Where did the Brownian particle go?

Consider the radial projection onto the unit sphere of the path a d-dimensional Brownian motion W, started at the center of the sphere and run for unit time. Given the occupation measure mu of this projected path, what can be said about the terminal point W(1), or about the range of the original path? In any dimension, for each Borel set A subseteq S^{d-1}, the conditional probability that the projection of W(1) is in A given mu(A) is just mu(A). Nevertheless, in dimension d>=3, both the range and the terminal point of W can be recovered with probability 1 from mu. In particular, for d>=3 the conditional law of the projection of W(1) given mu is not mu. In dimension~2 we conjecture that the projection of W(1) cannot be recovered almost surely from mu, and show that the conditional law of the projection of W(1) given mu is not mu.


Introduction
'This track, as you perceive, was made by a rider who was going from the direction of the school. ' 'Or towards it?' 'No, no, my dear Watson. The more deeply sunk impression is, of course, the hind wheel, upon which the weight rests. You perceive several places where it has passed across and obliterated the more shallow mark of the front one. It was undoubtedly heading away from the school.' From The Adventure of the Priory School, a Sherlock Holmes story by A. Conan Doyle.
The radial projection of a Brownian motion started at the origin and run for unit time in d dimensions defines a random occupation measure on the sphere S d−1 . Can we determine the endpoint of the Brownian path from this projected occupation measure? The problem of recovering data given a projection of the data is a common theme both inside and outside of probability theory. The title of this paper is adapted from a handout distributed by Peter Doyle, where the geometric problem of recovering from bicycle tracks the exit direction of the cyclist was posed.
An interesting feature of the present reconstruction problem is that the answer in low dimensions is different from the answer in dimensions d ≥ 3. This would not be too surprising, except that the behavior in the one-dimensional case involves a conditioning identity which does not seem inherently one-dimensional. This identity concerns the conditional distribution of the endpoint given the occupation measure. One of the aims of this paper is to understand why this identity breaks down in higher dimensions, and what version of this identity might hold even when the occupation measure determines the endpoint and indeed determines the entire unprojected path. In high dimensions, recovery of the endpoint (and entire path), while intuitively plausible, is somewhat tricky because, as described in [17, page 275], the particle "comes in spinning". In particular, the range of the projected path is a.s. a dense subset of the sphere (see remark at the end of this introduction). Thus some quantitative criterion on accumulation of measure is required even to recover the set of occupied points on the sphere from the occupation measure.
Throughout the paper d is a positive integer, and S d−1 ⊆ R d is the unit sphere. We often omit d in the notation for various spaces and mappings whose definition depends on d. Let π : R d → S d−1 be the spherical projection π(x) = x/|x| for x = 0, with some arbitrary conventional value for π(0). Let (W t , t ≥ 0) denote a standard Brownian motion in R d with W 0 = 0, which we take to be defined on some underlying probability space (Ω, F, P). For t ≥ 0 let Θ t := π(W t ), and let Θ := (Θ t , 0 < t ≤ 1). Let µ Θ denote the random occupation measure of Θ on S d−1 , that is for Borel subsets B of S d−1 . We may regard µ Θ as a random variable defined on (Ω, F, P) with values in the space (prob(S d−1 ), F 2 ) of Borel probability measure on S d−1 endowed with the σ-field generated by the measures of Borel sets.
The questions considered in this paper arose from the following identity: for each Borel subset B of S d−1 , we have (

1.2)
If d = 1 then S 0 = {−1, 1}, and µ Θ ({1}) and µ Θ ({−1}) are the times spent positive and negative respectively by a one-dimensional Brownian motion up to time 1. As observed in Pitman-Yor [28], formula (1.2) in this case can be read from Lévy's description [22] of the joint law of the arcsine distributed variable µ Θ ({1}) and the Bernoulli(1/2) distributed indicator 1 W 1 >0 . The truth of (1.2) in higher dimensions is not so easily checked, due to the lack of explicit formulae for the distribution of µ Θ (B) even for the simplest subsets B of S d−1 ; see for instance [4]. However, (1.2) can be deduced from the scaling property of Brownian motion, which implies that the process (Θ t , t ≥ 0) is 0-self-similar, meaning there is the equality in distribution for all c > 0. According to an identity of Pitman and Yor [30], recalled as Proposition 2.1 in Section 2, the identityi (1.2) holds for an arbitrary jointly measurable 0-self-similar process (Θ t , t ≥ 0) with values in an abstract measurable space, for any measurable subset B of that space.
