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

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 (*).