org/doi/10.1103/RevModPhys.29.377.
  • R. Graham, Covariant formulation of non-equilibrium statistical thermodynamics, Z. Phys. B Con. Matt. 26 (1977) pp. 397–405. Available at https://link.springer.com/article/10.1007/BF01570750.
  • R. Graham, Path integral formulation of general diffusion processes, Z. Phys. B Con. Matt. 26 (1977) pp. 281–290. Available at https://link.springer.com/article/10.1007/BF01312935.
  • L.F. Cugliandolo, V. Lecomte, and F. Van Wijland, Building a path-integral calculus: A covariant discretization approach, J. Phys. A: Math. Theor. 52 (2019) p. 50LT01. Available at https://dx.doi.org/10.1088/1751-8121/ab3ad5.
  • K.H. Lan, N. Ostrowsky, and D. Sornette, Brownian dynamics close to a wall studied by photon correlation spectroscopy from an evanescent wave, Phys. Rev. Lett. 57 (1986) pp. 17–20. Available at https://link.aps.org/doi/10.1103/PhysRevLett.57.17.
  • S. Andarwa, H. Basirat Tabrizi, and G. Ahmadi, Effect of correcting near-wall forces on nanoparticle transport in a microchannel, Particuology 16 (2014) pp. 84–90. Available at https://www.sciencedirect.com/science/article/pii/S1674200114000170.
  • S. Ghosh, F. Mugele, and M.H.G. Duits, Effects of shear and walls on the diffusion of colloids in microchannels, Phys. Rev. E 91 (2015) p. 052305. Available at https://link.aps.org/doi/10.1103/PhysRevE.91.052305.
  • S. Pieprzyk, D.M. Heyes, and A.C. Brańka, Spatially dependent diffusion coefficient as a model for ph sensitive microgel particles in microchannels, Biomicrofluidics 10 (2016) p. 054118. Available at https://doi.org/10.1063/1.4964935.
  • S. Marbach, D.S. Dean, and L. Bocquet, Transport and dispersion across wiggling nanopores, Nat. Phys. 14 (2018) pp. 1108–1113. Available at https://www.nature.com/articles/s41567-018-0239-0.
  • W. Coffey and Y.P. Kalmykov, The Langevin Equation: With Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering, Vol. 27, World Scientific, Singapore, 2012.
  • J.T. Lewis, J. McConnell, and B.K.P. Scaife, Relaxation effects in rotational brownian motion, P. Roy. Irish Acad. A 76 (1976) pp. 43–69. Available at http://www.jstor.org/stable/20489032.
  • W.F. Brown Jr, Thermal fluctuations of a single-domain particle, Phys. Rev. 130 (1963) p. 1677. Available at http://link.aps.org/doi/10.1103/PhysRev.130.1677.
  • M.E. Cates and J. Tailleur, Motility-induced phase separation, Ann. Rev. Condens. Matter Phys. 6 (2015) pp. 219–244. Available at https://dx.doi.org/10.1146/annurev-conmatphys-031214-014710.
  • B. Spagnolo, D. Valenti, and A. Fiasconaro, Noise in ecosystems: A short review, Math. Biosci. Eng.1 (2004) pp. 185–211. Available at http://www.aimspress.com/rticle/doi/10.3934/mbe.2004.1.185.
  • J. Hull, Options, Futures and Other Derivatives/John C. Hull, Prentice Hall, Upper Saddle River, NJ, 2009.
  • N.G. van Kampen, Itô versus Stratonovich, J. Stat. Phys. 24 (1981) pp. 175–187. Available at https://link.springer.com/article/10.1007/BF01007642.
  • H.K. Janssen, On the renormalized field theory of nonlinear critical relaxation, in From Phase Transitions to Chaos, World Scientific, 1992, pp. 68–91, Available at https://doi.org/10.1142/9789814355872_0007.
  • J. Dunkel and P. Hänggi, Theory of relativistic Brownian motion: The (1 + 1)-dimensional case, Phys. Rev. E 71 (2005) p. 016124. Available at https://doi.org/10.1103/PhysRevE.71.016124.
  • G. Mil'shtejn, Approximate Integration of Stochastic Differential Equations, Teor. Veroyatnost. i Primenen. 19 (1974) pp. 583–588. Available at http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tvp&paperid=2929&option_lang=eng.