Formula (1.2) led us to the following question, which we discuss further in Section 3: hold for all measurable subsets B of the range space of Θ?
To clarify the difference between (1.2) and (1.3), P (Θ 1 ∈ B | µ Θ (B)) on the LHS of (1.3) is a conditional probability given the σ-field generated by the real random variable µ Θ (B), whereas P (Θ 1 ∈ B | µ Θ ) on the LHS of (1.3) is a conditional probability given the σ-field generated by the random measure µ Θ , that is by all the random variables µ Θ (C) as C ranges over measurable subsets of the range space of Θ. For a general process Θ, formula (1.3) implies (1.2), but not conversely. Now let Θ be the spherical projection of Brownian motion. If d = 1 then the σ-field generated by µ Θ is identical to that generated by either We show this for d = 2 in Section 4 by some explicit estimates involving the occupation times of quadrants. For d ≥ 3 formula (1.3) fails even more dramatically. In Section 5 we show that if d ≥ 3 then Θ 1 is a.s. equal to a measurable function of µ Θ . Less formally, we say that Θ 1 can be recovered from µ Θ . This brings us to the question of what features of the path of the original Brownian motion W := (W t , 0 ≤ t ≤ 1) can be recovered from µ Θ . If d = 3 it is well known that the path W has self-intersections almost surely, so one can define a measure-preserving map T on the Brownian path space that reverses the direction of an appropriately selected closed loop in the path. Regarding µ Θ = µ Θ (W ) as a function on path space, we then have For d ≥ 4 it is a routine consequence of the almost sure parameterizability of a Brownian path by its quadratic variation that W can be recovered from range(W ). So Theorem 1.2 implies that W can be recovered from µ Θ in dimensions d ≥ 4. The only part of the proof of Theorem 1.2 which involves probabilistic estimates for Brownian motion is Lemma 5.1, the remainder being mostly point-set topology. We remark that the topological arguments below also show that in dimension d = 2, the range of W and the endpoint W 1 can be recovered from the occupation measure of the planar path W := (W t , 0 ≤ t ≤ 1) As usual, the hardest (and most interesting) dimension is two. We conjecture that the high-dimensional behavior does not extend to two dimensions, that is, When d = 2 it can be deduced from work of Bass and Khoshnevisan [3, Theorem 2.9] that µ Θ almost surely has a continuous density, call it the angular local time process. The problem of describing the conditional law of W given µ Θ for d = 2 is then analogous to the problem studied by Warren and Yor [36], who give an account of the randomness left in a one-dimensional Brownian motion after conditioning on its occupation measure up to a suitable random time. Aldous [1] and Knight [19] treat related questions involving the distribution of Brownian motion conditioned on its local time process. However, as far as we know there is no Ray-Knight type description available for the angular local time process, and this makes it difficult to settle the conjecture.
Remark. Let Θ t := W t /|W t | be the radial projection of Brownian motion in R d . It is a classical fact that for any ǫ > 0, the initial path segment {Θ t : 0 < t < ǫ} is dense in the unit sphere S d−1 . Since this fact motivates much of our work, we include an elementary explanation for it, which is valid in greater generality. It suffices to show that for any open set U on the sphere and any ǫ > 0, the probability of the event E(U, ǫ) that {Θ t : 0 < t < ǫ} intersects U , equals one. By compactness, some finite number N U of rotated copies of U cover the sphere, so by rotation invariance of Brownian motion, whence by the Blumenthal zero-one law, this probability must be 1.

