Dichotomy in a scaling limit under Wiener measure with density.

Funaki, Tadahisa

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

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Let $B^{H_{i}, K_{i}} = {B_{t}^{H_{i}, K_{i}}, t \geq 0}$ , $i = 1, 2$ 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 $ℓ_{T} = \int_{0}^{T} δ (B_{s}^{H_{1}, K_{1}} - B_{s}^{H_{2}, K_{2}}) d s, T > 0,$ where $δ$ denotes the Dirac delta function. By an elementary method we show that $ℓ_{T}$ is smooth in the sense of Meyer-Watanabe if and only if $min {H_{1} K_{1}, H_{2} K_{2}} < 1 / (d + 2)$ .