Czechoslovak Mathematical Journal, Vol. 62, No. 4, pp. 969-989, 2012

Smoothness for the collision local time of two multidimensional bifractional Brownian motions

Guangjun Shen, Litan Yan, Chao Chen

Guangjun Shen, Department of Mathematics, Anhui Normal University, 1 East Beijing Rd., Wuhu 241000, P. R. China, e-mail:; Litan Yan, Department of Mathematics, Donghua University, 2999 North Renmin Rd., Songjiang, Shanghai 201620, P. R. China, e-mail:; Chao Chen, Department of Mathematics, East China University of Science and Technology, 130 Mei Long Rd., Xuhui, Shanghai 200237, P. R. China

Abstract: Let $B^{H_i,K_i}=\{B^{H_i,K_i}_t, t\geq0 \}$, $i=1,2$ be two independent, $d$-dimensional bifractional Brownian motions with respective indices $H_i\in(0,1)$ and $K_i\in(0,1]$. Assume $d\geq2$. One of the main motivations of this paper is to investigate smoothness of the collision local time
\ell_T=\int_0^T\delta(B_s^{H_1,K_1}-B_s^{H_2,K_2})  d s, \qquad T>0,
where $\delta$ denotes the Dirac delta function. By an elementary method we show that $\ell_T$ is smooth in the sense of Meyer-Watanabe if and only if $\min\{H_1K_1,H_2K_2\}<1/{(d+2)}$.

Keywords: bifractional Brownian motion, collision local time, intersection local time, chaos expansion

Classification (MSC 2010): 60G15, 60G18, 60J55

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