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We review the theory of martingales as applied to stochastic thermodynamics and stochastic processes in physics more generally.
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Quasi-martingales and stochastic integrals
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Nagasawa, M · 1964
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The feynman lectures on physics; vol. i
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The fluctuation-dissipation theorem
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A new representation for stochastic integrals and equations
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On square integrable martingales
Kunita, H. & Watanabe, S · 1967
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Quelques applications de la formule de changement de variables pour les semimartingales
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On the probability functional of diffusion processes
Stratonovich, R. L · 1971
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Martingale central limit theorems
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Branching processes (Springer-Verlag, 1972)
Athreya, K. B. & Ney, P · 1972
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A certain identity for stochastic integrals
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Theory of rational option pricing
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Logical reversibility of computation
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Solvable model of a spin-glass
Sherrington, D. & Kirkpatrick, S · 1975
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Network theory of microscopic and macroscopic behavior of master equation systems
Schnakenberg, J · 1976
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On a problem of girsanov
Kazamaki, N · 1977
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Self-organization in nonequilibrium systems
Nicolis, G · 1977
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General properties of entropy
Wehrl, A · 1978
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Diffusions, Markov processes, and martingales. Vol. 1, Foundations (Wiley, 1979)
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Stochastic processes: Time evolution, symmetries and linear response
Hänggi, P. & Thomas, H · 1982
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Branching processes , vol. 3 (Springer, 1983)
Asmussen, S. & Hering, H · 1983
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The neutral theory of molecular evolution (Cambridge University Press, 1983)
Kimura, M · 1983
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Random walks and electric networks , vol. 22 (American Mathematical Soc., 1984)
Doyle, P. G. & Snell, J. L · 1984
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Nature of the spin-glass phase
Mézard, M., Parisi, G., Sourlas, N., Toulouse, G. & Virasoro, M · 1984
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Handbook of stochastic methods , vol. 3 (springer Berlin, 1985)
Gardiner, C. W. et al · 1985
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Stationary states and their stability of the stepping stone model involving mutation and selection
Shiga, T. & Uchiyama, K · 1986
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Reaction-diffusion equations for interacting particle systems
De Masi, A., Ferrari, P. A. & Lebowitz, J. L · 1986
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Non equilibrium fluctuations for a zero range process
Ferrari, P. A., Presutti, E. & Vares, M. E · 1988
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The fokker-planck equation. methods of solution and applications
Risken, H · 1989
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The dynamic origin of increasing entropy
Mackey, M. C · 1989
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Stochastic processes (John Wiley & Sons, 1990), wiley classics library edn
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Introduction to stochastic integration , vol. 2 (Springer, 1990)
Chung, K. L. & Williams, R. J · 1990
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Ito, stratonovich and kinetic forms of stochastic equations
Klimontovich, Y. L · 1990
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Sekimoto, K · 1990
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Equivalent martingale measures and no-arbitrage in stochastic securities market models
Dalang, R. C., Morton, A. & Willinger, W · 1990
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Probability with martingales (Cambridge university press, 1991)
Williams, D · 1991
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Information theoretical aspects of quantum mechanical entropy
Wehrl, A · 1991
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Stochastic processes in physics and chemistry , vol. 1 (Elsevier, 1992)
Van Kampen, N. G · 1992
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Probability and random processes (Oxford university press, 1992), 2nd edn
Grimmett, G., & Stirzaker, D · 1992
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Probability and stochastic processes with applications
Knill, O · 1994
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Dynamical ensembles in nonequilibrium statistical mechanics
Gallavotti, G. & Cohen, E. G. D · 1995
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Fokker-planck equation
Risken, H · 1996
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Against the gods: The remarkable story of risk (Wiley New York, 1996)
Bernstein, P. L. & Bernstein, P. L · 1996
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Probability (1996), 2nd edn
Shiryaev, A. N · 1996
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Criticism of feynman’s analysis of the ratchet as an engine
Parrondo, J. M. & Español, P · 1996
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Universality classes for extreme-value statistics
Bouchaud, J.-P. & Mézard, M · 1997
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Kinetic characterization of heat bath and the energetics of thermal ratchet models
Sekimoto, K · 1997
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Nonequilibrium equality for free energy differences
Jarzynski, C · 1997
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A general framework for non-equilibrium phenomena: The master equation and its formal consequences
Gaveau, B. & Schulman, L · 1997
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Markov chains
Norris, J. R · 1998
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Sufficient conditions for the uniform integrability of exponential martingales
Kramkov, D. & Shiryaev, A · 1998
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Langevin equation and thermodynamics
Sekimoto, K · 1998
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Fluctuation theorem for stochastic dynamics
Kurchan, J · 1998
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Steady state thermodynamics
Oono, Y. & Paniconi, M · 1998
