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Transactions of the American Mathematical Society

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ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Linear additive functionals of superdiffusions and related nonlinear P.D.E.
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by E. B. Dynkin and S. E. Kuznetsov PDF
Trans. Amer. Math. Soc. 348 (1996), 1959-1987 Request permission

Abstract:

Let $L$ be a second order elliptic differential operator in a bounded smooth domain $D$ in $\mathbb {R}^{d}$ and let $1<\alpha \le 2$. We get necessary and sufficient conditions on measures $\eta , \nu$ under which there exists a positive solution of the boundary value problem \begin{equation*}\begin {gathered} -Lv+v^{\alpha }=\eta \quad \text { in } D,\ v=\nu \quad \text { on } \partial D. \end{gathered}\tag {*} \end{equation*} The conditions are stated both analytically (in terms of capacities related to the Green’s and Poisson kernels) and probabilistically (in terms of branching measure-valued processes called $(L,\alpha )$-superdiffusions). We also investigate a closely related subject — linear additive functionals of superdiffusions. For a superdiffusion in an arbitrary domain $E$ in $\mathbb {R}^{d}$, we establish a 1-1 correspondence between a class of such functionals and a class of $L$-excessive functions $h$ (which we describe in terms of their Martin integral representation). The Laplace transform of $A$ satisfies an integral equation which can be considered as a substitute for (*).
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Additional Information
  • E. B. Dynkin
  • Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853-7901
  • Email: ebd1@cornell.edu
  • S. E. Kuznetsov
  • Affiliation: Central Economics and Mathematical Institute, Russian Academy of Sciences, 117418, Moscow, Russia
  • Address at time of publication: Department of Mathematics, Cornell University, Ithaca, New York 14853-7901
  • Email: sk47@cornell.edu
  • Received by editor(s): March 29, 1995
  • Additional Notes: Partially supported by National Science Foundation Grant DMS-9301315
  • © Copyright 1996 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 348 (1996), 1959-1987
  • MSC (1991): Primary 60J60, 35J65; Secondary 60J80, 31C15, 60J25, 60J55, 31C45, 35J60
  • DOI: https://doi.org/10.1090/S0002-9947-96-01602-9
  • MathSciNet review: 1348859