On Tanaka formulae for \((\alpha, d,\beta)\)-superprocesses
type: Item
instance of: scholarly article
title: On Tanaka formulae for \((\alpha, d,\beta)\)-superprocesses
zbMATH Open document ID: 1123.60066
published in: Science in China. Series A
publication date: 2007-05-29T00:00:00Z
review text: This work treats \((\alpha, d, \beta)\)-superprocesses, and the aim of this paper is to establish Tanaka formulae for \(( \alpha, d, \beta)\)-superprocesses in dimensions where the local time exists, under the optimal initial condition. More precisely, let \({\mathcal M}(\mathbb R^d)\) be the set of Radon measures on \(\mathbb R^d\), and \({\mathcal M}_r(\mathbb R^d)\) the whole set of all elements \(\mu \in {\mathcal M}(\mathbb R^d)\) such that \(\langle \mu, \phi_r \rangle < \infty\) with \(\varphi_r(x)=( 1 + | x |^2)^{- r}\), \(x\in \mathbb R^d\) for \(r > d/2\). \({\mathcal B}_b\) denotes the family of all bounded Borel measurable functions on \(\mathbb R^d\). A càdlàg \({\mathcal M}_r(\mathbb R^d)\)-valued process \(X =( X_t)_{t \geq 0}\) is called an \(( \alpha, d, \beta)\)-superprocess if it is a time homogeneous and strongly Markov process such that for all \(\varphi, \psi\in {\mathcal B}_b^+\), \[ {\mathbb E} \left\{ \exp \left. \left( - \langle X_t, \varphi \rangle - \int_0^t \langle X_s, \psi \rangle \,ds \right) \,\right|\, X_0 = \mu \right\} = \exp ( - \langle \mu, V_t ( \varphi, \psi ) \rangle ), \tag{1} \] where \(\mu \in {\mathcal M}_r(\mathbb R^d)\), \(\langle \mu, f \rangle\) denotes integral of \(f\) over \(\mathbb R^d\) with respect to measure \(\mu\), and \(V_t(\varphi, \psi)\) is the unique non-negative solution of \[ v_t = S_t^{\alpha} \varphi + \int_0^t S_s^{\alpha} \psi\, ds - \int_0^t S_{t-s}^{\alpha}( (v_s)^{1 + \beta})\, ds, \quad t \geq 0; \tag{2} \] see, e.g., [\textit{I. Iscoe}, Probab. Theory Relat. Fields 71, 85--116 (1986; Zbl 0555.60034)]. If the weighted occupation time \(Y_t(dx):=\int_0^t X_s\, ds\), \(\forall t \geq 0\), is absolutely continuous with respect to Lebesgue measure on \(\mathbb R^d\), then the local time of \(X\) at \(x \in \mathbb R^d\) is defined as \[ L_t^x := \begin{cases}\lim\limits_{n \to \infty} \langle Y_t, \varphi_{n,x} \rangle, &\text{if it exists}, \\ 0, & \text{otherwise}, \end{cases}\tag{3} \] where \(\…
reviewed by: Isamu Dôku
Mathematics Subject Classification ID: 60J55
Mathematics Subject Classification ID: 60G57
zbMATH DE Number: 5157273
zbMATH Keywords: Local time
zbMATH Keywords: Tanaka formula
zbMATH Keywords: superprocess
zbMATH Keywords: weighted occupation time
author: Kai-Nan Xiang
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5157273