Small deviations for the mutual intersection local time of Brownian motions (Q6633633)

Let \(B_s\) and \(\tilde{B}_s\) be two independent 1-dimensional Brownian motions with \({B_0=\tilde{B}_0=0}.\) The authors establish bounds of small deviations for their mutual intersection local time. Namely, for some constants \({0<c\leq C}\) and for small \({\varepsilon>0}\) \N\[\Nc\varepsilon^{3/2}\leq\mathbb{P}\left\{\int_0^1\int_0^1\delta_0(B_s-\tilde{B}_r)\,dsdr\leq\varepsilon\right\}\leq C\varepsilon^{3/2},\N\]\Nwhere \(\delta_0\) is the Dirac delta function in \(\mathbb{R}.\) For small deviations of the self-intersection local time see [\textit{R. van der Hofstad} et al., Ann. Probab. 31, No. 4, 2003--2039 (2003; Zbl 1052.60019)].