  • G. Mil'shtejn, Approximate Integration of Stochastic Differential Equations, Theor. Probab. Appl+19 (1975) pp. 557–562. Available at https://epubs.siam.org/doi/10.1137/1119062.
  • P.E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, 2nd ed., no. 23 in Applications of mathematics, Springer, Berlin, NY, 1995.
  • R. Mannella, Integration of stochastic differential equations on a computer, Int. J. Mod. Phys. C 13 (2002) pp. 1177–1194. Available at https://www.worldscientific.com/doi/abs/10.1142/S0129183102004042.
  • P.E. Kloeden, E. Platen, and H. Schurz, Numerical Solution of SDE Through Computer Experiments, Springer Science & Business Media, Heidelberg, 2012.
  • O. Farago and N. Grønbech-Jensen, Langevin dynamics in inhomogeneous media: Re-examining the itô-stratonovich dilemma, Phys. Rev. E. 89 (2014) p. 013301. Available at http://dx.doi.org/10.1103/PhysRevE.89.013301.
  • F. Langouche, D. Roekaerts, and E. Tirapegui, Functional integrals and the Fokker-Planck equation, Nuovo Ciment. B 53 (1979) pp. 135–159. Available at https://dx.doi.org/10.1007/BF02739307.
  • Z. González Arenas and D.G. Barci, Hidden symmetries and equilibrium properties of multiplicative white-noise stochastic processes, J. Stat. Mech. 2012 (2012) p. P12005. Available at https://dx.doi.org/10.1088/1742-5468/2012/12/P12005.
  • G. Parisi, Statistical Field Theory, Addison-Wesley, New York, 1988.
  • W. Horsthemke and R. Lefever, Noise Induced Phase Transitions, Springer, Berlin, 1984.
  • F. Sagués, J.M. Sancho, and J. García-Ojalvo, Spatiotemporal order out of noise, Rev. Mod. Phys. 79 (2007) p. 829. Available at https://doi.org/10.1103/RevModPhys.79.829.
  • R.L. Stratonovich, Nonlinear Nonequilibrium Thermodynamics I, Springer, Berlin, 1992.
  • Y. Klimontovich, Ito, stratonovich and kinetic forms of stochastic equations, Physica. A.163 (1990) p. 515. Available at https://doi.org/10.1016/0378-4371(90)90142-F.
  • K. Sekimoto, Stochastic Energetics, Vol. 799, Springer, Heidelberg, 2010.
  • D.W. McLaughlin and L.S. Schulman, Path Integrals in Curved Spaces, J. Math. Phys. 12 (1971) pp. 2520–2524. Available at http://aip.scitation.org/doi/10.1063/1.1665567.
  • U. Weiss, Operator ordering schemes and covariant path integrals of quantum and stochastic processes in Curved space, Z. Phys. B Con. Matt. 30 (1978) pp. 429–436. Available at https://link.springer.com/article/10.1007/BF01321096.
  • A.C. Hirshfeld, Canonical and covariant path integrals, Phys. Lett. A 67 (1978) pp. 5–8. Available at https://www.sciencedirect.com/science/article/pii/0375960178905509.
  • H.O. Girotti and T.J.M. Simões, A generalized treatment of point canonical transformations in the path integral, Nuovo Ciment. B (1971–1996) 74 (1983) pp. 59–66. Available at https://doi.org/10.1007/BF02721685.
  • P. Hänggi, On derivations and solutions of master equations and asymptotic representations, Z. Phys. B Con. Matt. 30 (1978) pp. 85–95. Available at https://doi.org/10.1007/BF01323672.
  • S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, Wiley, New York, 1972.
  • H. Grabert, R. Graham, and M.S. Green, Fluctuations and nonlinear irreversible processes. ii, Phys. Rev. A. 21 (1980) p. 2136. Available at https://doi.org/10.1103/PhysRevA.21.2136.
  • J.A. Hertz, Y. Roudi, and P. Sollich, Path integral methods for the dynamics of stochastic and disordered systems, J. Phys. A: Math. Theor. 50 (2016) p. 033001. Available at https://dx.doi.org/10.1088/1751-8121/50/3/033001.
  • F. Langouche, D. Roekaerts, and E. Tirapegui, Functional integral methods for random fields, in Stochastic Processes in Nonequilibrium Systems, Springer, 1978, pp. 316–329, Available at https://dx.doi.org/10.1007/BFb0016723.
