Stochastic thermodynamics

Short description: Field of statistical mechanics

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## Contents

* 1 Overview
* 1.1 Fluctuation theorem
* 1.2 Jarzynski equality
* 1.3 Stochastic energetics
* 2 History
* 3 Open research
* 3.1 Quantum stochastic thermodynamics
* 3.2 Active baths
* 4 References
* 4.1 Notes
* 4.2 Citations
* 4.3 Academic references
* 4.4 Press

## Overview

When a microscopic machine (e.g. a MEM) performs useful work it generates heat and entropy as a byproduct of the process, however it is also predicted that this machine will operate in "reverse" or "backwards" over appreciable short periods. That is, heat energy from the surroundings will be converted into useful work. For larger engines, this would be described as a violation of the second law of thermodynamics, as entropy is consumed rather than generated. Loschmidt's paradox[2] states that in a time reversible system, for every trajectory there exists a time-reversed anti-trajectory. As the entropy production of a trajectory and its equal anti-trajectory are of identical magnitude but opposite sign, then, so the argument goes, one cannot prove that entropy production is positive.[3]

For a long time, exact results in thermodynamics were only possible in linear systems capable of reaching equilibrium, leaving other questions like the Loschmidt paradox unsolved. During the last few decades fresh approaches have revealed general laws applicable to non-equilibrium system which are described by nonlinear equations, pushing the range of exact thermodynamic statements beyond the realm of traditional linear solutions. These exact results are particularly relevant for small systems where appreciable (typically non-Gaussian) fluctuations occur. Thanks to stochastic thermodynamics it is now possible to accurately predict distribution functions of thermodynamic quantities relating to exchanged heat, applied work or entropy production for these systems.[4]

### Fluctuation theorem

Main article: Physics:Fluctuation theorem

The mathematical resolution to Loschmidt's paradox is called the (steady state) fluctuation theorem (FT), which is a generalisation of the second law of thermodynamics. The FT shows that as a system gets larger or the trajectory duration becomes longer, entropy-consuming trajectories become more unlikely, and the expected second law behaviour is recovered.

The FT was first put forward by Evans et al. (1993)({{{1}}}, {{{2}}}) and much of the work done in developing and extending the theorem was accomplished by theoreticians and mathematicians interested in nonequilibrium statistical mechanics.[lower-alpha 2][3]

The first observation and experimental proof of Evan's fluctuation theorem (FT) was performed by Wang et al. (2002)({{{1}}}, {{{2}}})

### Jarzynski equality

Main article: Jarzynski equality

Seifert writes:[4]

Jarzynski (1997a, 1997b) proved a remarkable relation which allows to express the free energy difference between two equilibrium systems by a nonlinear average over the work required to drive the system in a non-equilibrium process from one state to the other. By comparing probability distributions for the work spent in the original process with the time-reversed one, Crooks found a "refinement" of the Jarzynski relation (JR),[5] now called the Crooks fluctuation theorem. Both, this relation and another refinement of the JR, the Hummer-Szabo relation became particularly useful for determining free energy differences and landscapes of biomolecules. These relations are the most prominent ones within a class of exact results (some of which found even earlier and then rediscovered) valid for non-equilibrium systems driven by time-dependent forces. A close analogy to the JR, which relates different equilibrium states, is the Hatano-Sasa relation that applies to transitions between two different non-equilibrium steady states.

This is shown to be a special case of a more general relation.

### Stochastic energetics

## History

Seifert writes:[4]

Classical thermodynamics, at its heart, deals with general laws governing the transformations of a system, in particular, those involving the exchange of heat, work and matter with an environment. As a central result, total entropy production is identified that in any such process can never decrease, leading, inter alia, to fundamental limits on the efficiency of heat engines and refrigerators.[4]

The thermodynamic characterisation of systems in equilibrium got its microscopic justification from equilibrium statistical mechanics which states that for a system in contact with a heat bath the probability to find it in any specific microstate is given by the Boltzmann factor. For small deviations from equilibrium, linear response theory allows to express transport properties caused by small external fields through equilibrium correlation functions. On a more phenomenological level, linear irreversible thermodynamics provides a relation between such transport coefficients and entropy production in terms of forces and fluxes. Beyond this linear response regime, for a long time, no universal exact results were available.[4]

During the last 20 years fresh approaches have revealed general laws applicable to non-equilibrium system thus pushing the range of validity of exact thermodynamic statements beyond the realm of linear response deep into the genuine non-equilibrium region. These exact results, which become particularly relevant for small systems with appreciable (typically non-Gaussian) fluctuations, generically refer to distribution functions of thermodynamic quantities like exchanged heat, applied work or entropy production.[4]

