{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T14:57:54Z","timestamp":1777561074311,"version":"3.51.4"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2020,2,11]],"date-time":"2020-02-11T00:00:00Z","timestamp":1581379200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,2,11]],"date-time":"2020-02-11T00:00:00Z","timestamp":1581379200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100000266","name":"EPSRC","doi-asserted-by":"crossref","award":["EP\/L016613\/1"],"award-info":[{"award-number":["EP\/L016613\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100000266","name":"EPSRC","doi-asserted-by":"crossref","award":["EP\/N023781\/1"],"award-info":[{"award-number":["EP\/N023781\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Schrodinger Scholarship","award":["Schrodinger Scholarship"],"award-info":[{"award-number":["Schrodinger Scholarship"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2020,8]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>\nWe extend the It\u00f4\u2013Wentzell formula for the evolution of a time-dependent stochastic field along a semimartingale to <jats:italic>k<\/jats:italic>-form-valued stochastic processes. The result is the Kunita\u2013It\u00f4\u2013Wentzell (KIW) formula for <jats:italic>k<\/jats:italic>-forms. We also establish a correspondence between the KIW formula for <jats:italic>k<\/jats:italic>-forms derived here and a certain class of stochastic fluid dynamics models which preserve the geometric structure of deterministic ideal fluid dynamics. This geometric structure includes Eulerian and Lagrangian variational principles, Lie\u2013Poisson Hamiltonian formulations and natural analogues of the Kelvin circulation theorem, all derived in the stochastic setting.\n<\/jats:p>","DOI":"10.1007\/s00332-020-09613-0","type":"journal-article","created":{"date-parts":[[2020,2,11]],"date-time":"2020-02-11T17:06:04Z","timestamp":1581440764000},"page":"1421-1454","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["Implications of Kunita\u2013It\u00f4\u2013Wentzell Formula for k-Forms in Stochastic Fluid Dynamics"],"prefix":"10.1007","volume":"30","author":[{"given":"Aythami Bethencourt","family":"de L\u00e9on","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6362-9912","authenticated-orcid":false,"given":"Darryl D.","family":"Holm","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Erwin","family":"Luesink","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"So","family":"Takao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,2,11]]},"reference":[{"issue":"8","key":"9613_CR1","doi-asserted-by":"publisher","first-page":"081507","DOI":"10.1063\/1.4893357","volume":"55","author":"M Arnaudon","year":"2014","unstructured":"Arnaudon, M., Chen, X., Cruzeiro, A.B.: Stochastic Euler\u2013Poincar\u00e9 reduction. J. Math. Phys. 55(8), 081507 (2014)","journal-title":"J. Math. Phys."},{"key":"9613_CR2","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1007\/s00332-017-9404-3","volume":"28","author":"A Arnaudon","year":"2018","unstructured":"Arnaudon, A., de Castro, A.L., Holm, D.D.: Noise and dissipation on coadjoint orbits. J. Nonlinear Sci. 28, 91\u2013145 (2018)","journal-title":"J. Nonlinear Sci."