Small Deviations for the Mutual Intersection Local Time of Brownian Motions

Xia Chen () and Jian Song ()
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Xia Chen: University of Tennessee
Jian Song: Shandong University

Journal of Theoretical Probability, 2025, vol. 38, issue 1, 1-19

Abstract: Abstract In this note, we establish the bounds $$\begin{aligned} c\varepsilon ^{\frac{2}{3}}\le P\bigg \{\int _0^1\!\!\int _0^1\delta _0(B_s-{\tilde{B}}_r)\hbox {d}s\hbox {d}r\le \varepsilon \bigg \} \le C \varepsilon ^{\frac{2}{3}} \end{aligned}$$ c ε 2 3 ≤ P { ∫ 0 1 ∫ 0 1 δ 0 ( B s - B ~ r ) d s d r ≤ ε } ≤ C ε 2 3 for the mutual intersection local time of two independent 1-dimensional Brownian motions B and $${{\tilde{B}}}$$ B ~ .

Keywords: Brownian motion; Random walk; Intersection local time; Small ball probability; 60J55; 60F05; 60B12; 60J65 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10959-024-01377-7

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