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Stochastic modelling in fluid dynamics: Itô versus Stratonovich

Suppose the observations of Lagrangian trajectories for fluid flow in some physical situation can be modelled sufficiently accurately by a spatially correlated Itô stochastic process (with zero mean) obtained from data which is taken in fixed Eulerian space. Suppose we also want to apply Hamilton’s...

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Autor principal: Holm, Darryl D.
Formato: Online Artículo Texto
Lenguaje:English
Publicado: The Royal Society Publishing 2020
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7277131/
https://www.ncbi.nlm.nih.gov/pubmed/32518504
http://dx.doi.org/10.1098/rspa.2019.0812
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author Holm, Darryl D.
author_facet Holm, Darryl D.
author_sort Holm, Darryl D.
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description Suppose the observations of Lagrangian trajectories for fluid flow in some physical situation can be modelled sufficiently accurately by a spatially correlated Itô stochastic process (with zero mean) obtained from data which is taken in fixed Eulerian space. Suppose we also want to apply Hamilton’s principle to derive the stochastic fluid equations for this situation. Now, the variational calculus for applying Hamilton’s principle requires the Stratonovich process, so we must transform from Itô noise in the data frame to the equivalent Stratonovich noise. However, the transformation from the Itô process in the data frame to the corresponding Stratonovich process shifts the drift velocity of the transformed Lagrangian fluid trajectory out of the data frame into a non-inertial frame obtained from the Itô correction. The issue is, ‘Will non-inertial forces arising from this transformation of reference frames make a difference in the interpretation of the solution behaviour of the resulting stochastic equations?’ This issue will be resolved by elementary considerations.
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spelling pubmed-72771312020-06-08 Stochastic modelling in fluid dynamics: Itô versus Stratonovich Holm, Darryl D. Proc Math Phys Eng Sci Research Article Suppose the observations of Lagrangian trajectories for fluid flow in some physical situation can be modelled sufficiently accurately by a spatially correlated Itô stochastic process (with zero mean) obtained from data which is taken in fixed Eulerian space. Suppose we also want to apply Hamilton’s principle to derive the stochastic fluid equations for this situation. Now, the variational calculus for applying Hamilton’s principle requires the Stratonovich process, so we must transform from Itô noise in the data frame to the equivalent Stratonovich noise. However, the transformation from the Itô process in the data frame to the corresponding Stratonovich process shifts the drift velocity of the transformed Lagrangian fluid trajectory out of the data frame into a non-inertial frame obtained from the Itô correction. The issue is, ‘Will non-inertial forces arising from this transformation of reference frames make a difference in the interpretation of the solution behaviour of the resulting stochastic equations?’ This issue will be resolved by elementary considerations. The Royal Society Publishing 2020-05 2020-05-27 /pmc/articles/PMC7277131/ /pubmed/32518504 http://dx.doi.org/10.1098/rspa.2019.0812 Text en © 2020 The Authors. http://creativecommons.org/licenses/by/4.0/ http://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.
spellingShingle Research Article
Holm, Darryl D.
Stochastic modelling in fluid dynamics: Itô versus Stratonovich
title Stochastic modelling in fluid dynamics: Itô versus Stratonovich
title_full Stochastic modelling in fluid dynamics: Itô versus Stratonovich
title_fullStr Stochastic modelling in fluid dynamics: Itô versus Stratonovich
title_full_unstemmed Stochastic modelling in fluid dynamics: Itô versus Stratonovich
title_short Stochastic modelling in fluid dynamics: Itô versus Stratonovich
title_sort stochastic modelling in fluid dynamics: itô versus stratonovich
topic Research Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7277131/
https://www.ncbi.nlm.nih.gov/pubmed/32518504
http://dx.doi.org/10.1098/rspa.2019.0812
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