},{"key":"9613_CR3","doi-asserted-by":"crossref","unstructured":"Arnaudon, A., Holm, D.D., Sommer, S.: A geometric framework for stochastic shape analysis. In: Foundations of Computational Mathematics (FoCM) (2018)","DOI":"10.1007\/s10208-018-9394-z"},{"key":"9613_CR4","unstructured":"Arnaudon, A., Holm, D.D., Sommer, S.: String methods for stochastic image and shape matching. J. Math. Imaging Vis. (JMIV) 60, 953\u2013967 (2018)"},{"issue":"1","key":"9613_CR5","first-page":"29","volume":"5","author":"VI Arnold","year":"1966","unstructured":"Arnold, V.I.: Sur un principe variationnel pour les \u00e9coulements stationnaires des liquides parfaits et ses applications aux problemes de stabilit\u00e9 non lin\u00e9aires. J. de m\u00e9canique 5(1), 29 (1966)","journal-title":"J. de m\u00e9canique"},{"key":"9613_CR6","doi-asserted-by":"publisher","first-page":"331","DOI":"10.1007\/BF00532124","volume":"55","author":"J-M Bismut","year":"1981","unstructured":"Bismut, J.-M.: A generalized formula of Ito and some other properties of stochastic flows. Z. Wahrscheinlichkeitstheorie verw. Gebiete 55, 331\u2013350 (1981)","journal-title":"Z. Wahrscheinlichkeitstheorie verw. Gebiete"},{"key":"9613_CR7","doi-asserted-by":"crossref","unstructured":"Bismut, J.-M.: M\u00e9canique al\u00e9atoire. In: Ecole d\u2019Et\u00e9 de Probabilit\u00e9s de Saint-Flour X-1980, pp. 1\u2013100. Springer, Berlin (1982)","DOI":"10.1007\/BFb0095618"},{"issue":"2","key":"9613_CR8","doi-asserted-by":"publisher","first-page":"421","DOI":"10.1093\/imanum\/drn018","volume":"29","author":"N Bou-Rabee","year":"2009","unstructured":"Bou-Rabee, N., Owhadi, H.: Stochastic variational integrators. IMA J. Numer. Anal. 29(2), 421\u2013443 (2009)","journal-title":"IMA J. Numer. Anal."},{"key":"9613_CR9","unstructured":"Cotter, C.J., Crisan, D., Holm, D.D., Pan, W., Shevchenko, I.: Modelling uncertainty using circulation-preserving stochastic transport noise in a 2-layer quasi-geostrophic mode. arXiv:1802.05711 (2018)"},{"key":"9613_CR10","doi-asserted-by":"crossref","unstructured":"Cotter, C.J., Crisan, D., Holm, D.D., Pan, W., Shevchenko, I.: Numerically modelling stochastic Lie transport in fluid dynamics. arXiv:1801.09729 (2018)","DOI":"10.1137\/18M1167929"},{"key":"9613_CR11","unstructured":"Chen, X., Cruzeiro, A.B., Ratiu, T.S.: Constrained and stochastic variational principles for dissipative equations with advected quantities. arXiv preprint arXiv:1506.05024 (2015)"},{"key":"9613_CR12","doi-asserted-by":"crossref","unstructured":"Crisan, D., Flandoli, F., Holm, D.D.: Solution properties of a 3D stochastic Euler fluid equation. J. Nonlinear Sci. 29, 813\u2013870 (2018)","DOI":"10.1007\/s00332-018-9506-6"},{"issue":"2205","key":"9613_CR13","doi-asserted-by":"publisher","first-page":"20170388","DOI":"10.1098\/rspa.2017.0388","volume":"473","author":"CJ Cotter","year":"2017","unstructured":"Cotter, C.J., Gottwald, G.A., Holm, D.D.: Stochastic partial differential fluid equations as a diffusive limit of deterministic Lagrangian multi-time dynamics. Proc. R. Soc. A Math. Phys. Eng. Sci. 473(2205), 20170388 (2017)","journal-title":"Proc. R. Soc. A Math. Phys. Eng. Sci."},{"issue":"2","key":"9613_CR14","doi-asserted-by":"publisher","first-page":"873","DOI":"10.1007\/s00220-017-3048-x","volume":"357","author":"AB Cruzeiro","year":"2018","unstructured":"Cruzeiro, A.B., Holm, D.D., Ratiu, T.S.: Momentum maps and stochastic Clebsch action principles. Commun. Math. Phys. 357(2), 873\u2013912 (2018)","journal-title":"Commun. Math. Phys."},{"issue":"1","key":"9613_CR15","first-page":"3","volume":"10","author":"P Catuogno","year":"2016","unstructured":"Catuogno, P., Stelmastchuk, S.N.: A stochastic transport theorem. Commun. Stoch. Anal. 10(1), 3 (2016)","journal-title":"Commun. Stoch. Anal."},{"key":"9613_CR16","unstructured":"Drivas, T.D., Holm, D.D.: Circulation and energy theorem preserving stochastic fluids. arXiv preprint arXiv:1808.05308 (2018)"},{"key":"9613_CR17","first-page":"407","volume":"24","author":"M \u00c9mery","year":"1990","unstructured":"\u00c9mery, M.: On two transfer principles in stochastic differential geometry. S\u00e9minaire de probabilit\u00e9s (Strasbourg) 24, 407\u2013441 (1990)","journal-title":"S\u00e9minaire de probabilit\u00e9s (Strasbourg)"},{"key":"9613_CR18","volume-title":"On the Geometry of Diffusion Operators and Stochastic Flows","author":"K David Elworthy","year":"2007","unstructured":"David Elworthy, K., Le Jan, Y., Li, X.