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ISSN 1088-6850(e) ISSN 0002-9947(p)

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

Author(s): E. B. Dynkin; 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: 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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