Identities for scalar self-similar processes
Recall that a real or vector-valued process Such processes were studied by Lamperti [20,21], who called them semi-stable. See [34] for a survey of the literature of these processes. The conditioning formula (1.2) for any 0-self-similar process (Θ t , t ≥ 0) is an immediate consequence of the following identity. To see the direct implication, take X(t, ω) to equal 1 (0,∞) (ω(t)).
Proposition 2.1 (Pitman and Yor [30]) Let (X t , t ≥ 0) be stochastic process with X : Then for every t > 0, Proof: We simplify slightly the proof in [30]. Due to (2.2) it suffices to prove (2.4) for t = 1. It also suffices to prove this on the event {X 1 = 0}, since this implies EX 1 1 X 1 =0 = EX 1 1 X 1 =0 and subtracting the relation EX 1 = EX 1 (a consequence of (2.2)) yields EX 1 1 X 1 =0 = 0. This is equivalent to proving for a suitably large class of functions f . Let ν be the law of for bounded measurable f may be approximated in L 2 (ν) by bounded functions vanishing in a neighborhood of zero and having bounded continuous derivative, this class suffices.
Fix such a function f and apply the chain rule for Lebesgue integrals (see, e.g., [32], Chapter 0, Prop. (4.6)), treating ω as fixed, to obtain Boundedness of f ′ allows the interchange of expectation with integration, so using (2.2) we get (2.5) from the following computation: For a different proof and variations of the identity see [29]. We see immediately that (2.2) holds for any 0-self-similar process X. We observe also: where C is any Borel set which is a cone, i.e., for λ > 0, x ∈ C ⇔ λx ∈ C. Then (X t ) satisfies (2.2), and hence P( Applying Bayes' rule to (2.6) yields the following corollary.

Corollary 2.3
Let {Y t } be any β-self-similar vector-valued process, and let V t = t 0 X s ds with X t := 1 {Yt∈C} for a fixed positive cone C. Then

Corollary 2.4
Under the hypotheses of the Corollary 2.3, suppose X t has a beta(a, b) distribution. Then the conditional distribution of X t given Y t ∈ C is beta(a + 1, b) and the conditional distribution of X t given Y t / ∈ C is beta(a, b + 1).

Example 2.5 Stable Lévy Processes.
Let {Y t } be a stable Lévy process that satisfies P(Y t > 0) = p for all t. It is well known [23,15] that the distribution of the total duration . It follows that the conditional distributions of V 1 given the sign of Y 1 are respectively Brownian motion started at 0, µ > 0 and (ℓ t , t ≥ 0) is the local time process of B at zero. F. Petit [27] showed that V − 1 : These results have been stated and proved in [37,Th. 8.3] and in [8]. A more general class of beta laws has been obtained for the times spent in R ± by doubly perturbed Brownian motions, that is to say solutions of the stochastic equation See, e.g., Carmona-Petit-Yor [9], Perman-Werner [26] and Chaumont-Doney [10].

Example 2.7
More about the Brownian case. Formula (1.2) has some surprising consequences even in the simplest case when d = 1. Consider the function for 0 < t ≤ 1 and 0 ≤ a ≤ 1, where B is a one-dimensional Brownian motion and Without attempting to compute f (t, a) explicitly, which appears to be quite difficult, let us presume that f can be chosen to be continuous in (t, a). Then where the first equality follows from (2.9) and the second equality is read from (1.2). On the other hand, it is easily seen that That is to say, given V 1 = a > 1 2 , there is some time t < 1 such that the BM is more likely to be positive at time t than it is at time 1.

Identities for self-similar processes in dimension d ≥ 2
Say that a jointly measurable process Θ := (Θ t , 0 < t ≤ 1) has the sampling property if for all measurable subsets B of the range space of Θ. The results of this section consist of two examples where the sampling property does hold, and a characterization of the sampling property in terms of exchangeability.
Proposition 3.1 Suppose that Θ takes values in a Borel space. Let U 1 , U 2 , · · · be a sequence of i.i.d. random variables with uniform distribution on (0, 1), independent of Θ. Then the following are equivalent: (i) (Θ t ) has the sampling property; (ii) for each n = 1, 2, 3, · · ·, Proof: Clearly (ii) is equivalent to for all measurable subsets B of the range space of Θ. To connect this to (i), observe that {Θ U j } ∞ j=2 is a sequence of i.i.d. picks from µ Θ . Hence this sequence is conditionally independent of Θ given µ Θ . Therefore, (3.3) can be rewritten as for all measurable B, which is equivalent to (i).
