Absolute continuity results for superdiffusions with applications to differential equations

Eugene B. Dynkin

doi:10.1016/j.crma.2004.01.028

Partial Differential Equations

Absolute continuity results for superdiffusions with applications to differential equations
[Résultats d'absolue continuité pour les superdiffusions et applications aux équations différentielles]

Présenté par : Haïm Brezis

Eugene B. Dynkin ¹

¹ Department of Mathematics, Cornell University, Ithaca, NY 14853, USA

Comptes Rendus. Mathématique, Volume 338 (2004) no. 8, pp. 605-610.

Résumé
Abstract

On établit un résultat qui, combiné à des travaux antérieurs de Dynkin, Kuznetsov et Mselati, conduit à une classification complète des solutions positives de l'équation Δu=u^α dans un domaine borné régulier E, pour 1<α⩽2.

Let X=(X_D,P_μ) be a superdiffusion in a domain $E \subset ℝ^{d}$ . We introduce a germ σ-algebra $ℱ_{E -}$ at the boundary of E and we prove that, on this σ-algebra, P_μ₁ is absolutely continuous with respect to P_μ₂ if μ₁ and μ₂ are concentrated on compact subsets of E. In combination with previous results of Dynkin, Kuznetsov and Mselati, this leads to a complete classification of positive solutions of equation Δu=u^α in a bounded domain E of class C⁴ for the case 1<α⩽2.

Reçu le : 2004-01-13
Accepté le : 2004-01-13
Publié le : 2004-03-25

DOI : 10.1016/j.crma.2004.01.028

Affiliations des auteurs :

Eugene B. Dynkin ¹

¹ Department of Mathematics, Cornell University, Ithaca, NY 14853, USA

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Eugene B. Dynkin. Absolute continuity results for superdiffusions with applications to differential equations. Comptes Rendus. Mathématique, Volume 338 (2004) no. 8, pp. 605-610. doi : 10.1016/j.crma.2004.01.028. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.01.028/

Bibliographie
Cité par

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[5] E.B. Dynkin, On upper bounds for positive solutions of semilinear equations, J. Funct. Anal. (2003), in press

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[8] E.B. Dynkin, S.E. Kuznetsov, $ℕ$ -measures for branching exit Markov systems and their applications to differential equations, Probab. Theory Related Fields, in press

[9] S.E. Kuznetsov, An upper bound for positive solutions of the equation Δu=u^α, in preparation

[10] B. Mselati, Classification et représentation probabiliste des solutions positives de Δu=u² dans un domaine, Thése de Doctorat de l'Université Paris 6, 2002

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