no. 8
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

Dynkin, Eugene B.1
1 Department of Mathematics, Cornell University, Ithaca, NY 14853, USA
Comptes Rendus. Mathématique, Tome 338 (2004) no. 8, pp. 605-610.

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=uα dans un domaine borné régulier E, pour 1<α⩽2.

Let X=(XD,Pμ) be a superdiffusion in a domain E⊂ℝ d . We introduce a germ σ-algebra ℱ E- at the boundary of E and we prove that, on this σ-algebra, Pμ1 is absolutely continuous with respect to Pμ2 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=uα in a bounded domain E of class C4 for the case 1<α⩽2.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2004.01.028
Dynkin, Eugene B. 1

1 Department of Mathematics, Cornell University, Ithaca, NY 14853, USA 
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Dynkin, Eugene B. Absolute continuity results for superdiffusions with applications to differential equations. Comptes Rendus. Mathématique, Tome 338 (2004) no. 8, pp. 605-610. doi : 10.1016/j.crma.2004.01.028. http://archive.numdam.org/articles/10.1016/j.crma.2004.01.028/

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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=uα, 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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