Superprocesses and projective limits of branching Markov process
The author presents an a.s. limit theorem to construct a class of measure-valued processes defined by Watanabe and Dawson [cf. \textit{S. Watanabe}, J. Math. Kyoto Univ. 8, 141-167 (1968; Zbl 0159.462)]. Consider a family of branching Markov processes indexed by \(\tau \in {\mathbb{R}}^+\). The paths start at x with a number of points following a Poisson law of mean \(\psi \tau^{-1/\beta}\), and points move independently according to semigroup \(P_ t\) and branch at exponential times of mean \(\tau\) according to a reproduction generating function \(\sum q_ ku^ k=(1-u)^{1+\beta}/(1+\beta)+u\) (\(\beta\in]0,1])\). Denoting by \(Z^{\tau}_ t\) the induced multiple-valued process and taking a projective limit, the author proves that, using a martingale approach, the measure \(\psi^{-1}\epsilon^{1/\beta}\tilde Z^{\epsilon}_{t-\epsilon}P_{\epsilon}\) converges a.s. to a measure \(\mu_ t\) as \(\epsilon\downarrow 0\) and that \(\mu_ t\) is a Markov process. If \(U_ tf(x)=-Log(E(\exp (-\mu_ t(f)))\) for any positive bounded measurable function f, then \(U_ t(f)\) is uniquely determined by the equation \[ U_ t(f)=P_ tf-\psi^{-\beta}\int^{t}_{0}P_ s(U_{t-s}(f)^{1+\beta})ds/(1+\beta). \]
scientific article; zbMATH DE number 4194892