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April 1997 Natural linear additive functionals of superprocesses
E. B. Dynkin, S. E. Kuznetsov
Ann. Probab. 25(2): 640-661 (April 1997). DOI: 10.1214/aop/1024404414


We investigate natural linear additive (NLA) functionals of a general critical $(\xi, K, \psi)$-superprocess $X$. We prove that all of them have only fixed discontinuities. All homogeneous NLA functionals of time-homogeneous superprocesses are continuous (this was known before only in the case of quadratic branching).

We introduce an operator $\mathscr{E}(u)$ defined in terms of $(\xi, K, \psi)$ and we prove that the potential $h$ and the log-potential $u$ of a NLA functional $A$ are connected by the equation $u + \mathscr{E}(u) = h$. The potential is always an exit rule for $\xi$ and the condition $h + \mathscr{E}(h) < \infty$ a.e. is sufficient for an exit rule $h$ to be a potential.

In an accompanying paper, these results are applied to boundary value problems for partial differential equations involving nonlinear operator $Lu = u^{\alpha}$ where $L$ is a second order elliptic differential operator and $1 < \alpha \leq 2$.


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E. B. Dynkin. S. E. Kuznetsov. "Natural linear additive functionals of superprocesses." Ann. Probab. 25 (2) 640 - 661, April 1997.


Published: April 1997
First available in Project Euclid: 18 June 2002

zbMATH: 0880.60079
MathSciNet: MR1443191
Digital Object Identifier: 10.1214/aop/1024404414

Primary: 60J60
Secondary: 31C45 , 60J25 , 60J55 , 60J80

Keywords: branching , log-potentials , Measure-valued processes , natural linear additive functionals , potentials

Rights: Copyright © 1997 Institute of Mathematical Statistics


Vol.25 • No. 2 • April 1997
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