Strong law of large number of a class of super-diffusions

Abstract

In this paper we prove that, under certain conditions, a strong law of large numbers holds for a class of super-diffusions $X$ corresponding to the evolution equation ${\partial }_t{u}_t=L{u}_t+\beta u_t-\psi (u_t)$ on a bounded domain $D$ in ${\mathbb{R}}^d$, where $L$ is the generator of the underlying diffusion and the branching mechanism $\psi (x,\lambda )=1/2\alpha (x){\lambda }^2+{\int }_0^{\infty }(e^{-\lambda r}-1+\lambda r)n(x,\mathrm{d}r)$ satisfies ${\sup }_{x\in D}{\int }_0^{\infty }(r\wedge r^2)n(x,\mathrm{d}r)<\infty$.

References in corpus (2)

• Intrinsic ultracontractivity of nonsymmetric diffusions with measure-valued drifts and potentials
• Stochastic integral representation and regularity of the density for the Exit measure of super-Brownian motion