Probability Seminar: Yan-Xia Ren

I will talk about our recent progress on strong law of large numbers for some classes of superprocesses $X$ corresponding to $\partial_t u_t=A u_t+\beta u_t-\psi(u_t)$ in a domain $D$ of ${\bf R}^d$, where $A$ is the generator of a diffusion or a stable process, and the branching mechanism $\psi(x,\lambda)=\beta\lambda+a\lambda^2+\int_0^\infty (e^{-\lambda r}-1+\lambda r)n(x, {\rm d}r)$ satisfies $\sup_{x\in D}\int_0^\infty (r\wedge r^2) n(x,{\rm d}r)<\infty$.

Recently many people have established limit theorems for branching Markov processes or super-processes using the principal eigenvalue and ground state of the linear part of the characteristic equations. All the papers above assumed that the processes satisfy a second moment condition or a $(1+\theta)$-moment condition with $\theta>0$. Asmussen and Hering (1976) established a Kesten-Stigum $L\log L$ type theorem for a class branching diffusion processes under a condition which is later called a positive regular property. We established Kesten-Stigum $L\log L$ type theorems for superdiffusions and branching Hunt processes respectively.

Recently, we established strong law of large numbers for a class of superdiffusions in a domain $D$ of ${\bf R}^d$ with general branching mechanism, and for super-$\alpha$-stable processes in ${\bf R}^d$ with $\psi(x, \lambda)=-\beta\lambda+\eta\lambda^2$, where $\beta$ and $\eta$ are positive constants. The main tool is the stochastic integral representation of superprocesses.