-M.: On the Geometry of Diffusion Operators and Stochastic Flows. Springer, BErlin (2007)"},{"key":"9613_CR19","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0346-0176-4","volume-title":"The Geometry of Filtering","author":"K David Elworthy","year":"2010","unstructured":"David Elworthy, K., Le Jan, Y., Li, X.-M.: The Geometry of Filtering. Springer, Berlin (2010)"},{"issue":"1","key":"9613_CR20","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1006\/aima.1998.1721","volume":"137","author":"DD Holm","year":"1998","unstructured":"Holm, D.D., Marsden, J.E., Ratiu, T.S.: The Euler\u2013Poincar\u00e9 equations and semidirect products with applications to continuum theories. Adv. Math. 137(1), 1\u201381 (1998)","journal-title":"Adv. Math."},{"issue":"2176","key":"9613_CR21","doi-asserted-by":"publisher","first-page":"20140963","DOI":"10.1098\/rspa.2014.0963","volume":"471","author":"DD Holm","year":"2015","unstructured":"Holm, D.D.: Variational principles for stochastic fluid dynamics. Proc. R. Soc. A 471(2176), 20140963 (2015)","journal-title":"Proc. R. Soc. A"},{"key":"9613_CR22","unstructured":"Kunita, H., Ghosh, M.K.: Lectures on stochastic flows and applications. Tata Institute of Fundamental Research, Bombay (1986)"},{"issue":"1\u20132","key":"9613_CR23","doi-asserted-by":"publisher","first-page":"295","DOI":"10.1007\/s00440-010-0275-x","volume":"150","author":"NV Krylov","year":"2011","unstructured":"Krylov, N.V.: On the It\u00f4-Wentzell formula for distribution-valued processes and related topics. Probab. Theory Relat. Fields 150(1\u20132), 295\u2013319 (2011)","journal-title":"Probab. Theory Relat. Fields"},{"key":"9613_CR24","doi-asserted-by":"crossref","unstructured":"Kunita, H.: Some extensions of Ito\u2019s formula. In: S\u00e9minaire de Probabilit\u00e9s XV 1979\/80, pp. 118\u2013141. Springer, Berlin (1981)","DOI":"10.1007\/BFb0088362"},{"key":"9613_CR25","doi-asserted-by":"crossref","unstructured":"Kunita, H.: Stochastic differential equations and stochastic flows of diffeomorphisms. In: Ecole d\u2019\u00e9t\u00e9 de probabilit\u00e9s de Saint-Flour XII-1982, pp. 143\u2013303. Springer, Berlin (1984)","DOI":"10.1007\/BFb0099433"},{"key":"9613_CR26","volume-title":"Stochastic Flows and Stochastic Differential Equations","author":"H Kunita","year":"1997","unstructured":"Kunita, H.: Stochastic Flows and Stochastic Differential Equations, vol. 24. Cambridge University Press, Cambridge (1997)"},{"key":"9613_CR27","unstructured":"L\u00e1zaro-Cam\u00ed, J.-A., Ortega, J.-P.: Stochastic Hamiltonian dynamical systems. arXiv preprint arXiv:math\/0702787 (2007)"},{"issue":"2","key":"9613_CR28","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1080\/03091929.2013.836190","volume":"108","author":"E M\u00e9min","year":"2014","unstructured":"M\u00e9min, E.: Fluid flow dynamics under location uncertainty. Geophys. Astrophys. Fluid Dyn. 108(2), 119\u2013146 (2014)","journal-title":"Geophys. Astrophys. Fluid Dyn."},{"issue":"5","key":"9613_CR29","doi-asserted-by":"publisher","first-page":"1250","DOI":"10.1137\/S0036141002409167","volume":"35","author":"R Mikulevicius","year":"2004","unstructured":"Mikulevicius, R., Rozovskii, B.L.: Stochastic Navier\u2013Stokes equations for turbulent flows. SIAM J. Math. Anal. 35(5), 1250\u20131310 (2004)","journal-title":"SIAM J. Math. Anal."