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Transactions of the American Mathematical Society

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



Linear additive functionals of
superdiffusions and related nonlinear P.D.E.

Authors: E. B. Dynkin and S. E. Kuznetsov
Journal: Trans. Amer. Math. Soc. 348 (1996), 1959-1987
MSC (1991): Primary 60J60, 35J65; Secondary 60J80, 31C15, 60J25, 60J55, 31C45, 35J60
MathSciNet review: 1348859
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Abstract | References | Similar Articles | Additional Information

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

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

Received by editor(s): March 29, 1995
Additional Notes: Partially supported by National Science Foundation Grant DMS-9301315
Article copyright: © Copyright 1996 American Mathematical Society