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

• Published in: Journal of inequalities and applications Vol. 2011; no. 1; pp. 1 - 16
• Main Author: Shen, Guangjun
• Format: Journal Article
• Language: English
• 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
• ISSN: 1029-242X; 1025-5834; 1029-242X
• DOI: 10.1186/1029-242X-2011-139

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.