Generating integrable one dimensional driftless diffusions

Peter Carr; Peter Laurence; Tai-Ho Wang

doi:10.1016/j.crma.2006.05.025

Lie Algebras/Partial Differential Equations

Generating integrable one dimensional driftless diffusions
[Une méthode pour génerer des diffusions intégrables dans le cas unidimensionelle]

Présenté par : Marc Yor

Peter Carr ¹ ; Peter Laurence ² ; Tai-Ho Wang ³

¹ Bloomberg LP, 731, Lexington Avenue, New York, NY 10022-1331, USA
² Dipartimento di Matematica, Piazzale Aldo Moro 2, 00185 Rome, Italy
³ Department of Mathematics, National Chung Cheng University, Chia-Yi 621, Taiwan

Comptes Rendus. Mathématique, Volume 343 (2006) no. 6, pp. 393-398.

Résumé
Abstract

Nous présentons une condition nécessaire et suffisante sur le coefficient de diffusion $g (x, t)$ d'une diffusion sans drift, afin que celle-ci puisse se réduire, par des transformations ponctuelles des variables dépendentes et indépendantes, à la forme canonique de Lie $u_{t} - \frac{1}{2} u_{x x} + \frac{A}{x^{2}} u = 0$ où $A \in R$ . Lie a démontré que celle-ci est la forme canonique d'une diffusion dont le groupe de symétrie est de dimension quatre ou six. Notre résultat complète donc celui de Lie, en donnant une condition locale intrinsèque sur g rendant possible une telle réduction, ainsi qu'une condition constructive, dans la mesure où elle nous permet de construire de façon explicite la solution fondamentale de l'équation correspondante.

A criterion on the diffusion coefficient is formulated that allows the classification of driftless time and state dependent diffusions that are integrable in closed form via point transformations. In the time dependent and state dependent case, a remarkable intertwining with the inhomogeneous Burger's equation is exploited. The criterion is constructive. It allows us to devise families of driftless diffusions parametrized by a rich class of several arbitrary functions for which the solution of any initial value problem can be expressed in closed form. We also derive an elegant form for the master equation for infinitesimal symmetries, previously considered only in the time homogeneous case.

Reçu le : 2005-09-26
Accepté le : 2006-04-06
Publié le : 2006-09-15

DOI : 10.1016/j.crma.2006.05.025

Affiliations des auteurs :

Peter Carr ¹ ; Peter Laurence ² ; Tai-Ho Wang ³

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     author = {Peter Carr and Peter Laurence and Tai-Ho Wang},
     title = {Generating integrable one dimensional driftless diffusions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {393--398},
     publisher = {Elsevier},
     volume = {343},
     number = {6},
     year = {2006},
     doi = {10.1016/j.crma.2006.05.025},
     language = {en},
}

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Peter Carr; Peter Laurence; Tai-Ho Wang. Generating integrable one dimensional driftless diffusions. Comptes Rendus. Mathématique, Volume 343 (2006) no. 6, pp. 393-398. doi : 10.1016/j.crma.2006.05.025. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.05.025/

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