Abstract
We are interested in path decompositions of a perturbed reflecting Brownian motion (PRBM) at the hitting times and at the minimum. Our study relies on the loop soups developed by Lawler and Werner (Probab Theory Relat Fields 4:197–217, 2004) and Le Jan (Ann Probab 38:1280–1319, 2010; Markov Paths, Loops and Fields. École d’été Saint-Flour XXXVIII 2008. Lecture Notes in Mathematics vol 2026. Springer, Berlin, 2011), in particular on a result discovered by Lupu (Mém Soc Math Fr (N.S.) 158, 2018) identifying the law of the excursions of the PRBM above its past minimum with the loop measure of Brownian bridges.
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Notes
- 1.
Under \(\mathfrak { n}^+ ( \cdot \, |\, \ell ^{m}(\mathfrak {e}) > x- y)\), an excursion up to the inverse local time x − y at position m is a three-dimensional Bessel process, up to the hitting time of m, followed by a Brownian motion starting at m stopped at local time at level m given by x − y, this Brownian motion being conditioned on not touching 0 during that time. By excursion theory, the time-reversed process is distributed as a Brownian motion starting at level m stopped at the hitting time of 0 conditioned on the local time at m being equal to x − y. We conclude by William’s time reversal theorem (Corollary VII.4.6 of [21]).
- 2.
For instance, we may show that for any T > 0, almost surely \(\sup _{0\le t \le T}| X^{-, m'}_t - X^{-, m}_t| \to 0\) as m′→ m. Let us give a proof by contradiction. Suppose there exists some ε 0 > 0, a sequence (t k) in [0, T] and m k → m such that \(| X^{-, m_k}_{t_k} - X^{-, m}_{t_k}|>\varepsilon _0\). Write for simplification \(s_k:=\alpha _{t_k}^{-, m}\) and \(s^{\prime }_k:=\alpha _{t_k}^{-, m_k}\). Consider the case m k > m (the other direction can be treated in a similar way). Then \(s_k\ge s^{\prime }_k\) and \(|X_{s^{\prime }_k}- X_{s_k}|> \varepsilon _0\) for all k. Since \(X_{s_k}\le m\) and \(X_{s^{\prime }_k}\le m_k\), either \(X_{s_k}\le m - \frac {\varepsilon _0}2\) or \(X_{s^{\prime }_k}\le m - \frac {\varepsilon _0}2\) for all large k. Consider for example the case \(X_{s^{\prime }_k}\le m - \frac {\varepsilon _0}2\). By the uniform continuity of X t on every compact, there exists some δ 0 > 0 such that X u ≤ m for all \(|u-s^{\prime }_k| \le \delta _0\) and k ≥ 1. Then for any \(s \ge s^{\prime }_k\), \(\int _{s^{\prime }_k}^s 1_{\{X_u \le m\}} \mathrm {d} u \ge \min (\delta _0, s-s^{\prime }_k)\). Note that by definition, \(\int _{s^{\prime }_k}^{s_k} 1_{\{X_u \le m\}} \mathrm {d} u =t_k- \int _{0}^{s^{\prime }_k} 1_{\{X_u \le m\}} \mathrm {d} u= \int _{m}^{m_k} L(s^{\prime }_k, x) \mathrm {d} x \le \zeta \, (m_k- m)\), with \(\zeta :=\sup _{x\in {\mathbb R}} L(\alpha _T^{-,m}, x)\). It follows that for all sufficiently large k, \(0\le s_k-s^{\prime }_k \le \zeta \, (m_k- m)\). Consequently \(X_{s_k}- X_{s^{\prime }_k} \to 0\) as m k → m, in contradiction with the assumption that \(|X_{s^{\prime }_k}- X_{s_k}|> \varepsilon _0\) for all k. This proves the continuity of m → X −, m.
- 3.
More precisely for any process (γ t, t ≥ 0), Φ(γ) is the process defined by \(\theta (\gamma _t)= \Phi (\gamma )\big (\int _0^t (\theta '(\gamma _s))^2 \mathrm {d} s\big )\).
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Acknowledgements
We are grateful to Titus Lupu for stimulating discussions on the link between the PRBM and the Brownian loop soup. We also thank an anonymous referee for useful suggestions on the paper. The project was partly supported by ANR MALIN.
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Aïdékon, E., Hu, Y., Shi, Z. (2021). Path Decompositions of Perturbed Reflecting Brownian Motions. In: Chaumont, L., Kyprianou, A.E. (eds) A Lifetime of Excursions Through Random Walks and Lévy Processes. Progress in Probability, vol 78. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-83309-1_2
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