On an explicit representation of the solution of linear stochastic partial differential equations with delays

Mathieu Galtier; Jonathan Touboul

doi:10.1016/j.crma.2012.01.004

Ordinary Differential Equations/Partial Differential Equations

On an explicit representation of the solution of linear stochastic partial differential equations with delays
[Une expression analytique pour la solution dʼéquations aux dérivées partielles linéaires stochastiques avec délais]

Présenté par : Jean-Michel Bony

Mathieu Galtier ¹ ; Jonathan Touboul ²

¹ NeuroMathComp Project Team, INRIA, 23, avenue dʼItalie, 75013 Paris, France
² The Mathematical Neuroscience Laboratory, CIRB/Collège de France (CNRS UMR 7241, INSERM U1050, UPMC ED 158, MEMOLIFE PSL

Comptes Rendus. Mathématique, Volume 350 (2012) no. 3-4, pp. 167-172.

Résumé
Abstract

En se basant sur lʼanalyse dʼune certaine classe dʼopérateurs linéraires dans des espaces de Banach, nous établissons une expression analytique pour la solution de certaines équations aux dérivées partielles linéaires avec des entrées non-autonomes, des délais et des termes stochastique, sous la forme dʼun développement en série.

Based on the analysis of a certain class of linear operators on a Banach space, we provide a closed form expression for the solutions of certain linear partial differential equations with non-autonomous input, time delays and stochastic terms, which takes the form of an infinite series expansion.

Reçu le : 2011-03-28
Accepté le : 2012-01-05
Publié le : 2012-01-25

DOI : 10.1016/j.crma.2012.01.004

Affiliations des auteurs :

Mathieu Galtier ¹ ; Jonathan Touboul ²

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     title = {On an explicit representation of the solution of linear stochastic partial differential equations with delays},
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     language = {en},
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Mathieu Galtier; Jonathan Touboul. On an explicit representation of the solution of linear stochastic partial differential equations with delays. Comptes Rendus. Mathématique, Volume 350 (2012) no. 3-4, pp. 167-172. doi : 10.1016/j.crma.2012.01.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.01.004/

Bibliographie
Cité par

[1] J. Brewer Kronecker products and matrix calculus in system theory, IEEE Trans. Circuits Syst., Volume CAS-25 (1978) no. 9

[2] M. Galtier, G. Wainrib, Multiscale analysis of slow–fast neuronal learning models with noise, submitted for publication.

[3] J.K. Hale; S.M.V. Lunel Introduction to Functional Differential Equations, Springer, 1993

[4] X. Mao Stochastic Differential Equations and Applications, Horwood Publishing, 2008

[5] G. Da Prato; J. Zabczyk Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, 1992

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