Small deviation for the mutual intersection local time of Brownian motions

Abstract

In this note, we establish the bounds \[ c\varepsilon^{\frac23}\le P\bigg\{\int_0^1\!\!\int_0^1δ_0(B_s-\tilde{B}_r)dsdr\le \varepsilon \bigg\} \le C \varepsilon^{\frac23},\] for the mutual intersection local time of two independent 1-dimensional Brownian motions $B$ and $\tilde{B}$.