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-measures for branching exit Markov systems and their applications to differential equations
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  • Published: 03 March 2004

-measures for branching exit Markov systems and their applications to differential equations

  • E.B. Dynkin1 &
  • S.E. Kuznetsov2 

Probability Theory and Related Fields volume 130, pages 135–150 (2004)Cite this article

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Abstract.

Semilinear equations Lu=ψ(u) where L is an elliptic differential operator and ψ is a positive function can be investigated by using (L,ψ)-superdiffusions. In a special case Δu=u2 a powerful probabilistic tool – the Brownian snake – introduced by Le Gall was successfully applied by him and his school to get deep results on solutions of this equation. Some of these results (but not all of them) were extended by Dynkin and Kuznetsov to general equations by applying superprocesses. An important role in the theory of the Brownian snake and its applications is played by measures x on the space of continuous paths. Our goal is to introduce analogous measures related to superprocesses (and to general branching exit Markov systems). They are defined on the space of measures and we call them -measures. Using -measures allows to combine some advantages of Brownian snakes and of superprocesses as tools for a study of semilinear PDEs.

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References

  1. Dawson, D.A.: Measure-valued Markov processes. Lecture Notes Math. 1541, Springer, 1993

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Author information

Authors and Affiliations

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

    E.B. Dynkin

  2. Department of Mathematics, University of Colorado, Boulder, CO 80309-0395, USA

    S.E. Kuznetsov

Authors
  1. E.B. Dynkin
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  2. S.E. Kuznetsov
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Corresponding author

Correspondence to E.B. Dynkin.

Additional information

Partially supported by National Science Foundation Grant DMS-0204237 and DMS-9971009

Mathematics Subject Classification (2000): Primary 31C15, Secondary 35J65, 60J60

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Cite this article

Dynkin, E., Kuznetsov, S. -measures for branching exit Markov systems and their applications to differential equations. Probab. Theory Relat. Fields 130, 135–150 (2004). https://doi.org/10.1007/s00440-003-0333-8

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  • Received: 31 March 2003

  • Revised: 29 November 2003

  • Published: 03 March 2004

  • Issue Date: September 2004

  • DOI: https://doi.org/10.1007/s00440-003-0333-8

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Keywords

  • Semilinear Elliptic PDEs
  • Infinitely divisible Random Measures
  • -measures
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