Sharp uniform-in-location bounds for the conditional shape law of planar Brownian complement components (Werner, 1994)

For a planar Brownian path run for a unit time, Werner (1994) proved that the normalized shape of the connected component containing a point $z$ in the complement of the path, conditioned on the path coming within distance $\varepsilon$ of $z$ before time 1, converges as $\varepsilon\to0$ to a universal limit law $\mathcal{L}_1$ — for each fixed $z$. The proof replaces the component by a rescaled window component between the first entrance into the $\varepsilon$-disc around $z$ and the first exit of the $|\log\varepsilon|^{-4}$-disc, and its decisive error term, the conditional probability $P(T_R>1\mid T_\varepsilon<1)$ that the window fails to close before time 1, is bounded by $Ce^{4|z|^2}|\log\varepsilon|^{-v}$. Werner notes that, conditionally on reaching a far target by the deadline, the entrance time concentrates at the deadline, so no bound tending to zero can hold uniformly over all $z$; the correct quantitative dependence on the location has remained undetermined. This problem asks for the sharp two-sided behavior of that conditional probability in the joint $(\varepsilon,|z|)$ regime — in particular whether the dependence is polynomial rather than exponential in $|z|$, and what the largest uniformity window $h(\varepsilon)\to\infty$ is — and for the resulting quantitative, uniform-in-location convergence rate of the conditional shape law. No later work is known to address this question for the fixed-time planar Brownian model; the nearest result is an analogue for two-dimensional Brownian random interlacements whose sharpness is itself open.

Background

Planar Brownian motion and its islands. Let $B=(B_t)_{t\ge0}$ be a standard two-dimensional Brownian motion started at the origin. For fixed $t>0$ the trace $B([0,t])$ is almost surely a compact connected set with empty interior, and its complement $\mathbb{R}^2\setminus B([0,t])$ consists of one unbounded component together with countably many bounded components, the Brownian islands. Single points are almost surely never visited by a planar Brownian path (they are polar), so for any $z\ne0$ the connected component $C_t^z$ of $z$ in $\mathbb{R}^2\setminus B([0,t])$ is almost surely well defined. Werner [1] studies the shape of such components, normalizing by translating $z$ to the origin and rescaling by the distance from $z$ to the path,

$D_t^z=\Big\{\frac{y-z}{\mathrm{dist}(z,B([0,t]))}:y\in C_t^z\Big\},\qquad \mathrm{dist}(z,B([0,t]))=\inf_{0\le s\le t}|B_s-z|,$

so that $D_t^z$ is a bounded simply connected open set containing the origin whose boundary touches the unit circle. The natural state space is $\mathcal{O}_0$, the set of bounded simply connected open subsets of the plane containing the origin, equipped with the sigma-field constructed in [1, Section 2].

The limit shape law. Werner proved three limit theorems. First, as $t\to\infty$ the law of $D_t^z$ converges to a law $\mathcal{L}_1$ on $\mathcal{O}_0$, identified in [1] with the renormalized inverse of the unbounded component of the standard planar Brownian loop (a Brownian path bridged to return to its starting point) ([1], Théorème 9; the paper is written in French, and its numbered results are cited here by their French labels, Théorème and Lemme meaning Theorem and Lemma). Second, at fixed time $t$ and for fixed $z$, the conditional law of $D_t^z$ given the event that the path comes within distance $\varepsilon$ of $z$ before time $t$ also converges to $\mathcal{L}_1$ as $\varepsilon\to0$ ([1], Théorème 6; this is the theorem of interest). Third, the empirical distribution of the shapes of all the islands of a single unit-time path, numbered by decreasing area, converges almost surely to an explicitly described transform of a companion law $\mathcal{L}_0$ ([1], Théorème 13). The related quantitative geometry of the islands (their out-radii and in-radii, i.e. the radii of the smallest circumscribed and largest inscribed discs) was later determined in [2].

