This paper gives probabilistic expressions of theminimal and maximal positive solutions of the partial differential equation -1/2△v(x) + γ(x)v(x)α = 0 in D, where D is a regular domain in Rd(d ≥ 3) such that its complement Dc is compact, γ(x) is a positive bounded integrable function in D, and 1 ＜α≤ 2. As an application, some necessary and sufficient conditions for a compact set to be S-polar are presented.
We simply call a superprocess conditioned on non-extinction a conditioned superprocess. In this study, we investigate some properties of the conditioned superprocesses (subcritical or critical). Firstly, we give an equivalent description of the probability of the event that the total occupation time measure on a compact set is finite and some applications of this equivalent description. Our results are extensions of those of Krone (1995) from particular branching mechanisms to general branching mechanisms. We also prove a claim of Krone for the cases of d = 3, 4. Secondly, we study the local extinction property of the conditioned binary super-Brownian motion {Xt, P μ∞ }. When d = 1, as t goes to infinity, Xt/√t converges to ηλ in weak sense under P μ∞ , where η is a nonnegative random variable and λ is the Lebesgue measure on R. When d 2, the conditioned binary super-Brownian motion is locally extinct under P μ∞ .
假定 X 是一超级 -- 在 R (d) 的α - 马厩过程，(0 < α < 2 ) ，谁的分叉的率功能是 dt，并且分叉机制具有形式Ψ(z )= z (1+ β) 。让 X τ和 Y τ表示出口措施和全部的加权的职业时间在围住的光滑的领域 D 的 X 的措施分别地。X τ a nd Y τ的绝对连续性被讨论。给词调音：超级 -- α - 马厩过程；绝对连续性；出口措施；
