Rough path

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In probability theory, a rough path is an analytical object allowing one 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 of 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 the theory of regularity structures.[9] For this work he was awarded a Fields medal in 2014 .


  1. ^ Lyons, T. (1998). "Differential equations driven by rough signals". Revista Matemática Iberoamericana: 215–310. doi:10.4171/RMI/240. 
  2. ^ Lyons, Terry; Qian, Zhongmin (2002). "System Control and Rough Paths". Oxford Mathematical Monographs. Oxford: Clarendon Press. doi:10.1093/acprof:oso/9780198506485.001.0001. ISBN 9780198506485. Zbl 1029.93001. 
  3. ^ Lyons, Terry; Caruana, Michael; Levy, Thierry (2007). Differential equations driven by rough paths, vol. 1908 of Lecture Notes in Mathematics. Springer. 
  4. ^ Lejay, A. (2003). "An Introduction to Rough Paths". Séminaire de Probabilités XXXVII. Lecture Notes in Mathematics 1832. pp. 1–1. doi:10.1007/978-3-540-40004-2_1. ISBN 978-3-540-20520-3. 
  5. ^ Gubinelli, Massimiliano (2004). "Controlling rough paths". Journal of Functional Analysis 216 (1): 86–140. doi:10.1016/j.jfa.2004.01.002. 
  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. ^ Hairer, Martin (2013). "Solving the KPZ equation". Annals of Mathematics 178 (2): 559–664. doi:10.4007/annals.2013.178.2.4. 
  9. ^ Hairer, Martin (2014). "A theory of regularity structures". Inventiones mathematicae. doi:10.1007/s00222-014-0505-4.