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The Brownian snake and solutions of Δu=u 2 in a domain
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  • Published: September 1995

The Brownian snake and solutions of Δu=u 2 in a domain

  • Jean-François Le Gall1 

Probability Theory and Related Fields volume 102, pages 393–432 (1995)Cite this article

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Summary

We investigate the connections between the path-valued process called the Brownian snake and nonnegative solutions of the partial differential equation Δu=u 2 in a domain of ℝd. In particular, we prove two conjectures recently formulated by Dynkin. The first one gives a complete characterization of the boundary polar sets, which correspond to boundary removable singularities for the equation Δu=u 2. The second one establishes a one-to-one correspondence between nonnegative solutions that are bounded above by a harmonic function, and finite measures on the boundary that do not charge polar sets. This correspondence can be made explicit by a probabilistic formula involving a special class of additive functionals of the Brownian snake. Our proofs combine probabilistic and analytic arguments. An important role is played by a new version of the special Markov property, which is of independent interest.

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Authors and Affiliations

  1. Institut Universitaire de France et Laboratoire de Probabilités, Université Paris 6, 4, Place Jussieu, 75252, Paris Cedex 05, France

    Jean-François Le Gall

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  1. Jean-François Le Gall
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Le Gall, JF. The Brownian snake and solutions of Δu=u 2 in a domain. Probab. Th. Rel. Fields 102, 393–432 (1995). https://doi.org/10.1007/BF01192468

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  • Received: 23 September 1994

  • Revised: 14 March 1995

  • Issue Date: September 1995

  • DOI: https://doi.org/10.1007/BF01192468

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Mathematics Subject Classification (1991)

  • 60J25
  • 60J45
  • 60J80
  • 60J55
  • 35J60
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