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Thermodynamics and an introduction to thermostatistics (1998)
Callen, H. B · 1998
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Scaling limits of interacting particle systems , vol. 320 (Springer Science & Business Media, 1998)
Kipnis, C. & Landim, C · 1998
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The importance of strictly local martingales; applications to radial ornstein–uhlenbeck processes
Elworthy, K. D., Li, X.-M. & Yor, M · 1999
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Elements of information theory (John Wiley & Sons, 1999)
Cover, T. M · 1999
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The fluctuation theorem as a gibbs property
Maes, C · 1999
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A gallavotti–cohen-type symmetry in the large deviation functional for stochastic dynamics
Lebowitz, J. L. & Spohn, H · 1999
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Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences
Crooks, G. E · 1999
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Essentials of stochastic finance: facts, models, theory , vol. 3 (World scientific, 1999)
Shiryaev, A. N · 1999
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Lévy processes and infinitely divisible distributions , vol. 68 (Cambridge university press, 1999)
Ken-Iti, S · 1999
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Symmetric path integrals for stochastic equations with multiplicative noise
Arnold, P · 2000
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Path-ensemble averages in systems driven far from equilibrium
Crooks, G. E · 2000
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An introduction to econophysics (Cambridge University Press, Cambridge, 2000)
Stanley, H. E. & Mantegna, R. N · 2000
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Theory of financial risks , vol. 4 (Cambridge University Press, Cambridge From Statistical Physics to Risk …, 2000)
Bouchaud, J.-P. & Potters, M · 2000
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Realistic statistical modelling of financial data
Rydberg, T. H · 2000
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Nonequilibrium statistical mechanics (Oxford university press, 2001)
Zwanzig, R · 2001
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A guide to first-passage processes (Cambridge university press, 2001)
Redner, S · 2001
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Markov chains: Gibbs fields, Monte Carlo simulation, and queues , vol. 31 (Springer Science & Business Media, 2001)
Brémaud, P · 2001
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Drift without flux: Brownian walker with a space-dependent diffusion coefficient
Lançon, P., Batrouni, G., Lobry, L. & Ostrowsky, N · 2001
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Steady-state thermodynamics of langevin systems
Hatano, T. & Sasa, S.-i · 2001
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Free energy reconstruction from nonequilibrium single-molecule pulling experiments
Hummer, G. & Szabo, A · 2001
Growth, competition and cooperation in spatial population genetics
Pigolotti, S. et al · 2013
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Paul Wilmott on quantitative finance (John Wiley & Sons, 2013)
Wilmott, P · 2013
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Decisions on the fly in cellular sensory systems
Siggia, E. D. & Vergassola, M · 2013
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Sequential analysis: Hypothesis testing and changepoint detection (CRC press, 2014)
Tartakovsky, A., Nikiforov, I. & Basseville, M · 2014
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Stochastic processes and applications: diffusion processes, the Fokker-Planck and Langevin equations , vol. 60 (Springer, 2014)
Pavliotis, G. A · 2014
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Stochastic differential equations and diffusion processes (Elsevier, 2014)
Ikeda, N. & Watanabe, S · 2014
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Probability and random processes (Oxford university press, 2001)
Grimmett, G. R. & Stirzaker, D. R · 2001
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Martingale models for quantum state reduction
Adler, S. L., Brody, D., Brun, T. & Hughston, L · 2001
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Neural computations that underlie decisions about sensory stimuli
Gold, J. I. & Shadlen, M. N · 2001
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Maxwell’s demon and the second law
Leff, H. S. & Rex, A. F · 2002
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Stochastic differential equations (Springer, 2003)
Øksendal, B · 2003
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Interacting particles, the stochastic fisher–kolmogorov–petrovsky–piscounov equation, and duality
Doering, C. R., Mueller, C. & Smereka, P · 2003
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Non-stationary extreme value analysis in a changing climate
Cheng, L., AghaKouchak, A., Gilleland, E. & Katz, R. W · 2014
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Martingale limit theory and its application (Academic press, 2014)
Hall, P. & Heyde, C. C · 2014
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Modern thermodynamics: from heat engines to dissipative structures (John Wiley & Sons, 2014)
Kondepudi, D. & Prigogine, I · 2014
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Affinity and fluctuations in a mesoscopic noria
Bauer, M. & Cornu, F · 2014
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Universal features in the energetics of symmetry breaking
Roldán, É., Martinez, I. A., Parrondo, J. M. & Petrov, D · 2014
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Nonequilibrium equalities in absolutely irreversible processes
Murashita, Y., Funo, K. & Ueda, M · 2014
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Decision making in the arrow of time
Roldán, É., Neri, I., Dörpinghaus, M., Meyr, H. & Jülicher, F · 2015
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Thermodynamic uncertainty relation for biomolecular processes
Barato, A. C. & Seifert, U · 2015
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The exclusion process: A paradigm for non-equilibrium behaviour
Mallick, K · 2015
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Effective langevin equations for constrained stochastic processes
Majumdar, S. N. & Orland, H · 2015
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Nonequilibrium markov processes conditioned on large deviations
Chetrite, R. & Touchette, H · 2015
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A formal view on level 2.5 large deviations and fluctuation relations
Barato, A. C. & Chetrite, R · 2015
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Ensemble and trajectory thermodynamics: A brief introduction
Van den Broeck, C. & Esposito, M · 2015
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Inflow rate, a time-symmetric observable obeying fluctuation relations
Baiesi, M. & Falasco, G · 2015
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Thermodynamics of error correction