The conditional theorem and its proof. Write $T_\varepsilon=\inf\{t\ge0:|B_t-z|\le\varepsilon\}$ for the first entrance into the closed $\varepsilon$-disc around $z$, and, for $R>\varepsilon$, $T_R=\inf\{t>T_\varepsilon:|B_t-z|>R\}$ for the first exit of the $R$-disc after that entrance; by Brownian scaling one takes $t=1$. The proof of Théorème 6 replaces the full-time component $C_1^z$ by the window component $C^{z,[T_\varepsilon,T_R]}$ of $z$ in the complement of the path increment $B([T_\varepsilon,T_R])$, with the choice $R=R(\varepsilon)=|\log\varepsilon|^{-4}$, so that $R\to0$ but $R/\varepsilon\to\infty$. After the entrance, the increment is rescaled by $\varepsilon$ using the strong Markov property (the fact that Brownian motion restarts afresh, with independent stationary increments, at stopping times), and the law of the normalized window component is controlled by a quantitative estimate ([1], Proposition 5): for every bounded measurable $F$ on $\mathcal{O}_0$,

$\Big|E\big[F\big(D^{[T_{\varepsilon},T_R]}\big)\big]-\int_{\mathcal{O}_0}F\,d\mathcal{L}_1\Big|\;\le\;k'\|F\|\Big(\frac{\varepsilon}{|z|-\varepsilon}+\eta(\varepsilon,R)\Big),$

where $\eta$ collects logarithmic terms tending to $0$; the location-dependent factor $\varepsilon/(|z|-\varepsilon)$ is benign, since it decreases as the target recedes (it comes from the Cauchy law of the winding angle, $E(e^{in\theta_\varepsilon})=(\varepsilon/|z|)^{|n|}$). The remaining step is to show that the window faithfully represents the full-time component. [1], Lemme 7, bounds the conditional replacement error by three terms,

$P\big(C_1^z\neq C^{z,[T_\varepsilon,T_R]}\mid T_\varepsilon<1\big)\;\le\;e_1\;+\;P\big(T_R>1\mid T_\varepsilon<1\big)\;+\;e_3,$

where $e_1$ (the window component stays inside a tiny disc around $z$) and $e_3$ (after leaving the $R$-disc the path does not return to the $\varepsilon$-disc before time 1) are local, $z$-independent, and of order $\log|\log\varepsilon|/|\log\varepsilon|$. The middle term is the crux: the event $\{T_R>1\}$ that the window has not closed by the deadline. Werner bounds it by

$P\big(T_R>1\mid T_\varepsilon<1\big)\;\le\;C\,e^{4|z|^2}\,|\log\varepsilon|^{-v},$

the exponential factor arising from the rare-event lower bound $P(T_\varepsilon<1)\ge\varepsilon^2e^{-4|z|^2}/(4k''|\log\varepsilon|)$, itself a heat-kernel estimate ($\int_0^{1/2}p_s(0,2|z|)\,ds\ge e^{-4|z|^2}/4$, where $p_s$ denotes the heat kernel, i.e. the Gaussian transition density of planar Brownian motion). He immediately notes the structural obstruction: as $|z|\to\infty$, the conditional law of $T_\varepsilon$ given $\{T_\varepsilon<1\}$ degenerates to the Dirac mass $\delta_1$ at the deadline — given that a far target is reached in time, the entrance essentially happens at the deadline — so, in his words, there are no estimates uniform in $z$. Consequently a bound tending to $0$ cannot hold uniformly over all $z$: the true uniformity domain must be an $\varepsilon$-dependent window $|z|\le h(\varepsilon)$ with $h(\varepsilon)\to\infty$. The paper itself poses no further quantitative question; whether the exponential $e^{4|z|^2}$ location dependence can be improved — to polynomial in $|z|$, as the two-scale structure suggests — and what the largest such window is, is the quantitative form of the non-uniformity gap that [1] flags. No convergence rate for the conditional shape law, uniform or not, was derived in [1].

