In this note, we establish the bounds \[ cε²³≤ P\{∫_0^1\!\!∫_0^1δ_0(B_s-B̃_r)dsdr≤ ε \} ≤ C ε²³,\] for the mutual intersection local time of two independent 1-dimensional Brownian motions B and B.
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Chen et al. (2024) studied this question.