Convergence of time changed skew product diffusion processes
type: Item
instance of: scholarly article
title: Convergence of time changed skew product diffusion processes
zbMATH Open document ID: 1285.60082
author: Tomoko Takemura
published in: Potential Analysis
publication date: 2013-02-15T00:00:00Z
review text: This paper treats a limit theorem for the time changed skew product diffusion processes. Roughly speaking, skew product diffusion processes \(X_n\) are given by one-dimensional diffusion processes \(R_n\) and the spherical Brownian motion \(\Theta\), and the time change \(\Phi_n(t)\) is based on a positive continuous additive functional, related to the local time \(\ell(t)\) of \(R_n\). In this paper, the author considers the time changed skew product diffusion processes \(X_n(t)\) \(( n \in {\mathbb N})\) which are associated with the following Dirichlet forms \({\mathcal E}_n\) on \(L^2( I \times S^{d-1}, \psi_n)\): \[ {\mathcal E}_n (u,v) = \int_{ S^{d-1}} {\mathcal E}^{ R_n } ( u( \cdot, \theta), v( \cdot, \theta) ) d m^{ \Theta}(\theta) + \int_I {\mathcal E}^{ \Theta} ( u(r, \cdot), v(r, \cdot)) d \nu_n(r). \tag{1} \] Here \({\mathcal E}^{R_n}\) is the Dirichlet form corresponding to the one-dimensional diffusion process \(R_n(t)\) on an interval \(I\) \(=\) \(( \ell_1, \ell_2)\) with the scale function \(s_n\) and the speed measure \(m_n\), and it is actually given by \[ {\mathcal E}^{R_n}(u,v) = \int_I \frac{ du}{d s_n} \frac{dv}{ d s_n} d s_n. \tag{2} \] \({\mathcal E}^{\Theta}\) is the Dirichlet form correponding to the spherical Brownian motion \(\Theta(t)\) on \(S^{d-1}\), and \(m^{\Theta}\) stands for the spherical element of \(S^{d-1}\). \(\psi_n\) denotes a bounded measure on \(I \times S^{d-1}\) and \(\nu_n\) is a Borel measure on \(I\). Assume that \newline (A.1) \(s_n(x) \to s(x)\), \(x \in I\); \newline (A.2) \(m_n(x) \to m(x)\), \(x \in {\mathcal C}(m)\) (= the set of all continuity points of \(m\)); \newline (A.3) \(\nu_n \to \nu\), vaguely; \newline (A.4) \(\psi_n \to \mu \otimes m^{ \Theta}\), vaguely, \newline where \(s\) is an increasing continuous function, \(m\) is an increasing right continuous function, \(\nu\) is a Borel measure satisfying \(\text{supp} [ \nu ] = I\), and \(\mu\) is a Borel measure on \(I\). The author is interested in t…
reviewed by: Isamu Dôku
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60J75
Mathematics Subject Classification ID: 58J65
zbMATH DE Number: 6136705
zbMATH Keywords: skew product
zbMATH Keywords: diffusion process
zbMATH Keywords: Dirichlet form
zbMATH Keywords: convergence theorem
zbMATH Keywords: jumping measure
MaRDI profile type: Publication
full work available at URL: s11118-011-9262-9
OpenAlex ID: W2088563510
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DOI: 10.1007/S11118-011-9262-9
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description: scientific article; zbMATH DE number 6136705