Natural linear additive functionals of superprocesses

Abstract

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 $1 < \alpha \leq 2$.