Small deviation for the mutual intersection local time of Brownian motions

In this note, we establish the bounds $$c{\epsilon }^{\frac{2}{3}}\le P\{{\int }_0^1\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}{\int }_0^1{\delta }_0(B_s-{\tilde{B}}_r)dsdr\le \epsilon \}\le C{\epsilon }^{\frac{2}{3}},$$ for the mutual intersection local time of two independent 1-dimensional Brownian motions $B$ and $\tilde{B}$.