A new inequality for superdiffusions and its applications to nonlinear differential equations

Author:
E. B. Dynkin

Journal:
Electron. Res. Announc. Amer. Math. Soc. **10** (2004), 68-77

MSC (2000):
Primary 60H30; Secondary 35J60, 60J60

Published electronically:
August 2, 2004

Comment(s):
Additional information about this paper

MathSciNet review:
2075898

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Abstract: Our motivation is the following problem: to describe all positive solutions of a semilinear elliptic equation with in a bounded smooth domain . In 1998 Dynkin and Kuznetsov solved this problem for a class of solutions which they called -moderate. The question if all solutions belong to this class remained open. In 2002 Mselati proved that this is true for the equation in a domain of class . His principal tool--the Brownian snake--is not applicable to the case . In 2003 Dynkin and Kuznetsov modified most of Mselati's arguments by using superdiffusions instead of the snake. However a critical gap remained. A new inequality established in the present paper allows us to close this gap.

**[Dy91]**E. B. Dynkin,*A probabilistic approach to one class of nonlinear differential equations*, Probab. Theory Related Fields**89**(1991), no. 1, 89–115. MR**1109476**, 10.1007/BF01225827**[Dy02]**E. B. Dynkin,*Diffusions, superdiffusions and partial differential equations*, American Mathematical Society Colloquium Publications, vol. 50, American Mathematical Society, Providence, RI, 2002. MR**1883198****[Dy04a]**E. B. Dynkin,*On upper bounds for positive solutions of semilinear equations*, J. Funct. Anal.**210**(2004), no. 1, 73–100. MR**2051633**, 10.1016/S0022-1236(03)00147-2**[Dy04b]**-,*Superdiffusions and positive solutions of nonlinear partial differential equations*, Uspekhi Matem. Nauk**59**(2004), to appear.**[Dy04c]**Eugene B. Dynkin,*Absolute continuity results for superdiffusions with applications to differential equations*, C. R. Math. Acad. Sci. Paris**338**(2004), no. 8, 605–610 (English, with English and French summaries). MR**2056468**, 10.1016/j.crma.2004.01.028**[Dy04d]**-,*Superdiffusions and positive solutions of nonlinear partial differential equations*, American Mathematical Society, Providence, RI, 2004, to appear.**[DK03]**E. B. Dynkin and S. E. Kuznetsov,*Poisson capacities*, Math. Res. Lett.**10**(2003), no. 1, 85–95. MR**1960126**, 10.4310/MRL.2003.v10.n1.a9**[DK04]**-, -*measures for branching exit Markov systems and their applications to differential equations*, Probab. Theory and Related Fields, to appear.**[Ku04]**S. E. Kuznetsov,*An upper bound for positive solutions of the equation*, Amer. Math. Soc., Electronic Research Announcements, to appear.**[MV04]**M. Marcus and L. Véron,*Capacitary estimates of positive solutions of semilinear elliptic equations with absorbtion*, J. European Math. Soc., to appear.**[Ms02]**B. Mselati,*Classification et représentation probabiliste des solutions positives de dans un domaine*, Thése de Doctorat de l'Université Paris 6, 2002.**[Ms04]**B. Mselati,*Classification and probabilistic representation of the positive solutions of a semilinear elliptic equation*, Memoirs of the American Mathematical Society**168**(2004), no. 798, to appear.

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Additional Information

**E. B. Dynkin**

Affiliation:
Department of Mathematics, Cornell University, Ithaca, NY 14853

Email:
ebd1@cornell.edu

DOI:
http://dx.doi.org/10.1090/S1079-6762-04-00131-3

Keywords:
Positive solutions of semilinear elliptic PDEs,
superdiffusions,
conditional diffusions,
$\mathbb{N}$-measures

Received by editor(s):
April 23, 2004

Published electronically:
August 2, 2004

Additional Notes:
Partially supported by the National Science Foundation Grant DMS-0204237

Communicated by:
Mark Freidlin

Article copyright:
© Copyright 2004
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.