Stochastic thermodynamics combines the stochastic energetics introduced by Sekimoto (1998)[6] with the idea that entropy can consistently be assigned to a single fluctuating trajectory.[6]

## Open research

### Quantum stochastic thermodynamics

Main article: Physics:Quantum thermodynamics

Stochastic thermodynamics can be applied to driven (i.e. open) quantum systems whenever the effects of quantum coherence can be ignored. The dynamics of an open quantum system is then equivalent to a classical stochastic one. However, this is sometimes at the cost of requiring unrealistic measurements at the beginning and end of a process.[lower-alpha 3][7]

Understanding non-equilibrium quantum thermodynamics more broadly is an important and active area of research. The efficiency of some computing and information theory tasks can be greatly enhanced when using quantum correlated states; quantum correlations can be used not only as a valuable resource in quantum computation, but also in the realm of quantum thermodynamics.[8] New types of quantum devices in non-equilibrium states function very differently to their classical counterparts. For example, it has been theoretically shown that non-equilibrium quantum ratchet systems function far more efficiently then that predicted by classical thermodynamics.[lower-alpha 4][9] It has also been shown that quantum coherence can be used to enhance the efficiency of systems beyond the classical Carnot limit. This is because it could be possible to extract work, in the form of photons, from a single heat bath. Quantum coherence can be used in effect to play the role of Maxwell's demon[10] though the broader information theory based interpretation of the second law of thermodynamics is not violated.[lower-alpha 5][13]

Quantum versions of stochastic thermodynamics have been studied for some time[lower-alpha 6] and the past few years have seen a surge of interest in this topic.[lower-alpha 3] Quantum mechanics involves profound issues around the interpretation of reality (e.g. the Copenhagen interpretation, many-worlds, de Broglie-Bohm theory etc are all competing interpretations that try to explain the unintuitive results of quantum theory) . It is hoped that by trying to specify the quantum-mechanical definition of work, dealing with open quantum systems, analyzing exactly solvable models, or proposing and performing experiments to test non-equilibrium predictions,[lower-alpha 7] important insights into the interpretation of quantum mechanics and the true nature of reality will be gained.[15]

Applications of non-equilibrium work relations, like the Jarzynski equality, have recently been proposed for the purposes of detecting quantum entanglement (Hide Vedral) and to improving optimization problems (minimize or maximize a function of multivariables called the cost function) via quantum annealing (Ohzeki Nishimori).[15]

### Active baths

Until recently thermodynamics has only considered systems coupled to a thermal bath and, therefore, satisfying Boltzmann statistics. However, some systems do not satisfy these conditions and are far from equilibrium such as living matter, for which fluctuations are expected to be non-Gaussian.[16]

Active particle systems are able to take energy from their environment and drive themselves far from equilibrium. An important example of active matter is constituted by objects capable of self propulsion. Thanks to this property, they feature a series of novel behaviours that are not attainable by matter at thermal equilibrium, including, for example, swarming and the emergence of other collective properties.[17] A passive particle is considered in an active bath when it is in an environment where a wealth of active particles are present. These particles will exert nonthermal forces on the passive object so that it will experience non-thermal fluctuations and will behave widely different from a passive Brownian particle in a thermal bath. The presence of an active bath can significantly influence the microscopic thermodynamics of a particle. Experiments have suggested that the Jarzynski equality does not hold in some cases due to the presence of non-Boltzmann statistics in active baths.[lower-alpha 8] This observation points towards a new direction in the study of non-equilibrium statistical physics and stochastic thermodynamics, where also the environment itself is far from equilibrium.[19]

Active baths are a question of particular importance in biochemistry. For example, biomolecules within cells are coupled with an active bath due to the presence of molecular motors within the cytoplasm, which leads to striking and largely not yet understood phenomena such as the emergence of anomalous diffusion (Barkai et al., 2012). Also, protein folding might be facilitated by the presence of active fluctuations (Harder et al., 2014b) and active matter dynamics could play a central role in several biological functions (Mallory et al., 2015; Shin et al., 2015; Suzuki et al., 2015). It is an open question to what degree stochastic thermodynamics can be applied to systems coupled to active baths.[16]

