The Annals of Probability

Natural linear additive functionals of superprocesses

E. B. Dynkin and S. E. Kuznetsov

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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 $\alpha \leq 2$.

Article information

Ann. Probab., Volume 25, Number 2 (1997), 640-661.

First available in Project Euclid: 18 June 2002

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Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65]
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J25: Continuous-time Markov processes on general state spaces 60J55: Local time and additive functionals 31C45: Other generalizations (nonlinear potential theory, etc.)

Measure-valued processes branching natural linear additive functionals potentials log-potentials


Dynkin, E. B.; Kuznetsov, S. E. Natural linear additive functionals of superprocesses. Ann. Probab. 25 (1997), no. 2, 640--661. doi:10.1214/aop/1024404414.

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