Small Deviations for the Mutual Intersection Local Time of Brownian Motions

In this note, we establish the bounds \(\begin{aligned} c\varepsilon ^{\frac{2}{3}}\le P\bigg \{\int _0^1\!\!\int _0^1\delta _0(B_s-{\tilde{B}}_r)\hbox {d}s\hbox {d}r\le \varepsilon \bigg \} \le C \varepsilon ^{\frac{2}{3}} \end{aligned}\) $$\begin{array}{c}\hfill c{\epsilon }^{\frac{2}{3}}\le P\{{\int }_0^1{\int }_0^1{\delta }_0(B_s-{\tilde{B}}_r)\text{d}s\text{d}r\le \epsilon \}\le C{\epsilon }^{\frac{2}{3}}\end{array}$$ for the mutual intersection local time of two independent 1-dimensional Brownian motions B and \({{\tilde{B}}}\) $\tilde{B}$ .