},{"issue":"1","key":"9613_CR30","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1214\/009117904000000630","volume":"33","author":"R Mikulevicius","year":"2005","unstructured":"Mikulevicius, R., Rozovskii, B.L., et al.: Global $$L^2$$-solutions of stochastic Navier\u2013Stokes equations. Ann. Probab. 33(1), 137\u2013176 (2005)","journal-title":"Ann. Probab."},{"key":"9613_CR31","volume-title":"Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems","author":"JE Marsden","year":"2013","unstructured":"Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, vol. 17. Springer, Berlin (2013)"},{"key":"9613_CR32","doi-asserted-by":"crossref","unstructured":"Rezakhanlou, F.: Stochastically symplectic maps and their applications to the Navier\u2013Stokes equation. In: Annales de l\u2019Institut Henri Poincare (C) Non Linear Analysis, vol. 33, pp. 1\u201322. Elsevier, Amsterdam (2016)","DOI":"10.1016\/j.anihpc.2014.09.001"},{"issue":"3","key":"9613_CR33","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1080\/03091929.2017.1310210","volume":"111","author":"V Resseguier","year":"2017","unstructured":"Resseguier, V., M\u00e9min, E., Chapron, B.: Geophysical flows under location uncertainty, Part I random transport and general models. Geophys. Astrophys. Fluid Dyn. 111(3), 149\u2013176 (2017)","journal-title":"Geophys. Astrophys. Fluid Dyn."},{"issue":"3","key":"9613_CR34","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1080\/03091929.2017.1312101","volume":"111","author":"V Resseguier","year":"2017","unstructured":"Resseguier, V., M\u00e9min, E., Chapron, B.: Geophysical flows under location uncertainty, Part II Quasi-geostrophy and efficient ensemble spreading. Geophys. Astrophys. Fluid Dyn. 111(3), 177\u2013208 (2017)","journal-title":"Geophys. Astrophys. Fluid Dyn."},{"issue":"3","key":"9613_CR35","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1080\/03091929.2017.1312102","volume":"111","author":"V Resseguier","year":"2017","unstructured":"Resseguier, V., M\u00e9min, E., Chapron, B.: Geophysical flows under location uncertainty, Part III SQG and frontal dynamics under strong turbulence conditions. Geophys. Astrophys. Fluid Dyn. 111(3), 209\u2013227 (2017)","journal-title":"Geophys. Astrophys. Fluid Dyn."},{"key":"9613_CR36","doi-asserted-by":"publisher","first-page":"888","DOI":"10.1017\/jfm.2017.467","volume":"826","author":"V Resseguier","year":"2017","unstructured":"Resseguier, V., M\u00e9min, E., Heitz, D., Chapron, B.: Stochastic modelling and diffusion modes for proper orthogonal decomposition models and small-scale flow analysis. J. Fluid Mech. 826, 888\u2013917 (2017)","journal-title":"J. Fluid Mech."}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-020-09613-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00332-020-09613-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-020-09613-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,2,10]],"date-time":"2021-02-10T07:11:11Z","timestamp":1612941071000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00332-020-09613-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,11]]},"references-count":36,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2020,8]]}},"alternative-id":["9613"],"URL":"https:\/\/doi.org\/10.1007\/s00332-020-09613-0","relation":{},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"value":"0938-8974","type":"print"},{"value":"1432-1467","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,2,11]]},"assertion":[{"value":"10 May 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 January 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 February 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}