## References

### Notes

1. ↑ See Seifert (2008),({{{1}}}, {{{2}}}) Seifert (2012)({{{1}}}, {{{2}}}) and Jarzynski (2011)({{{1}}}, {{{2}}}) for academic review articles of stochastic thermodynamics.
2. ↑ Evan's original numerical analysis was shown heuristically and related to a thermostatted shear-driven fluid in contact with a heat bath. It was later mathematically proven for a large class of systems using concepts from chaotic dynamics by Gallavotti & Cohen (1995), for driven Langevin dynamics by Kurchan (1998) and for driven diffusive dynamics by Lebowitz & Spohn (1999). A variant, a transient fluctuation theorem valid for relaxation towards the steady state was later found by Evans & Searles (1994).[4]
3. ↑ 3.0 3.1 See Esposito et al. (2009) and Campisi et al. (2011) for academic review articles on non-equilibrium quantum fluctuations[7]
4. ↑ See for example Yukawa et al. (1997), Reimann et al. (1997), Tatara et al. (1998)[9]
5. ↑ See for example Scully (2001),({{{1}}}, {{{2}}}) Scully et al. (2003),[10] Dillenschneider & Lutz (2009),[11] Roßnagel et al. (2014),({{{1}}}, {{{2}}}) and Roßnagel et al. (2016)[12]
6. ↑ See for example Yukawa (2000)({{{1}}}, {{{2}}}) and Mukamel (2003)({{{1}}}, {{{2}}})
7. ↑ See for example Huber et al. (2008)({{{1}}}, {{{2}}}) and An et al. (2014)[14]
8. ↑ See for example Argun et al. (2016)[18]

### Citations

1. ↑ Seifert 2012, p. 7.
2. ↑ Loschmidt 1876.
3. ↑ 3.0 3.1 Wang et al. 2002, p. 050601-1.
4. ↑ 4.0 4.1 4.2 4.3 4.4 4.5 4.6 Seifert 2012, p. 6.
5. ↑ Crooks 1999.
6. ↑ 6.0 6.1 Seifert 2008, p. 1.
7. ↑ 7.0 7.1 Seifert 2012, p. 9.
8. ↑ Dillenschneider & Lutz 2009, p. 6.
9. ↑ 9.0 9.1 Yukawa 2000, p. 1.
10. ↑ 10.0 10.1 Maruyama et al. 2009, p. 20.
11. ↑ Modi et al. 2012, p. 43.
12. ↑ Cartlidge 2015.
13. ↑ Dillenschneider & Lutz 2009, pp. 5–6.
14. ↑ Roßnagel et al. 2016, p. 1.
15. ↑ 15.0 15.1 Jarzynski 2011, p. 348.
16. ↑ 16.0 16.1 Bechinger et al. 2016, p. 45.
17. ↑ Bechinger et al. 2016, p. 2.
18. ↑ Bechinger et al. 2016, p. 12, 26, 45.
19. ↑ Bechinger et al. 2016, p. 26.