Later literature. The location-uniformity question for the fixed-time planar Brownian model has not been revisited. Holden, Nacu, Peres and Salisbury [2] quantified the roundness of all islands of a unit-time path (moment bounds for out-radii and a divergence result for in-radii), building on the empirical theorem rather than the conditional one. Comets and Popov [3] introduced two-dimensional Brownian random interlacements (BRI), a model built from Brownian trajectories conditioned never to hit a fixed disc via an explicit change of measure (the Doob $h$-transform) — a transient analogue of the planar Brownian path that makes far-field estimates for conditioned Brownian motion tractable. Collin and Popov [4] proved the BRI analogue of the conditional shape theorem: the component of a point $x$ of the BRI vacant set (the region left uncovered by the interlacement trajectories) is close in distribution to a rescaled Brownian amoeba (the component around the origin for the conditioned path) when $x$ is close to the interlacement trajectories, with explicit dependence on that distance, and they flag the sharpness of these estimates as open. Guillaume [5] applied the unit-time trace's self-similarity to the persistent homology (a tool from topological data analysis that records the lifespans of holes in a set as it is progressively thickened) of the planar Wiener sausage (the radius-$r$ neighborhood of the trace). None of these works returns to the original fixed-time model to determine the correct dependence of the conditional estimates on the location, and no improvement of the $e^{4|z|^2}$ bound is known.

Problem Statement

Let $B=(B_t)_{t\ge0}$ be a standard two-dimensional Brownian motion started at the origin, and fix $z\in\mathbb{R}^2\setminus\{0\}$. For $t>0$ let $C_t^z$ be the connected component of $z$ in $\mathbb{R}^2\setminus B([0,t])$ and $D_t^z=\{(y-z)/\mathrm{dist}(z,B([0,t])):y\in C_t^z\}$ its normalization, where $\mathrm{dist}(z,B([0,t]))=\inf_{0\le s\le t}|B_s-z|$. Let $T_\varepsilon=\inf\{t\ge0:|B_t-z|\le\varepsilon\}$ and, for $R>\varepsilon$, $T_R=\inf\{t>T_\varepsilon:|B_t-z|>R\}$. Let $\mathcal{O}_0$ denote the space of bounded simply connected open subsets of the plane containing the origin, and let $\mathcal{L}_1$ denote the probability law on $\mathcal{O}_0$ defined as the limit, as $t\to\infty$, of the law of $D_t^z$ — a limit shown in [1] to exist and not to depend on $z\ne0$, and identified there with the renormalized inverse of the unbounded component of the standard planar Brownian loop (a Brownian path bridged to return to its starting point). Set $R(\varepsilon)=|\log\varepsilon|^{-4}$ (the choice made in [1]). Define

$q(\varepsilon,z)\;=\;P\big(T_{R(\varepsilon)}>1\;\big|\;T_\varepsilon<1\big).$

Determine the sharp behavior of $q(\varepsilon,z)$ in the joint regime $\varepsilon\to0$, $|z|\to\infty$ (with $\varepsilon\le|z|$), and convert it into a quantitative, uniform-in-location version of the conditional shape theorem. The problem consists of three linked parts.

(a) Upper bound with polynomial location dependence. Prove that there exist constants $C<\infty$, $p<\infty$, $\varepsilon_0>0$ and an explicit function $g(\varepsilon)\to0$ (ideally a negative power of $|\log\varepsilon|$, up to explicit $\log|\log\varepsilon|$ corrections) such that

$q(\varepsilon,z)\;\le\;C\,(1+|z|)^p\,g(\varepsilon)\qquad\text{for all }\varepsilon<\varepsilon_0,\ \varepsilon\le|z|\le h(\varepsilon),$

where $h(\varepsilon)\to\infty$ is explicit — ideally $h(\varepsilon)=|\log\varepsilon|^{4-\delta}$ for every $\delta>0$. The scale suggested by the two-scale structure of the problem (conditionally on $\{T_\varepsilon<1\}$, the remaining time $1-T_\varepsilon$ is typically of order $|z|^{-2}$, while closing the window requires time of order $R(\varepsilon)^2=|\log\varepsilon|^{-8}$) is $q(\varepsilon,z)\asymp|z|^2|\log\varepsilon|^{-8}$; this is a target, not an assumption. Alternatively, if no polynomial-in-$|z|$ bound holds, establish the optimal dependence on $|z|$ with the same rigor and specificity.

