Measurable moving targets for Werner's one-dimensional Brownian comparison: extending the circle-valued avoidance inequality beyond continuous targets to rederive Beurling's projection theorem for arbitrary compact sets within the one-dimensional framework

Werner (1996) proved Beurling's projection theorem — planar Brownian motion is at least as likely to hit a compact set $K$ in the unit disk as to hit its circular projection — through a purely one-dimensional comparison: among all continuous moving targets on the unit circle, the constant antipodal point is the one that circle-valued Brownian motion $t\mapsto e^{iB_t}$ is most likely to avoid. The derivation requires the target curve to be continuous, which restricts the theorem to compact sets that are countable unions of closed path-connected components, excluding compact sets with uncountably many components. This task asks whether the circle comparison extends to arbitrary measurable (Borel) moving targets and whether the accompanying reduction can be carried out, so that the one-dimensional method yields the theorem directly for every compact set.

Background

Planar Brownian motion $B=(B_t)_{t\ge 0}$ is the random-process model for diffusion in the plane; started at a point $z$ in the unit disk $\Delta=\{w\in\mathbb{C}:|w|<1\}$, it almost surely exits the disk in finite time. For a set $A$ write $T_A=\inf\{t\ge 0:B_t\in A\}$ for the first hitting time of $A$, and let $C=\{w:|w|=1\}$ denote the unit circle. The harmonic measure of a compact set $K\subset\Delta$ at $z$ (relative to $\Delta$) is the hitting probability $w(z,\Delta,K)=\mathbb{P}_z(T_K<T_C)$; it is the probabilistic solution of the Dirichlet problem in $\Delta\setminus K$ and a central quantity of potential theory and conformal mapping. The circular (radial) projection of $K$ is the compact subset $\Pi_K=\{-|w|:w\in K\}$ of the negative real axis. Beurling's projection theorem, from his 1933 Uppsala thesis, states that planar Brownian motion is at least as likely to hit $K$ as to hit its projection:

$$w(z,\Delta,K)\ \ge\ w(|z|,\Delta,\Pi_K)\qquad\text{for all }z\in\Delta;$$

equivalently, among compact sets with a given radial range, the straight radial segment is the one Brownian motion is least likely to hit. The statement for general compact sets is classical, with analytic proofs in standard monographs and stochastic proofs via two-dimensional potential theory (Øksendal 1983), and it has been refined in higher dimensions, in non-tangentially accessible domains (NTA domains — domains whose boundary points are approachable by non-tangential cones in a quantitative sense, a standard regularity class from harmonic analysis), and in two-point form (Essén–Haliste 1989; Knopf 2001; Betsakos 2003).

Werner (1996) gave an alternative proof built entirely from one-dimensional estimates. The key object is circle-valued Brownian motion, the process $t\mapsto e^{iB_t}$ obtained by exponentiating a one-dimensional Brownian motion $B$ started at $0$, which winds around the unit circle while its real-valued counterpart diffuses on the line. Werner's Proposition 2 compares avoidance probabilities for this process: for every open time set $I\subset[0,\infty)$ and every continuous target curve $M:I\to C$,

$$\mathbb{P}\big(\forall t\in I:\ e^{iB_t}\neq M(t)\big)\ \le\ \mathbb{P}\big(\forall t\in I:\ e^{iB_t}\neq -1\big),$$

that is, among all continuous moving targets on the circle, the constant antipodal point $-1$ is the one the winding motion is most likely to avoid (hardest to hit). The proof discretizes time, uses a periodic rearrangement inequality showing that among discretized moving barriers the symmetric one maximizes the avoidance probability, and passes to the continuous limit. Beurling's theorem is then recovered through the skew-product decomposition of planar Brownian motion, which factors it into a radial part and an independent angular part: on each radial excursion across an annulus, the angular trace of $K$ is compared with the single point $-1$ via Proposition 2. Assembling the angular traces into the continuous target curves required by Proposition 2 (through polygonal approximation) forces the restriction that $K$ be a countable union of closed path-connected sets. This hypothesis is not vacuous: a countable union of closed path-connected sets has at most countably many connected components (each component contains at least one of the sets), whereas a general compact plane set may have uncountably many components — uncountably many point components, for instance Cantor-type subsets of the plane with positive logarithmic capacity (i.e. non-polar: compact sets that planar Brownian motion hits with positive probability), and uncountably many positive-diameter components, for instance a product of a Cantor set with an interval — and the paper's derivation does not cover such sets. The source paper identifies the removal of the continuity assumption on the target curve — proving the analogue of Proposition 2 for arbitrary measurable targets — as a technical obstacle left open.

Problem Statement

The primary question is (a): prove or refute the measurable-target extension of the one-dimensional circle comparison of Werner (1996). Part (b) is posed under the hypothesis that (a) holds, and asks to derive Beurling's projection theorem for arbitrary compact sets directly within that one-dimensional framework, using the extension.