  • F. Langouche, D. Roekaerts, and E. Tirapegui, Functional Integration and Semiclassical Expansions, Kluwer Academic Publishers, Dordrecht, 1982.
  • F. Langouche, D. Roekaerts, and E. Tirapegui, General Langevin equations and functional integration, in Field Theory, Quantization and Statistical Physics: In Memory of Bernard Jouvet, Mathematical Physics and Applied Mathematics, Springer, Dordrecht, 1981, pp. 295–318, Available at https://doi.org/10.1007/978-94-009-8368-7_17.
  • M. Omote, Point canonical transformations and the path integral, Nucl. Phys. B 120 (1977) pp. 325–332. Available at https://doi.org/10.1016/0550-3213(77)90047-5.
  • A.W.C. Lau and T.C. Lubensky, State-dependent diffusion: Thermodynamic consistency and its path integral formulation, Phys. Rev. E 76 (2007) pp. 011123. Available at http://link.aps.org/doi/10.1103/PhysRevE.76.011123.
  • C. Aron, D.G. Barci, L.F. Cugliandolo, Z. González-Arenas, and G.S. Lozano, Magnetization dynamics: Path-integral formalism for the stochastic landau-lifshitz-gilbert equation, J. Stat. Mech. 2014 (2014) p. P09008. Available at https://doi.org/10.1088/1742-5468/2014/09/P09008.
  • R.L. Stratonovich, On the probability functional of diffusion processes, in Sixth All-Union Conf. Theory Prob. and Math. Statist. (Vilnius, 1960) (Russian), Gosudarstv. Izdat. Politic˘esk. i Nauc˘n. Lit. Litovsk. SSR, Vilnius, 1962, pp. 471–482.
  • R.L. Stratonovich, On the probability functional of diffusion processes, Selected Trans. Math. Stat. Prob. 10 (1971) pp. 273–286.
  • W. Horsthemke and A. Bach, Onsager-Machlup Function for one dimensional nonlinear diffusion processes, Z. Phys. B Con. Matt.22 (1975) pp. 189–192. Available at https://link.springer.com/article/10.1007/BF01322364.
  • F. Langouche, D. Roekaerts, and E. Tirapegui, Short derivation of Feynman Lagrangian for general diffusion processes, J. Phys. A: Math. Gen. 13 (1980) p. 449. Available at https://dx.doi.org/10.1088/0305-4470/13/2/013.
  • R. Adler, M. Bazin, and M. Schiffer, Introduction to General Relativity, 2nd ed., International series in pure and applied physics, McGraw-Hill, New York, 1975.
  • K. Cheng, Quantization of a general dynamical system by Feynman's path integration formulation, J. Math. Phys. 13 (1972) pp. 1723–1726. Available at https://doi.org/10.1063/1.1665897.
  • C. Morette, On the definition and approximation of Feynman's path integrals, Phys. Rev. 81 (1951) p. 848. Available at https://dx.doi.org/10.1103/PhysRev.81.848.
  • H. Dekker, Path integrals in Riemannian spaces, Phys. Lett. A 76 (1980) pp. 8–10. Available at http://www.sciencedirect.com/science/article/pii/0375960180901322.
  • H. Dekker, Proof of identity of graham and dekker covariant lattice propagators, Phys. Rev. A. 24 (1981) p. 3182. Available at https://doi.org/10.1103/PhysRevA.24.3182.
  • H. Dekker, On the functional integral for generalized Wiener processes and nonequilibrium phenomena, Physica. A. 85 (1976) pp. 598–606. Available at http://www.sciencedirect.com/science/article/pii/0378437176900285.
  • P. Arnold, Symmetric path integrals for stochastic equations with multiplicative noise, Phys. Rev. E61 (2000) pp. 6099–6102. Available at https://link.aps.org/doi/10.1103/PhysRevE.61.6099.
  • P.S. Riseborough and P. Hanggi, Diffusion on surfaces of finite size: Mössbauer effect as a probe, Surf. Sci. 122 (1982) pp. 459–473. Available at https://doi.org/10.1016/0039-6028(82)90096-6.