### Academic references

* An, Shuoming; Zhang, Jing-Ning; Um, Mark; Lv, Dingshun; Lu, Yao; Zhang, Junhua; Yin, Zhang-Qi; Quan, H. T. et al. (2014). "Experimental test of the quantum Jarzynski equality with a trapped-ion system". Nature Physics 11 (2): 193–199. doi:10.1038/nphys3197. ISSN 1745-2473. Bibcode: 2015NatPh..11..193A.
* Argun, Aykut; Moradi, Ali-Reza; Pince, Erçağ; Bagci, Gokhan Baris; Volpe, Giovanni (2016). "Experimental evidence of the failure of Jarzynski equality in active bath". arXiv:1601.01123 [cond-mat.soft].
* Bechinger, Clemens; Di Leonardo, Roberto; Löwen, Hartmut; Reichhardt, Charles; Volpe, Giorgio; Volpe, Giovanni (2016). "Active Particles in Complex and Crowded Environments". Reviews of Modern Physics 88 (4). doi:10.1103/RevModPhys.88.045006. ISSN 0034-6861. Bibcode: 2016RvMP...88d5006B.
* Bertini, Lorenzo; De Sole, Alberto; Gabrielli, Davide; Jona-Lasinio, Giovanni; Landim, Claudio (2015). "Macroscopic fluctuation theory". Reviews of Modern Physics 87 (2): 593–636. doi:10.1103/RevModPhys.87.593. ISSN 0034-6861. Bibcode: 2015RvMP...87..593B.
* Campisi, Michele; Hänggi, Peter; Talkner, Peter (2011). "Colloquium: Quantum fluctuation relations: Foundations and applications". Reviews of Modern Physics 83 (3): 771–791. doi:10.1103/RevModPhys.83.771. ISSN 0034-6861. Bibcode: 2011RvMP...83..771C.
* Crooks, Gavin E. (1999). "Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences". Physical Review E 60 (3): 2721–2726. doi:10.1103/PhysRevE.60.2721. PMID 11970075. Bibcode: 1999PhRvE..60.2721C. https://link.aps.org/doi/10.1103/PhysRevE.60.2721.
* Dillenschneider, R.; Lutz, E. (2009). "Energetics of quantum correlations". EPL (Europhysics Letters) 88 (5). doi:10.1209/0295-5075/88/50003. ISSN 0295-5075. Bibcode: 2009EL.....8850003D.
* Esposito, Massimiliano; Harbola, Upendra; Mukamel, Shaul (2009). "Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems". Reviews of Modern Physics 81 (4): 1665–1702. doi:10.1103/RevModPhys.81.1665. ISSN 0034-6861. Bibcode: 2009RvMP...81.1665E. http://orbilu.uni.lu/handle/10993/12322.
* Evans, Denis J.; Cohen, E. G. D.; Morriss, G. P. (1993). "Probability of second law violations in shearing steady states". Physical Review Letters 71 (15): 2401–2404. doi:10.1103/PhysRevLett.71.2401. ISSN 0031-9007. PMID 10054671. Bibcode: 1993PhRvL..71.2401E. https://unsworks.unsw.edu.au/bitstreams/afc89775-141a-4615-b510-d41787576a5c/download.
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* Horodecki, Ryszard; Horodecki, Paweł; Horodecki, Michał; Horodecki, Karol (17 June 2009). "Quantum entanglement". Reviews of Modern Physics 81 (2): 865–942. doi:10.1103/revmodphys.81.865. Bibcode: 2009RvMP...81..865H.
* Huber, Gerhard; Schmidt-Kaler, Ferdinand; Deffner, Sebastian; Lutz, Eric (2008). "Employing Trapped Cold Ions to Verify the Quantum Jarzynski Equality". Physical Review Letters 101 (7). doi:10.1103/PhysRevLett.101.070403. ISSN 0031-9007. PMID 18764513. Bibcode: 2008PhRvL.101g0403H.
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* Maruyama, Koji; Nori, Franco; Vedral, Vlatko (2009). "Colloquium: The physics of Maxwell's demon and information". Reviews of Modern Physics 81 (1): 1–23. doi:10.1103/RevModPhys.81.1. ISSN 0034-6861. Bibcode: 2009RvMP...81....1M.
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* Pal, P. S.; Deffner, Sebastian (2020). "Stochastic thermodynamics of relativistic Brownian motion". New Journal of Physics 22 (7): 073054. doi:10.1088/1367-2630/ab9ce6. Bibcode: 2020NJPh...22g3054P.
* Peliti, Luca; Pigolotti, Simone (2021). Stochastic Thermodynamics: An Introduction. Princeton: Princeton University Press. ISBN 978-0-521-85103-9.
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* Roßnagel, J.; Abah, O.; Schmidt-Kaler, F.; Singer, K.; Lutz, E. (2014). "Nanoscale Heat Engine Beyond the Carnot Limit". Physical Review Letters 112 (3). doi:10.1103/PhysRevLett.112.030602. ISSN 0031-9007. PMID 24484127. Bibcode: 2014PhRvL.112c0602R.
* Roßnagell, J.; Dawkins, S. T.; Tolazzi, K. N.; Abah, O.; Lutz, E.; Schmidt-Kaler, F.; Singer, K. (2016). "A single-atom heat engine". Science 352 (6283): 325–329. doi:10.1126/science.aad6320. ISSN 0036-8075. PMID 27081067. Bibcode: 2016Sci...352..325R.
* Scully, Marlan O. (2001). "Extracting Work from a Single Thermal Bath via Quantum Negentropy". Physical Review Letters 87 (22). doi:10.1103/PhysRevLett.87.220601. ISSN 0031-9007. PMID 11736390. Bibcode: 2001PhRvL..87v0601S.
* Scully, Marlan O.; Zubairy, M. Suhail; Agarwal, Girish S.; Walther, Herbert. (2003). "Extracting Work from a Single Heat Bath via Vanishing Quantum Coherence". Science 299 (5608): 862–864. doi:10.1126/science.1078955. ISSN 0036-8075. PMID 12511655. Bibcode: 2003Sci...299..862S.
* Seifert, U. (2008). "Stochastic thermodynamics: principles and perspectives". The European Physical Journal B 64 (3–4): 423–431. doi:10.1140/epjb/e2008-00001-9. ISSN 1434-6028. Bibcode: 2008EPJB...64..423S.
* Ohzeki, Masayuki; Nishimori, Hidetoshi (2011). "Quantum annealing with Jarzynski equality". Computer Physics Communications 182 (1): 257–259. doi:10.1016/j.cpc.2010.07.008. ISSN 0010-4655. PMID 20867896. Bibcode: 2011CoPhC.182..257O.
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* Tatara, Gen; Kikuchi, Makoto; Yukawa, Satoshi; Matsukawa, Hiroshi (1998). "Dissipation Enhanced Asymmetric Transport in Quantum Ratchets". Journal of the Physical Society of Japan 67 (4): 1090–1093. doi:10.1143/JPSJ.67.1090. ISSN 0031-9015. Bibcode: 1998JPSJ...67.1090T.
* Wang, G. M.; Sevick, E. M.; Mittag, Emil; Searles, Debra J.; Evans, Denis J. (2002). "Experimental Demonstration of Violations of the Second Law of Thermodynamics for Small Systems and Short Time Scales". Physical Review Letters 89 (5). doi:10.1103/PhysRevLett.89.050601. ISSN 0031-9007. PMID 12144431. Bibcode: 2002PhRvL..89e0601W. https://espace.library.uq.edu.au/data/UQ_282878/UQ282878_OA.pdf?Expires=1485941550&Signature=GMF-QnYunEhzAIL7LoBi5JXUu6w6yy5nTAmd1p6~pi~3hWHN5ZpsVyoF~mRY92XtSTK6zyOYVxRnICr1QrD7RXx1Fyga19kMXa0qGS6bIVka~I0alobclJBOtI81XUcrUZcKYIHRXc5NJMA7RHNw2xwUVwu4WhGOqxUwN3K-83rzry83FWyG3Oxzaszh2fY1VOHH-vFUNM-CiJM63PIPG-rHCeZIhYcLgxU7O2TUqY3D1kuhpDxOQJL21lyQL74LmgsCIJ9ZOb2zwNpZ9SlXFxJFST6aQiq26OCeU~Ddou6nqXGYBRCyzSjUfMrNtRSoIBi9Qh4~N0jDQd6fLV-KPQ__&Key-Pair-Id=APKAJKNBJ4MJBJNC6NLQ.
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### Press

