Hölder continuity for spatial and path processes via spectral analysis
type: Item
instance of: scholarly article
title: Hölder continuity for spatial and path processes via spectral analysis
zbMATH Open document ID: 0980.60092
author: Douglas Blount
author: Michael A. Kouritzin
publication date: 2002-02-24T00:00:00Z
review text: Let \(\nu( d \theta)\) be a \(\sigma\)-finite Borel measure on \(\mathbb R^d\), and \(L^2( \nu(d\theta))\) the usual \(L^2\)-space of all measurable functions \(f(\theta)\) on \(\mathbb R^d\) with respect to \(d \nu\) with the norm \(\|\cdot \|\). \(\lambda(\theta)\geq \Lambda >0\) denotes a Borel measurable function of \(\theta \in\mathbb R^d\). \(m(t,\theta)\), \(\theta \in \mathbb R^d\), is a continuous martingale with quadratic variation process \[ [ m(\cdot, \theta)](t) = \int_0^t g(s,\theta) ds \quad \text{for fixed } \theta \in \mathbb R^d, \] where \(g(w, s, \theta)\) is measurable in \((w,s,\theta)\) with \(w\in \Omega\) of a probability space \((\Omega, {\mathcal F}, P)\), satisfying that \[ \sup_{0 \leq u \leq t} E[ g(u,\theta)^r ]^{1/r} \leq G(r,t,\theta). \] Here \(G\) is a measurable function in \(\theta\) for each \(t \leq T\) and \(r \geq 1\), such that \[ \int_{\lambda(\theta) \leq a} G(r,T,\theta) \nu(d \theta) < \infty \quad \text{and} \quad \int_{\lambda(\theta)> a} G(r, T, \theta) \lambda^{\delta-1}(\theta) \nu(d\theta) < \infty \] for \(\exists a \geq 0\) and \(\exists \delta \in (0,1]\). Then the \(L^2( \nu(d\theta))\)-valued stochastic process \(Y(t)\) is defined by \(Y(t)= y(t, \cdot)\) with \(E \|Y(t) \|^2< \infty\) for each \(t \geq 0\). The authors prove timewise Hölder continuity and maximal inequalities for \(Y\). Namely, Theorem A. Let \(r \delta > 1\). Then \(Y\) has a modification \(\widetilde{Y}\) satisfying \[ E \Bigl[ \sup_{0 \leq s < t \leq T} ( \|\widetilde{Y}(t) - \widetilde{Y}(s) \|/ |t-s |^{\beta})^{2r} \Bigr] < \infty \] for any \(\beta \in [ 0, (\delta/2) - (1/2r))\). Theorem B. (a) For \(r \delta > 1\), \(0 \leq s \leq t \leq T\), \[ E \Bigl[ \sup_{s \leq u, v \leq t} \|Y(v) - Y(u) \|^{2r} \Bigr] \leq C(T, \delta, r, a, \Lambda) |t-s |^{ r \delta}. \] (b) Furthermore, assume \(\sup_{0 \leq u < \infty} E[ g(u,\theta)^r ]^{1/r}\leq G(r,\theta)\) where \(G\) satisfies some conditions. Then \[ E\Bigl[ \sup_{0 \leq t \leq T} \|…
reviewed by: Isamu Dôku
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60G57
Mathematics Subject Classification ID: 60G17
zbMATH DE Number: 1618290
zbMATH Keywords: Hölder continuity
zbMATH Keywords: stochastic partial differential equations
zbMATH Keywords: superprocess
zbMATH Keywords: Fourier analysis
zbMATH Keywords: maximal inequality
MaRDI profile type: Publication
DOI: 10.1007/S004400000117
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 1618290