  • G. Falsone, Stochastic differential calculus for Gaussian and non-Gaussian noises: A critical review, Comm. Nonlinear Sci. 56 (2018) pp. 198–216. Available at https://doi.org/10.1016/j.cnsns.2017.08.001.
  • R. Kubo, K. Matsuo, and K. Kitahara, Fluctuation and relaxation of macrovariables, J. Stat. Phys.9 (1973) pp. 51–96. Available at https://link.springer.com/article/10.1007/BF01016797.
  • C. De Dominicis, Techniques de renormalisation de la théorie des champs et dynamique des phénomènes critiques, J. Phys. Colloq. 37 (1976) pp. C1 85C1–247 –C1–253. Available at http://dx.doi.org/10.1051/jphyscol:1976138.
  • H.K. Janssen, On a Lagrangean for classical field dynamics and renormalization group calculations of dynamical critical properties, Z. Phys. B Con. Matt. 23 (1976) pp. 377–380. Available at https://doi.org/10.1007/BF01316547.
  • C. De Dominicis and L. Peliti, Field-theory renormalization and critical dynamics above Tc: Helium, antiferromagnets, and liquid-gas systems, Phys. Rev. B 18 (1978) pp. 353–376. Available at http://link.aps.org/doi/10.1103/PhysRevB.18.353.
  • P.C. Martin, E.D. Siggia, and H.A. Rose, Statistical Dynamics of Classical Systems, Phys. Rev. A 8 (1973) pp. 423–437. Available at http://link.aps.org/doi/10.1103/PhysRevA.8.423.
  • C. Aron, D.G. Barci, L.F. Cugliandolo, Z. González Arenas, and G.S. Lozano, Dynamical symmetries of Markov processes with multiplicative white noise, arXiv:1412.7564v1 [cond-mat] (2014), Available at http://arxiv.org/abs/1412.7564v1, arXiv: 1412.7564v1.
  • R. Kubo, The fluctuation-dissipation theorem, Rep. Prog. Phys. 29 (1966) p. 255. Available at https://dx.doi.org/10.1088/0034-4885/29/1/306.
  • D.J. Evans, E.G.D. Cohen, and G.P. Morriss, Probability of second law violations in shearing steady states, Phys. Rev. Lett. 71 (1993) pp. 2401–2404. Available at http://link.aps.org/doi/10.1103/PhysRevLett.71.2401.
  • G. Gallavotti and E.G.D. Cohen, Dynamical Ensembles in Nonequilibrium Statistical Mechanics, Phys. Rev. Lett.74 (1995) pp. 2694–2697. Available at http://link.aps.org/doi/10.1103/PhysRevLett.74.2694.
  • G. Gallavotti and E.G.D. Cohen, Dynamical ensembles in stationary states, J. Stat. Phys. 80 (1995) pp. 931–970. Available at http://link.springer.com/article/10.1007/BF02179860.
  • J. Kurchan, Fluctuation theorem for stochastic dynamics, J. Phys. A: Math. Gen. 31 (1998) pp. 3719–3729. Available at http://iopscience.iop.org/0305-4470/31/16/003.
  • J.L. Lebowitz and H. Spohn, A Gallavotti–Cohen-Type Symmetry in the Large Deviation Functional for Stochastic Dynamics, J. Stat. Phys. 95 (1999) pp. 333–365. Available at https://doi.org/10.1023/A:1004589714161.
  • D.J. Evans and D.J. Searles, The fluctuation theorem, Adv. Phys. 51 (2002) p. 1529. Available at https://doi.org/10.1080/00018730210155133.
  • E.M. Sevick, R. Prabhakar, S.R. Williams, and D.J. Searles, Fluctuation theorems, Ann. Rev. Phys. Chem. 59 (2008) p. 603. Available at https://doi.org/10.1146/annurev.physchem.58.032806.104555.
  • J. Tailleur, J. Kurchan, and V. Lecomte, Mapping out-of-equilibrium into equilibrium in one-dimensional transport models, J. Phys. A: Math. Theor. 41 (2008) p. 505001. Available at https://dx.doi.org/10.1088/1751-8113/41/50/505001.
  • R. Phythian, The functional formalism of classical statistical dynamics, J. Phys. A: Math. General10 (1977) p. 777. Available at https://dx.doi.org/10.1088/0305-4470/10/5/011.