* Cartlidge, Edwin (21 October 2015). "Scientists build heat engine from a single atom". https://www.science.org/content/article/scientists-build-heat-engine-single-atom.
* Chalmers, Matthew (19 July 2002). "Second law of thermodynamics "broken"". https://www.newscientist.com/article/dn2572-second-law-of-thermodynamics-broken/.
* Gerstner, Ed (23 July 2002). "Second law broken: Small-scale energy fluctuations could limit minaturization". Nature. doi:10.1038/news020722-2. http://www.nature.com/news/1998/020722/full/news020722-2.html.
* "Prototype of single ion heat engine created". 3 February 2014. https://www.sciencedaily.com/releases/2014/02/140203101031.htm.
* Whitehouse, David (18 July 2002). "Beads of doubt". https://news.bbc.co.uk/2/hi/science/nature/2135779.stm.
* Zyga, Lisa (27 January 2014). "Nanoscale heat engine exceeds standard efficiency limit". https://phys.org/news/2014-01-nanoscale-standard-efficiency-limit.html.

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* Maxwell relations

Models|

* Ferromagnetism models
* Ising
* Potts
* Heisenberg
* percolation
* Particles with force field
* depletion force
* Lennard-Jones potential

Mathematical approaches|

* Boltzmann equation
* H-theorem
* Vlasov equation
* BBGKY hierarchy
* stochastic process
* mean field theory and conformal field theory

Critical phenomena|

* Phase transition
* Critical exponents
* correlation length
* size scaling

Entropy|

* Boltzmann
* Shannon
* Tsallis
* Rényi
* von Neumann

Applications|

* Statistical field theory
* elementary particle
* superfluidity
* condensed matter physics
* complex system
* chaos
* information theory
* Boltzmann machine

* v
* t
* e

Stochastic processes

Discrete time|

* Bernoulli process
* Branching process
* Chinese restaurant process
* Galton–Watson process
* Independent and identically distributed random variables
* Markov chain
* Moran process
* Random walk
* Loop-erased
* Self-avoiding
* Biased
* Maximal entropy

