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 $\alpha \leq 2$.
Publié le : 1997-04-14
Classification:
Measure-valued processes,
branching,
natural linear additive functionals,
potentials,
log-potentials,
60J60,
60J80,
60J25,
60J55,
31C45
@article{1024404414,
author = {Dynkin, E. B. and Kuznetsov, S. E.},
title = {Natural linear additive functionals of superprocesses},
journal = {Ann. Probab.},
volume = {25},
number = {4},
year = {1997},
pages = { 640-661},
language = {en},
url = {http://dml.mathdoc.fr/item/1024404414}
}
Dynkin, E. B.; Kuznetsov, S. E. Natural linear additive functionals of superprocesses. Ann. Probab., Tome 25 (1997) no. 4, pp. 640-661. http://gdmltest.u-ga.fr/item/1024404414/