  • P. Hänggi, Path integral solutions for non-Markovian processes, Z. Phys. B Con. Matt. 75 (1989) pp. 275–281. Available at https://doi.org/10.1007/BF01308011.
  • P. Hänggi, Path integral solution for nonlinear generalized langevin equations, in Path Integrals From Mev To Mev: Tutzing '92 (Proceedings Of The Fouth International Conference), World Scientific, Singapore, London, Hong-Kong, 1993, pp. 289–301.
  • G. Parisi and N. Sourlas, Supersymmetric field theories and stochastic differential equations, Nucl. Phys. B 206 (1982) pp. 321–332. Available at http://www.sciencedirect.com/science/article/pii/0550321382905387.
  • M.V. Feigel'man and A.M. Tsvelik, Hidden supersymmetry of stochastic dissipative dynamics, Sov. Phys. JETP (Engl. Transl.) 56(4) (1982) p.849. Available at http://www.jetp.ras.ru/cgi-bin/e/index/e/56/4/p823?a=list.
  • B. Marguet, E. Agoritsas, L. Canet, and V. Lecomte, Supersymmetries in nonequilibrium Langevin dynamics, Phys. Rev. E 104 (2021) p. 044120. Available at https://link.aps.org/doi/10.1103/PhysRevE.104.044120.
  • J. Kappler and R. Adhikari, Stochastic action for tubes: Connecting path probabilities to measurement, Phys. Rev. Res. 2 (2020) pp. 023407. Available at https://link.aps.org/doi/10.1103/PhysRevResearch.2.023407.
  • M. Capitaine, On the Onsager-Machlup functional for elliptic diffusion processes, in Séminaire de Probabilités XXXIV, J. Azéma, M. Ledoux, M. Èmery, and M. Yor, eds., Springer Berlin Heidelberg, Berlin, Heidelberg, 2000, pp. 313–328, Available at https://doi.org/10.1007/BFb0103810.
  • D.S. Dean, B. Miao, and R. Podgornik, Path integrals for higher derivative actions, J. Phys. A: Math. Theor. 52 (2019) p. 505003. Available at https://doi.org/10.1088/1751-8121/ab54df.
  • J.H. Wilson and V. Galitski, Breakdown of the coherent state path integral: Two simple examples, Phys. Rev. Lett. 106 (2011) p. 110401. Available at https://doi.org/10.1103/PhysRevLett.106.110401.
  • F. Bruckmann and J.D. Urbina, Rigorous construction of coherent state path integrals through dualization, arXiv:1807.10462 (2018), Available at http://arxiv.org/abs/1807.10462.
  • A. Rançon, Hubbard–Stratonovich transformation and consistent ordering in the coherent state path integral: Insights from stochastic calculus, J. Phys. A: Math. Theor. 53 (2020) p. 105302. Available at https://doi.org/10.1088/1751-8121/ab6d3b.
  • P. Grassberger and M. Scheunert, Fock-space methods for identical classical objects, Fortschr. Phys.28 (1980) pp. 547–578. Available at https://doi.org/10.1002/prop.19800281004.
  • L. Peliti, Renormalisation of fluctuation effects in the A+A to A reaction, J. Phys. A: Math. Gen.19 (1986) pp. L365. Available at https://dx.doi.org/10.1088/0305-4470/19/6/012.
  • J. Cardy, Renormalisation group approach to reaction-diffusion problems, arXiv preprint cond-mat/9607163 (1996), Available at https://doi.org/10.48550/arXiv.cond-mat/9607163.
  • M. Doi, Second quantization representation for classical many-particle system, J. Phys. A: Math. Gen. 9 (1976) p. 1465. Available at http://iopscience.iop.org/0305-4470/9/9/008.
  • L. Peliti, Path integral approach to birth-death processes on a lattice, J. Phys. 46 (1985) p. 15. Available at https://dx.doi.org/10.1051/jphys:019850046090146900.
  • A. Andreanov, G. Biroli, J.P. Bouchaud, and A. Lefèvre, Field theories and exact stochastic equations for interacting particle systems, Phys. Rev. E 74 (2006) p. 030101. Available at https://doi.org/10.1103/PhysRevE.74.030101.
  • P. Grassberger, On phase transitions in Schlögl's second model, Z. Phys. B Con. Matt. 47 (1982) pp. 365–374. Available at https://doi.org/10.1007/BF01313803.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.