The conditions (3.2) increase in strength as n increases. For n = 2, (3.2) is just Proposition 3.2 If the distribution of (Θ s , Θ t ) depends only on t/s then the conditions (3.5) and (3.6) are equivalent.
Proof: Construct U 1 and U 2 as follows. Let Y and Z be independent with Y uniform on [0, 1] and Z having density 2x on [0, 1]. Let X be an independent ±1 fair coin-flip and set ( By the assumption on Θ this is one half the law of (Θ 1 , Θ U 2 ) plus one half the law of (Θ U 2 , Θ 1 ). This and (3.6) imply (3.5).
We note that the spherical projection of Brownian motion in R d satisfies (3.6) for all d. So this condition is not enough to imply the sampling property for a 0-self-similar process Θ. When Θ is not 0-self-similar it is easy to find cases where (3.6) holds but not (3.5).
2 , these two probabilities are not equal.
We now mention some interesting examples of 0-self-similar processes which do have the sampling property.
Example 3.4 Walsh's Brownian motions. Let B be a one-dimensional BM started at 0. Suppose that each excursion of B away from 0 is assigned a random angle in [0, 2π) according to some arbitrary distribution, independently of all other excursions. Let Θ t be the angle assigned to the excursion in progress at time t, with the convention Θ t = 0 if B t = 0. So (|B t |, Θ t ) is Walsh's singular Brownian motion in the plane [35,2]. As shown in [28,Section 4], the process (Θ t ) is a 0-self-similar process with the sampling property, and the same is true of (Θ t ) defined similarly for a δ-dimensional Bessel process instead of |B| for arbitrary 0 < δ < 2.
The proof of the sampling property of the angular part (Θ t ) of Walsh's Brownian motion is based on the following lemma, which is implicit in arguments of [28,Section 4] and [30, formula (24)].
Lemma 3.5 Let Z be a random closed subset of [0, 1] with Lebesgue measure zero. For 0 ≤ t ≤ 1 let N t − 1 be the number of component intervals of the set [0, t]\Z whose length exceeds t − G t , where G t = sup{s : s < t, s ∈ Z}. So N t has values in {1, 2, · · · , ∞}. Given Z, let (Θ t ) be a process constructed by assigning each complementary interval of Z an independent angle according to some arbitrary distribution on [0, 2π), and letting Θ t = 0 if t ∈ Z. If (N t ) has the sampling property, then so does (Θ t ).
According to [28,Theorem 1.2] and [30, formula (24)], for Z the zero set of a Brownian motion, or more generally the range of a stable(α) subordinator for 0 < α < 1, the process (N t ) has the sampling property, hence so does the angular part (Θ t ) of Walsh's Brownian motion whose radial part is a Bessel process of dimension δ for arbitrary 0 < δ < 2.
Example 3.6 A Dirichlet Distribution. Let Z be the set of points of a Poisson random measure on (0, ∞) with intensity measure θx −1 dx, x > 0. Construct (Θ t ) from Z as in Lemma 3.5. So between each pair of points of the Poisson process, an independent angle is assigned, with some common distribution H of angles on [0, 2π). It was shown in [30] that (N t ) derived from this Z has the sampling property, hence so does (Θ t ) derived from this Z. In this example µ Θ is a Dirichlet random measure governed by θH as studied in [14,18,16,33].