Continuous time|

* Bessel process
* Birth–death process
* Brownian motion
* Bridge
* Excursion
* Fractional
* Geometric
* Meander
* Cauchy process
* Contact process
* Continuous-time random walk
* Cox process
* Diffusion process
* Empirical process
* Feller process
* Fleming–Viot process
* Gamma process
* Geometric process
* Hunt process
* Interacting particle systems
* Itô diffusion
* Itô process
* Jump diffusion
* Jump process
* Lévy process
* Local time
* Markov additive process
* McKean–Vlasov process
* Ornstein–Uhlenbeck process
* Poisson process
* Compound
* Non-homogeneous
* Point process
* Schramm–Loewner evolution
* Semimartingale
* Sigma-martingale
* Stable process
* Superprocess
* Telegraph process
* Variance gamma process
* Wiener process
* Wiener sausage

Both|

* Branching process
* Galves–Löcherbach model
* Gaussian process
* Hidden Markov model (HMM)
* Markov process
* Martingale
* Differences
* Local
* Sub-
* Super-
* Random dynamical system
* Regenerative process
* Renewal process
* Stochastic chains with memory of variable length
* White noise

Fields and other|

* Dirichlet process
* Gaussian random field
* Gibbs measure
* Hopfield model
* Ising model
* Potts model
* Boolean network
* Markov random field
* Percolation
* Pitman–Yor process
* Point process
* Cox
* Poisson
* Random field
* Random graph

Time series models|

* Autoregressive conditional heteroskedasticity (ARCH) model
* Autoregressive integrated moving average (ARIMA) model
* Autoregressive (AR) model
* Autoregressive–moving-average (ARMA) model
* Generalized autoregressive conditional heteroskedasticity (GARCH) model
* Moving-average (MA) model

Financial models|

* Black–Derman–Toy
* Black–Karasinski
* Black–Scholes
* Chen
* Constant elasticity of variance (CEV)
* Cox–Ingersoll–Ross (CIR)
* Garman–Kohlhagen
* Heath–Jarrow–Morton (HJM)
* Heston
* Ho–Lee
* Hull–White
* LIBOR market
* Rendleman–Bartter
* SABR volatility
* Vašíček
* Wilkie

Actuarial models|

* Bühlmann
* Cramér–Lundberg
* Risk process
* Sparre–Anderson

Queueing models|

* Bulk
* Fluid
* Generalized queueing network
* M/G/1
* M/M/1
* M/M/c

Properties|

* Càdlàg paths
* Continuous
* Continuous paths
* Ergodic
* Exchangeable
* Feller-continuous
* Gauss–Markov
* Markov
* Mixing
* Piecewise deterministic
* Predictable
* Progressively measurable
* Self-similar
* Stationary
* Time-reversible

Limit theorems|

* Central limit theorem
* Donsker's theorem
* Doob's martingale convergence theorems
* Ergodic theorem
* Fisher–Tippett–Gnedenko theorem
* Large deviation principle
* Law of large numbers (weak/strong)
* Law of the iterated logarithm
* Maximal ergodic theorem
* Sanov's theorem

Inequalities|

* Burkholder–Davis–Gundy
* Doob's martingale
* Kunita–Watanabe

Tools|

* Cameron–Martin formula
* Convergence of random variables
* Doléans-Dade exponential
* Doob decomposition theorem
* Doob–Meyer decomposition theorem
* Doob's optional stopping theorem
* Dynkin's formula
* Feynman–Kac formula
* Filtration
* Girsanov theorem
* Infinitesimal generator
* Itô integral
* Itô's lemma
* Karhunen–Loève_theorem
* Kolmogorov continuity theorem
* Kolmogorov extension theorem
* Lévy–Prokhorov metric
* Malliavin calculus
* Martingale representation theorem
* Optional stopping theorem
* Prokhorov's theorem
* Quadratic variation
* Reflection principle
* Skorokhod integral
* Skorokhod's representation theorem
* Skorokhod space
* Snell envelope
* Stochastic differential equation
* Tanaka
* Stopping time
* Stratonovich integral
* Uniform integrability
* Usual hypotheses
* Wiener space
* Classical
* Abstract

Disciplines|

* Actuarial mathematics
* Econometrics
* Ergodic theory
* Extreme value theory (EVT)
* Large deviations theory
* Mathematical finance
* Mathematical statistics
* Probability theory
* Queueing theory
* Renewal theory
* Ruin theory
* Statistics
* System on Chip design
* Stochastic analysis
* Time series analysis
* Machine learning

* List of topics
* Category

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