Smoothness for the collision local time of two multidimensional bifractional Brownian motions. (English) Zbl 1274.60119

Summary: Let \(B^{H_{i},K_i}=\{B^{H_{i},K_i}_t,\, t\geq 0 \}\), \(i=1,2\) be two independent, \(d\)-dimensional bifractional Brownian motions with respective indices \(H_i\in (0,1)\) and \(K_i\in (0,1]\). Assume \(d\geq 2\). One of the main motivations of this paper is to investigate smoothness of the collision local time \[ \ell _T=\int _{0}^{T}\delta (B_{s}^{H_{1},K_1}-B_{s}^{H_{2},K_2})\,\text{d} s, \quad T>0, \] 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 \(\min \{H_{1}K_1,H_{2}K_2\}<{1}/{(d+2)}\).


60G15 Gaussian processes
60G18 Self-similar stochastic processes
60G22 Fractional processes, including fractional Brownian motion
60J65 Brownian motion
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