We close this section by rewriting Proposition 2.1 as a statement concerning stationary processes. Let (X t ) be a jointly measurable process and Y t = X e t . The process (X t ) being 0-self-similar is equivalent to the process (Y t ) being stationary, so a change of variables turns Proposition 2.1 into: Corollary 3.7 (Pitman-Yor [29]) Fix λ > 0 and define Y λ := ∞ 0 λe −λt Y t dt, where {Y t : t ∈ R} is a stationary process and E|Y 0 | < ∞. Then The following proposition provides a partial converse: Assuming either F has a density or F is a lattice distribution, the identity E(Y 0 | Y F ) = Y F holds for every such process {Y t } if and only if F has density λe −λt for some λ ∈ (0, ∞) or F = δ 0 .
we find that Taking Y to be an Ornstein-Uhlenbeck process shows that this holds for r(t) = e −αt , so that |ξ 1 − ξ 2 | has the same Laplace transform, hence the same distribution, as |ξ 1 |.
Assuming that F is concentrated on [0, ∞) and has a density, Puri and Rubin [31] showed that this condition implies F is an exponential. If F is a lattice distribution, they showed it must be δ 0 or 1 2 δ 0 + 1 2 δ a or a times a geometric for some a > 0. It is easy to construct examples ruling out the nondegenerate discrete cases.
Changing back to X t := Y log t , Proposition 3.8 yields:

Quadrants and the two-dimensional case
In this section we establish the following Proposition.
is not almost surely equal to µ(Q k ).
Fix the dimension d = 2 throughout, and denote by A ǫ the event that µ(Q 2 ) ∈ [ǫ, 2ǫ], µ(Q 3 ) ∈ [ǫ, 2ǫ], and µ(Q 4 ) ∈ [ǫ, 2ǫ]. Thus, if A ǫ occurs, then the Brownian motion W spends only a small amount of time in Q 2 , Q 3 , and Q 4 . The idea behind the proof is that if Brownian motion spends most of its time in Q 1 , then it is very unlikely to be in Q 3 at time 1, since Q 1 and Q 3 do not share a common boundary. More precisely, we will show that there is a constant C for which for sufficiently small ǫ > 0, which clearly implies Proposition 4.1. The estimate (4.1) follows immediately from the lower bound for P(A ǫ ) and the upper bound for P({W 1 ∈ Q 3 } ∩ A ǫ ) given in Lemmas 4.3 and 4.4 below. Proof: The first limit results from the fact that min t∈[0,1] B t has density 2φ(x) on (−∞, 0] where φ is the standard normal density of B 1 . The second follows easily from the reflection principle, which shows that the probability involved equals

Lemma 4.3
There exists a constant C 1 > 0 such that for sufficiently small ǫ > 0.

Lemma 4.4 There exists a constant
for sufficiently small ǫ > 0.
By splitting up the event {W 1 ∈ Q 3 } ∩ A ǫ according to the value of T δ , we see that if {W 1 ∈ Q 3 } ∩ A ǫ occurs, then either R 1 , R 2 , or R 3 must occur. We will prove the lemma by establishing upper bounds on P(R 1 ), P(R 2 ), and P(R 3 ).
A bound for P(R 2 ) follows from the observation that on A ǫ , there must be some t ∈ [1 − 6ǫ, 1] for which W t ∈ Q 1 . Thus on R 2 , one of the two coordinate processes has an oscillation of at least δ on the time interval [1 − 6ǫ, 1]. This implies that one of the coordinate processes strays by at least δ/2 from its starting value in the interval [1 − 6ǫ, 1], hence by the Markov property, By choice of C 4 > 12, this is o(ǫ 3 ).
A bound on P(R 1 ) may be obtained in a similar way. Observe that on A ǫ , there must be some t ∈ [T δ , T δ + 6ǫ] for which W t ∈ Q 1 . Thus one of the coordinates increases by at least δ from its starting value on the time interval [T δ , T δ + 6ǫ]. The strong Markov property yields As before, the choice of C 4 implies that P(R 1 ) = o(ǫ 3 ) and summing the upper bounds on P(R 1 ), P(R 2 ) and P(R 3 ) proves the lemma.
Proof of Proposition 4.1: The inequality (4.1), and the theorem, follow directly from Lemmas 4.3 and 4.4: for sufficiently small ǫ > 0
The following lemma contains the probabilistic content of the argument and is proved at the end of this section. Facts (2) and (3) are true when d = 2 as well, which is all that is needed to establish the remark after Theorem 1.2.
