báo cáo hóa học: Necessary and sufficient condition for the smoothness of intersection local time of subfractional Brownian motions

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báo cáo hóa học:"Necessary and sufficient condition for the smoothness of intersection local time of subfractional Brownian motions"Journal of Inequalities andApplications This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon.Necessary and sufficient condition for the smoothness of intersection local time of subfractional Brownian motions Journal of Inequalities and Applications 2011, 2011:139 doi:10.1186/1029-242X-2011-139 Guangjun Shen (guangjunshen@yahoo.com.cn) ISSN 1029-242X Article type Research Submission date 6 September 2011 Acceptance date 19 December 2011 Publication date 19 December 2011 Article URL http://www.journalofinequalitiesandapplications.com/content/2011/1/ This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). For information about publishing your research in Journal of Inequalities and Applications go to http://www.journalofinequalitiesandapplications.com/authors/instructions/ For information about other SpringerOpen publications go to http://www.springeropen.com © 2011 Shen ; licensee Springer.This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Necessary and sufficient condition for the smoothness of intersection local time of subfractional Brownian motions Guangjun Shen Department of Mathematics, Anhui Normal University, Wuhu 241000, China Email address: guangjunshen@yahoo.com.cn Abstract Let S H and S H be two independent d-dimensional sub- fractional Brownian motions with indices H ∈ (0, 1). Assume d ≥ 2, we investigate the intersection local time of subfractional Brownian motions TT H H = δ St − Ss dsdt, T > 0, T 0 0 where δ denotes the Dirac delta function at zero. By elementary inequalities, we show that T exists in L2 if and only if Hd < 2 and it is smooth in the sense of the Meyer-Watanabe if and only if 2 H < d+2 . As a related problem, we give also the regularity of the intersection local time process. 2010 Mathematics Subject Classification: 60G15; 60F25; 60G18; 60J55. Keywords: subfractional Brownian motion; intersection local time; Chaos expansion. 1. Introduction The intersection properties of Brownian motion paths have been in-vestigated since the forties (see [1]), and since then, a large numberof results on intersection local times of Brownian motion have beenaccumulated (see Wolpert [2], Geman et al. [3], Imkeller et al. [4], deFaria et al. [5], Albeverio et al. [6] and the references therein). Theintersection local time of independent fractional Brownian motions hasbeen studied by Chen and Yan [7], Nualart et al. [8], Rosen [9], Wu andXiao [10] and the references therein. As for applications in physics, theEdwards, model of long polymer molecules by Brownian motion pathsuses the intersection local time to model the ‘excluded volume’ effect:different parts of the molecule should not be located at the same pointin space, while Symanzik [11], Wolpert [12] introduced the intersectionlocal time as a tool in constructive quantum field theory. 12 Intersection functionals of independent Brownian motions are usedin models handling different types of polymers (see, e.g., Stoll [13]).They also occur in models of quantum fields (see, e.g., Albeverio [14]). As an extension of Brownian motion, recently, Bojdecki et al. [15]introduced and studied a rather special class of self-similar Gaussianprocesses, which preserves many properties of the fractional Brown-ian motion. This process arises from occupation time fluctuations ofbranching particle systems with Poisson i ...