(b) Lower bound, degeneracy, and the maximal window. Prove rigorously that the conditional law of $T_\varepsilon$ given $\{T_\varepsilon<1\}$ converges to $\delta_1$ as $|z|\to\infty$ with $\varepsilon$ fixed (stated without proof as a remark in [1]), and establish a matching lower bound, ideally $q(\varepsilon,z)\ge c\,\min\{1,|z|^2R(\varepsilon)^2\}$ or the correct sharp form. Consequently identify the maximal uniformity window through the critical growth exponent

$a^*\;=\;\sup\Big\{a\ge0:\ \sup_{\varepsilon\le|z|\le|\log\varepsilon|^a}q(\varepsilon,z)\to0\ \text{as }\varepsilon\to0\Big\},$

the maximal window being then $|\log\varepsilon|^{a^*}$ up to $|\log\varepsilon|^{o(1)}$ factors (conjecturally $a^*=4$), and show in particular that $q(\varepsilon,z)\not\to0$ uniformly over all $z$, so that [1]'s factor $e^{4|z|^2}$ is quantitatively suboptimal on the window.

(c) Quantitative uniform conditional shape theorem. Prove that for every bounded measurable $F:\mathcal{O}_0\to\mathbb{R}$ there is an explicit $\mathrm{Err}(\varepsilon,|z|)\to0$ with

$\Big|E\big[F(D_1^z)\mid T_\varepsilon<1\big]-\int_{\mathcal{O}_0}F\,d\mathcal{L}_1\Big|\;\le\;\mathrm{Err}(\varepsilon,|z|)\qquad\text{for all }\varepsilon\le|z|\le h(\varepsilon),$

and that the corresponding uniform statement fails on any window growing faster than $|\log\varepsilon|^{a^*+o(1)}$.

All constants, exponents, and admissible logarithmic corrections must be explicit. Parts (a) and (b) together constitute the sharp two-sided description of the conditional window-closing probability — the quantity whose exponential location dependence is the gap left by [1] — and part (c) is its transfer to the shape law underpinning Théorème 6. Answers may equally establish that the conjectured polynomial scale is wrong, provided the correct dependence is then determined with the same rigor and specificity.

References

1. [1] Wendelin Werner, "Sur la forme des composantes connexes du complémentaire de la courbe brownienne plane", Probability Theory and Related Fields 98 (1994), no. 3, pp. 307-337, doi:10.1007/BF01192257, https://doi.org/10.1007/BF01192257.
2. [2] Nina Holden, Șerban Nacu, Yuval Peres, Thomas S. Salisbury, "How round are the complementary components of planar Brownian motion?", Ann. Inst. Henri Poincaré Probab. Stat. 55 (2019), no. 2, pp. 882-908, doi:10.1214/18-AIHP902, https://doi.org/10.1214/18-AIHP902; preprint arXiv:1609.06627, https://arxiv.org/abs/1609.06627.
3. [3] Francis Comets, Serguei Popov, "Two-dimensional Brownian random interlacements", Potential Analysis 53 (2020), no. 2, pp. 727-771, doi:10.1007/s11118-019-09786-8, https://doi.org/10.1007/s11118-019-09786-8; preprint arXiv:1804.05967, https://arxiv.org/abs/1804.05967.
4. [4] Orphée Collin, Serguei Popov, "On the shape of the connected components of the complement of two-dimensional Brownian random interlacements", Communications in Mathematics 33 (2025), no. 1, article no. 14455, doi:10.46298/cm.14455, https://doi.org/10.46298/cm.14455; preprint arXiv:2410.08944, https://arxiv.org/abs/2410.08944.
5. [5] Tristan Guillaume, "Persistent homology of the planar Wiener sausage: Brownian scaling and a logarithmic expectation law", preprint arXiv:2606.11248 (2026), https://arxiv.org/abs/2606.11248.