To construct ψ from Lemma 5.1 and fact (1), let C j be a finite cover of S d−1 by balls of radius 2 −j and let Cen(D) denote the center of the ball D. The set of limit points of a sequence {S n } of elements in cb-sets, defined by {x : lim inf n d(x, S n ) = 0}, is called the Hausdorff limsup, denoted lim sup n→∞ S n . Observe that if S n are cb-sets-valued random variables, then lim sup n→∞ S n is measurable as well. Then ψ := lim sup j→∞ A j satisfies (1.4): for almost every ω.
Remark: In fact, from the proof we see that ψ = lim A j almost surely when µ = µ(ω) and ω is chosen from P.
The construction of ϕ from here uses two further non-probabilistic lemmas. Proof: It suffices to show this when S is a subset of the unit ball. It will be convenient to have a nested sequence of sets GRID 1 ⊆ GRID 2 ⊆ · · · such that GRID j is 2 −j−1 -dense in the unit ball. (To construct this, inductively choose GRID j to be a maximal set with no two points within distance 2 −j−1 .) The sets BALLS j defined to be the set of balls of radius 2 −j centered at points of GRID j , form a sequence of covers of the unit ball such that each element of BALLS j+1 is contained in an element of BALLS j .
For each j and each S ∈ cb-sets let Let P j be the set of connected components of X j (S) viewed as subsets of BALLS j . In other words, P j (S) = {C ⊆ BALLS j : C is a component of X j (S)}. By the finiteness of BALLS j , we see that each P j is measurable. Since each D ⊆ X j (S) is contained in a ball D ′ ∈ BALLS j−1 also intersecting S, X j ⊆ X j−1 and hence each component of X j is contained in a unique component of X j−1 . This defines a map parent j : P j → P j−1 which is measurable since it depends only on P j and P j−1 . Letting P j,δ be the subset of P j consisting of components of diameter at least δ, it is clear that parent j maps P j,δ to P j−1,δ and that these are measurable.
Claim: N δ (S) is the cardinality of the inverse limit of the system {P j,δ , parent j : j ≥ 1}. Indeed, suppose that {x j for all j. Conversely, if x and y are points of S lying in different connected components, then S is contained in a disjoint union X ǫ ∪ Y ǫ for some sets X, Y with x ∈ X, y ∈ Y (where Z ǫ denotes the set of points within ǫ of the set Z). It follows that for each j there is an x j ∈ P j,δ with x ∈ x j , there is a y j ∈ P j,δ with y ∈ y j , and that for 2 −j < ǫ, x j = y j .
Finally, the cardinality of the inverse limit is easily seen to be measurable. Say x j ∈ P j,δ is a survivor if for each k > j there is some y k ∈ P k,δ with y k ⊆ x j . The set of survivors is clearly measurable, and the cardinality of the inverse limit is the increasing limit of the number of survivors in the set P j,δ as j → ∞.
The endpoint ω(1) will be recovered from ω[0, 1] as the only nonzero limit point of cutpoints, which is not a cutpoint itself. To justify measurability of this operation, the following definition and lemma are useful.
Definition 5.5 Let cut δ (S) denote the set of δ-cutpoints of S, that is, those x ∈ S such that S \ x has at least two components of diameter at least δ (note: if S is not connected this may be all of S). For each positive integer j and each δ > 0, define the measurable function A δ,j : cb-sets → cb-sets by Proof: Suppose first that x is a δ ′ -cutpoint of S. Let T and U be two components of S \ x of diameter at least δ. If D is a ball of radius ǫ < (δ ′ − δ)/2 containing x, then S \ D will have at least two components of diameter at least δ ′ − 2ǫ. Thus x ∈ A δ,j for 2 −j < ǫ, hence x ∈ A δ .
Suppose now that x ∈ A δ and let {D n } be balls converging to x in the Hausdorff metric, such that each intersects S and has N δ (S \ D n ) ≥ 2. Let {D ′ n } be balls with diameters going to zero such that ∞ j=n D j ⊆ D ′ n . Then N δ ′′ (S \ D ′ n ) ≥ 2 when n is large enough so that the diameter of D ′ n is at most δ − δ ′′ .
