Necessary and sufficient condition for the smoothness of intersection local time of subfractional Brownian motions

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 ℓ T = ∫ 0 T ∫ 0 T δ S t H - S ̃ s H d s d t , T > 0 , where δ denotes the Dirac delta function at...

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Published inJournal of inequalities and applications Vol. 2011; no. 1; pp. 1 - 16
Main Author Shen, Guangjun
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 19.12.2011
Springer Nature B.V
BioMed Central Ltd
Subjects
Analysis
Applications of Mathematics
Brownian motion
Classification
Delta function
Inequalities
Intersections
Mathematical analysis
Mathematics
Mathematics and Statistics
Regularity
Smoothness
intersection local time
Chaos expansion
subfractional Brownian motion
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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 ℓ T = ∫ 0 T ∫ 0 T δ S t H - S ̃ s H d s d t , T > 0 , where δ denotes the Dirac delta function at zero. By elementary inequalities, we show that ℓ T exists in L 2 if and only if Hd < 2 and it is smooth in the sense of the Meyer-Watanabe if and only if H < 2 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.
AbstractList Let S ^sup H^ and [InlineEquation not available: see fulltext.] 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 [Equation not available: see fulltext.] where δ denotes the Dirac delta function at zero. By elementary inequalities, we show that ^sub T^ exists in L ^sup 2^ if and only if Hd < 2 and it is smooth in the sense of the Meyer-Watanabe if and only if [InlineEquation not available: see fulltext.]. As a related problem, we give also the regularity of the intersection local time process. 2010 Mathematics Subject Classification: 60G15; 60F25; 60G18; 60J55.[PUBLICATION ABSTRACT]
Let S super( )Hand [InlineEquation not available: see fulltext.] be two independent d-dimensional sub-fractional Brownian motions with indices H (0, 1). Assume d greater than or equal to 2, we investigate the intersection local time of subfractional Brownian motions [Equation not available: see fulltext.] where delta denotes the Dirac delta function at zero. By elementary inequalities, we show that sub( )Texists in L super(2) if and only if Hd < 2 and it is smooth in the sense of the Meyer-Watanabe if and only if [InlineEquation not available: see fulltext.]. As a related problem, we give also the regularity of the intersection local time process. 2010 Mathematics Subject Classification: 60G15; 60F25; 60G18; 60J55.
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 ℓ T = ∫ 0 T ∫ 0 T δ S t H - S ̃ s H d s d t , T > 0 , where δ denotes the Dirac delta function at zero. By elementary inequalities, we show that ℓ T exists in L 2 if and only if Hd < 2 and it is smooth in the sense of the Meyer-Watanabe if and only if H < 2 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.
Let SH and S̃H 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 motionsℓT= ∫ 0T∫ 0TδStH-S̃sHdsdt,T>0,where δ denotes the Dirac delta function at zero. By elementary inequalities, we show that ℓT exists in L2 if and only if Hd < 2 and it is smooth in the sense of the Meyer-Watanabe if and only if H<2d+2. As a related problem, we give also the regularity of the intersection local time process.2010 Mathematics Subject Classification: 60G15; 60F25; 60G18; 60J55.
ArticleNumber 139
Author Shen, Guangjun
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Cites_doi 10.1016/0022-1236(78)90061-7
10.1512/iumj.1974.23.23006
10.1016/0047-259X(87)90176-X
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10.1016/0022-1236(78)90062-9
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ContentType Journal Article
Copyright Shen; licensee Springer. 2011. This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Springer International Publishing AG 2011
Copyright_xml – notice: Shen; licensee Springer. 2011. This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Keywords intersection local time
Chaos expansion
subfractional Brownian motion
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Snippet 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...
Let S ^sup H^ and [InlineEquation not available: see fulltext.] be two independent d-dimensional sub-fractional Brownian motions with indices H (0, 1). Assume...
Let S super( )Hand [InlineEquation not available: see fulltext.] be two independent d-dimensional sub-fractional Brownian motions with indices H (0, 1). Assume...
Let SH and S̃H be two independent d-dimensional sub-fractional Brownian motions with indices H ∈ (0, 1). Assume d ≥ 2, we investigate the intersection local...
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SubjectTerms Analysis
Applications of Mathematics
Brownian motion
Classification
Delta function
Inequalities
Intersections
Mathematical analysis
Mathematics
Mathematics and Statistics
Regularity
Smoothness
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Title Necessary and sufficient condition for the smoothness of intersection local time of subfractional Brownian motions
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