Sartori, P. & Pigolotti, S · 2015
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Thermodynamics of information
Parrondo, J. M., Horowitz, J. M. & Sagawa, T · 2015
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A neural implementation of wald’s sequential probability ratio test
Kira, S., Yang, T. & Shadlen, M. N · 2015
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Thermodynamics of information
Parrondo, J. M., Horowitz, J. M. & Sagawa, T · 2015
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Universal bounds on current fluctuations
Pietzonka, P., Barato, A. C. & Seifert, U · 2016
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Dissipation bounds all steady-state current fluctuations
Gingrich, T. R., Horowitz, J. M., Perunov, N. & England, J. L · 2016
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Percentile indices for assessing changes in heavy precipitation events
Schär, C. et al · 2016
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Elephant random walks and their connection to pólya-type urns
Baur, E. & Bertoin, J · 2016
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Temporal correlations of the running maximum of a brownian trajectory
Bénichou, O., Krapivsky, P., Mejía-Monasterio, C. & Oshanin, G · 2016
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Arbitrarily slow, non-quasistatic, isothermal transformations
Gavrilov, M. & Bechhoefer, J · 2016
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Statistics of infima and stopping times of entropy production and applications to active molecular processes
Neri, I., Roldán, É. & Jülicher, F · 2017
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Generic properties of stochastic entropy production
Pigolotti, S., Neri, I., Roldán, É. & Jülicher, F · 2017
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A martingale approach for the elephant random walk
Bercu, B · 2017
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Rules of calculus in the path integral representation of white noise langevin equations: the onsager–machlup approach
Cugliandolo, L. F. & Lecomte, V · 2017
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Probability and measure (John Wiley & Sons, 2017)
Billingsley, P · 2017
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Central limit theorem and related results for the elephant random walk
Coletti, C. F., Gava, R. & Schütz, G. M · 2017
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Fundamental bounds on first passage time fluctuations for currents
Gingrich, T. R. & Horowitz, J. M · 2017
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Experimental realization of a minimal microscopic heat engine
Argun, A. et al · 2017
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Experimental measurement of binding energy, selectivity, and allostery using fluctuation theorems
Camunas-Soler, J., Alemany, A. & Ritort, F · 2017
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Anisotropic superattenuation of capillary waves on driven glass interfaces
Bresson, B. et al · 2017
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Statistics of infima and stopping times of entropy production and applications to active molecular processes
Neri, I., Roldán, E. & Jülicher, F · 2017
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The marginally stable bethe lattice spin glass revisited
Parisi, G · 2017
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Run and tumble particle under resetting: a renewal approach
Evans, M. R. & Majumdar, S. N · 2018
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Steady state, relaxation and first-passage properties of a run-and-tumble particle in one-dimension
Malakar, K. et al · 2018
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Pérégrinations sur les phénomènes aléatoires dans la nature
Chétrite, R · 2018
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Life efficiency does not always increase with the dissipation rate
Baiesi, M. & Maes, C · 2018
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Quantum fluctuation theorems for arbitrary environments: adiabatic and nonadiabatic entropy production
Manzano, G., Horowitz, J. M. & Parrondo, J. M · 2018
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First-passage duality
Krapivsky, P. & Redner, S · 2018
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Universal trade-off between power, efficiency, and constancy in steady-state heat engines
Pietzonka, P. & Seifert, U · 2018
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Asymmetry relations and effective temperatures for biased brownian gyrators
Cerasoli, S., Dotsenko, V., Oshanin, G. & Rondoni, L · 2018
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Progressive quenching - ising chain models
Etienne, M. & Sekimoto, K · 2018
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Progressive quenching: Globally coupled model
Ventéjou, B. & Sekimoto, K · 2018
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Testing optimality of sequential decision-making
Dörpinghaus, M., Neri, I., Roldán, É., Meyr, H. & Jülicher, F · 2018
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The full replica symmetry breaking in the ising spin glass on random regular graph
Concetti, F · 2018
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Integral fluctuation relations for entropy production at stopping times
Neri, I., Roldán, É., Pigolotti, S. & Jülicher, F · 2019
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Martingale theory for housekeeping heat
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Extreme reductions of entropy in an electronic double dot
Singh, S. et al · 2019
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Universal property of the housekeeping entropy production
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Quantum martingale theory and entropy production
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Gumbel central limit theorem for max-min and min-max
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Trade-offs between error, speed, noise, and energy dissipation in biological processes with proofreading
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Efficiency fluctuations in microscopic machines
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Second law of thermodynamics at stopping times
Neri, I · 2020
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Infima statistics of entropy production in an underdamped brownian motor
Cheng, K., Dong, J.-Q., Han, W.-H., Liu, F. & Huang, L · 2020
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Extreme-value statistics of stochastic transport processes
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Extreme value statistics of correlated random variables: a pedagogical review
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Memory through a hidden martingale process in progressive quenching
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A mechanism for hunchback promoters to readout morphogenetic positional information in less than a minute
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