(a) Circle comparison for measurable targets. Let $B$ be linear Brownian motion started at $0$ and let $Z_t=e^{iB_t}$ be circle-valued Brownian motion on the unit circle $C$. Establish or refute: for every open time set $I\subset[0,\infty)$ and every Borel-measurable target $M:I\to C$ (with the avoidance event shown to be measurable),

$$\mathbb{P}\big(\forall t\in I:\ Z_t\neq M(t)\big)\ \le\ \mathbb{P}\big(\forall t\in I:\ Z_t\neq -1\big).$$

(b) Direct reduction to arbitrary compact sets. Using (a) and the skew-product (radial-excursion) decomposition of planar Brownian motion, derive

$$\mathbb{P}_z(T_K<T_C)\ \ge\ \mathbb{P}_{|z|}(T_{\Pi_K}<T_C)\qquad\text{for every compact }K\subset\Delta\text{ and every }z\in\Delta,$$

directly from the circle comparison — that is, for arbitrary compact sets themselves, rather than through approximations of $K$ by polygonal sets or by outer approximations with countably many path-connected components. This must cover compact sets with uncountably many point components, such as non-polar Cantor-type subsets of the plane, which the source paper's hypothesis excludes.

References

1. Wendelin Werner, 'Beurling's projection theorem via one-dimensional Brownian motion', Mathematical Proceedings of the Cambridge Philosophical Society 119(4), 729–738 (1996). DOI: 10.1017/S0305004100074557, https://doi.org/10.1017/S0305004100074557.
2. Wendelin Werner, 'Bounds for Disconnection Exponents', Electronic Communications in Probability 1 (1996). DOI: 10.1214/ECP.V1-974, https://doi.org/10.1214/ECP.V1-974. (Uses Proposition 2 in the continuous-target form.)
3. Gregory F. Lawler, 'A Lower Bound on the Growth Exponent for Loop-Erased Random Walk in Two Dimensions', ESAIM: Probability and Statistics 3, 1–21 (1999). DOI: 10.1051/ps:1999100, https://doi.org/10.1051/ps:1999100; preprint arXiv:math/9803034, https://arxiv.org/abs/math/9803034. (Improves the discrete random-walk Beurling estimate, which concerns lattice walks rather than the continuous one-dimensional comparison.)
4. Christian Beneš, 'Some Estimates for Planar Random Walk and Brownian Motion', preprint (2006), arXiv:math/0611127, https://arxiv.org/abs/math/0611127. (States the Beurling projection theorem (Theorem 4.1) for arbitrary closed subsets of a disk and attributes the proof to classical monographs (probabilistic and analytic).)
5. Phanuel Mariano and Hugo Panzo, 'Conformal Skorokhod embeddings and related extremal problems', Electronic Communications in Probability 25 (2020). DOI: 10.1214/20-ECP324, https://doi.org/10.1214/20-ECP324; preprint arXiv:2001.12008, https://arxiv.org/abs/2001.12008. (Applies Øksendal's 1983 version of the projection theorem to general compact sets cut from general domains.)
6. Nina Holden, Şerban Nacu, Yuval Peres and Thomas S. Salisbury, 'How round are the complementary components of planar Brownian motion?', Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 55, 882–908 (2019). DOI: 10.1214/18-AIHP902, https://doi.org/10.1214/18-AIHP902; preprint arXiv:1609.06627, https://arxiv.org/abs/1609.06627. (States Beurling's theorem (Theorem A.1) for an arbitrary set.)
7. Nathanaël Berestycki and Isao Sauzedde, 'Brownian paths as loop-decorated SLEs', preprint (2026), arXiv:2602.04673, https://arxiv.org/abs/2602.04673. (Applies Beurling's projection principle (Lemma 5.11) to an arbitrary compact exhaustion.)
8. Bernt Øksendal, 'Projection estimates for harmonic measure', Arkiv för Matematik 21, 191–203 (1983). DOI: 10.1007/BF02384309, https://doi.org/10.1007/BF02384309. (Stochastic proofs of the Beurling and Hall projection theorems with d-dimensional versions (zbMATH review Zbl 0537.31002, https://zbmath.org/?q=an:0537.31002).)
9. Matts Essén and Kersti Haliste, 'On Beurling's theorem for harmonic measure and the rings of Saturn', Complex Variables, Theory and Applications 12, 137–152 (1989). DOI: 10.1080/17476938908814360, https://doi.org/10.1080/17476938908814360. (Higher-dimensional version of Beurling's theorem for closed subsets.)
10. Peter M. Knopf, 'Projection Theorems for Harmonic Measure in NTA Domains', Potential Analysis 14, 107–122 (2001). DOI: 10.1023/A:1008791022455, https://doi.org/10.1023/A:1008791022455. (Projection theorem in non-tangentially accessible domains.)
11. Dimitrios Betsakos, 'Two-point projection estimates for harmonic measure', Bulletin of the London Mathematical Society 35, 473–478 (2003). DOI: 10.1112/S0024609303002108, https://doi.org/10.1112/S0024609303002108. (Two-point refinement; the Beurling–Nevanlinna lower bound holds for the harmonic measure of any compact set in the plane, with no connectivity restriction (zbMATH review Zbl 1030.30021, https://zbmath.org/?q=an:1030.30021).)
12. Arne Beurling, 'Études sur un problème de majoration', thesis, Uppsala (1933). No DOI exists for this thesis; the zbMATH identifier is Zbl 0008.31802, https://zbmath.org/?q=an:0008.31802. (Origin of the projection theorem.)
13. Alma Sarai Hernández Torres, 'A macroscopic view of two discrete random models', PhD thesis, University of British Columbia (2020). DOI: 10.14288/1.0394111, https://doi.org/10.14288/1.0394111. (Applications-only citation of a Beurling-type estimate.)