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In probability theory, a rough path is an analytical object allowing to define solutions to differential equations controlled by irregular paths such as a Wiener process. The theory was developed in the 1990s by Terry Lyons. ^[1] ^[2] ^[3] Several accounts on the theory are available.^[4] ^[5] ^[6] ^[7] Martin Hairer used rough paths to solve the KPZ equation ^[8]. He then proposed a significant generalization known as theory of regularity structures ^[9] for which he was awarded in 2014 the Fields medal.

References

^ Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.4171/RMI/240, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.4171/RMI/240 instead.
^ Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1093/acprof:oso/9780198506485.001.0001, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1093/acprof:oso/9780198506485.001.0001 instead.
^ Lyons, Terry; Caruana, Michael; Levy, Thierry (2007). Differential equations driven by rough paths, vol. 1908 of Lecture Notes in Mathematics. Springer.
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^ Friz, Peter K.; Victoir, Nicolas (2010). Multidimensional Stochastic Processes as Rough Paths: Theory and Applications (Cambridge Studies in Advanced Mathematics ed.). Cambridge University Press.
^ Friz, Peter K.; Hairer, Martin (2014). A Course on Rough Paths, with an introduction to regularity structures. Springer.
^ Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.4007/annals.2013.178.2.4, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.4007/annals.2013.178.2.4 instead.
^ Hairer, Martin (2014). "A theory of regularity structures". Inventiones mathematicae. doi:10.1007/s00222-014-0505-4.

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[1] Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.4171/RMI/240, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.4171/RMI/240 instead.

[2] Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1093/acprof:oso/9780198506485.001.0001, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1093/acprof:oso/9780198506485.001.0001 instead.

[3] Lyons, Terry; Caruana, Michael; Levy, Thierry (2007). Differential equations driven by rough paths, vol. 1908 of Lecture Notes in Mathematics. Springer.

[lejay-4] Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1007/978-3-540-40004-2_1, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1007/978-3-540-40004-2_1 instead.

[gubinelli-5] Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1016/j.jfa.2004.01.002, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1016/j.jfa.2004.01.002 instead.

[6] Friz, Peter K.; Victoir, Nicolas (2010). Multidimensional Stochastic Processes as Rough Paths: Theory and Applications (Cambridge Studies in Advanced Mathematics ed.). Cambridge University Press.

[7] Friz, Peter K.; Hairer, Martin (2014). A Course on Rough Paths, with an introduction to regularity structures. Springer.

[8] Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.4007/annals.2013.178.2.4, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.4007/annals.2013.178.2.4 instead.

[9] Hairer, Martin (2014). "A theory of regularity structures". Inventiones mathematicae. doi:10.1007/s00222-014-0505-4.

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-In [[probability theory]], a '''rough path''' is an analytical object allowing to define solutions to [[differential equations]] controlled by irregular paths such as a [[Wiener process]]. The theory was developed in the 1990s by [[Terry Lyons (mathematician)|Terry Lyons]].<ref>{{cite journal|last1=Lyons|first1=Terry|title=Differential equations driven by rough paths'|journal=Rev. Mat. Iberoamericana|date=1998|issue=14|pages=215-310}}</ref>
+In [[probability theory]], a '''rough path''' is an analytical object allowing to define solutions to [[differential equations]] controlled by irregular paths such as a [[Wiener process]]. The theory was developed in the 1990s by [[Terry Lyons (mathematician)|Terry Lyons]]. <ref>{{cite doi|10.4171/RMI/240|noedit}}</ref>
+<ref>{{cite doi|10.1093/acprof:oso/9780198506485.001.0001|noedit}}</ref>
-<ref>{{cite doi|10.1093/acprof:oso/9780198506485.001.0001|noedit}}</ref>. <ref>{{cite book|last1=Lyons|first1=Terry|last2=Caruana|first2=Michael|last3=Levy|first3=Thierry|title=Differential equations driven by rough paths, vol. 1908 of Lecture Notes in Mathematics|date=2007|publisher=Springer}}</ref>Several accounts on the theory are available.<ref name="lejay">{{cite doi|10.1007/978-3-540-40004-2_1|noedit}}</ref><ref>{{cite book|last1=Friz|first1=Peter K.|last2=Victoir|first2=Nicolas|title=Multidimensional Stochastic Processes as Rough Paths: Theory and Applications|date=2010|publisher=Cambridge University Press|edition=Cambridge Studies in Advanced Mathematics}}</ref><ref>{{cite book|last1=Friz|first1=Peter K.|last2=Hairer|first2=Martin|title=A Course on Rough Paths, with an introduction to regularity structures|date=2014|publisher=Springer}}</ref>.
+<ref>{{cite book|last1=Lyons|first1=Terry|last2=Caruana|first2=Michael|last3=Levy|first3=Thierry|title=Differential equations driven by rough paths, vol. 1908 of Lecture Notes in Mathematics|date=2007|publisher=Springer}}</ref>
+Several accounts on the theory are available.<ref name="lejay">{{cite doi|10.1007/978-3-540-40004-2_1|noedit}}</ref>
+<ref name="gubinelli">{{cite doi|10.1016/j.jfa.2004.01.002|noedit}}</ref>
+<ref>{{cite book|last1=Friz|first1=Peter K.|last2=Victoir|first2=Nicolas|title=Multidimensional Stochastic Processes as Rough Paths: Theory and Applications|date=2010|publisher=Cambridge University Press|edition=Cambridge Studies in Advanced Mathematics}}</ref> <ref>{{cite book|last1=Friz|first1=Peter K.|last2=Hairer|first2=Martin|title=A Course on Rough Paths, with an introduction to regularity structures|date=2014|publisher=Springer}}</ref>
+[[Martin Hairer|Martin Hairer]] used rough paths to solve the [[Kardar–Parisi–Zhang equation|KPZ equation]] <ref>{{cite doi|10.4007/annals.2013.178.2.4|noedit}}</ref>. He then proposed a significant generalization known as theory of [[regularity structures|regularity structures]] <ref>{{cite journal|last1=Hairer|first1=Martin|title=A theory of regularity structures|journal=Inventiones mathematicae|date=2014|doi=10.1007/s00222-014-0505-4}}</ref> for which he was awarded in 2014 the [[Fields Medal|Fields medal]].
 ==References==