Claim: there are points x 1 , . . . , x k and an N 0 such that for n ≥ N 0 , each component of S \ D ′ n of diameter at least δ ′′ contains one of x 1 , . . . , x k .
decomposes into a countable set of intervals. At most k := ⌊1/ǫ⌋ of these intervals (u j , v j ), j = 1, . . . , k can have v − u ≥ ǫ, and these are the only ones containing times t with |f (t) − x| ≥ δ ′′ . Since S is connected, every component G of S \ D ′ n intersects ∂D ′ n , and if G has diameter at least δ ′′ then G must contain one of the k sojourns f (u j , v j ). Choose x j ∈ f (u j , v j ).
Since N δ ′′ (S \ D ′ n ) ≥ 2 for all n ≥ N 0 , there are i < j ≤ k such that infinitely many of the sets S \ D n have distinct components G n and H n of size at least δ ′′ containing x i and x j respectively. The increasing limits G n and H n must then be contained in distinct components of S \ {x}, showing that x ∈ cut δ ′′ (S).
Proof of Theorem 1.2 assuming Lemma 5.1: Clearly the sets A δ increase as δ → 0. Define We have shown that ψ • µ(ω) = ω[0, 1] almost surely with respect to P, and it follows from Lemma 5.6 that ϕ(S) ∪ {0} is the topological boundary of the set of cut-points of S. Fact (2) then implies that ω(1) ∈ ϕ(S). On the other hand, let x = ω(t) be any limit of cut-points, where 0 < t < 1. Thus there are times t j → t with ω(t j ) a cut-point. By fact (3), the sets ω(t j , 1) and ω(0, t j ) are disjoint, and each of them is connected. For t j > t, the set ω(t j , 1) is disjoint from ω(0, t) so if t j ↓ t, then ω(t, 1) is disjoint from ω(0, t). Likewise if t j ↑ t then ω(0, t) is disjoint from ω(t, 1), hence t is a cut-time. This shows that x / ∈ ϕ(S), so the only limits of cut-points that are not cutpoints are ω(0) and ω(1), which completes the proof.
To prove Lemma 5.1 we state several more lemmas. The cases d = 3 and d ≥ 4 differ slightly in that the estimates required for two-dimensional balls (d = 3) include logarithmic terms. Since recovery of the endpoint in dimension d ≥ 4 can be reduced to the three-dimensional case, and since the estimates for two-dimensional balls are strictly harder than for higher-dimensional balls, we assume for the remainder of the proof that d = 3. The formula for ρ D in this case is given by: We remark that when d > 3, the term log 2 r(D ′ ) is replaced by | log r(D ′ )| and the constant 2 in the denominator changes as well; this is due to the different normalization needed for "thick points" in dimension 3 and higher, see [11].
We begin by quoting two results from Dembo, Peres, Rosen and Zeitouni [12]. . Let Z t = t 0 1 D (W t ) dt be the occupation time of a standard two-dimensional Brownian motion up to time t in a ball D of radius r. Then for each t > 0 there is some λ > 0 not depending on r for which Ee λZt/(r 2 | log r|) < ∞. Consequently, P(Z t > Ar 2 log(1/r) < Ce −γA for some positive C and γ.
Proof: Dembo et al prove the result when the Brownian motion is started at radius r (in their notation r = r 1 = r 2 ) and the time t is instead the time to hit a ball of fixed radius r 3 = O(1). Accomodating these changes is trivial.
We now state three more lemmas which together imply Lemma 5.1.
Proof of Lemma 5.9: Covering D with small balls, it suffices to assume r(D) < δ and prove an upper bound of (1 + o(1)) times the RHS of (5.5) as δ → 0. Let β : S 2 → R 2 be a conformal map with Jacobian going to 1 near Cen(D). For example, take β to be stereographic projection from the antipode to Cen(D) to a plane (identified with R 2 ) tangent to S 2 at Cen(D). The path {π(W t ) : t ≥ ǫ} is a time-changed Brownian motion on S 2 , and in particular, π(W G(t) )) is a Brownian motion started from π(W 1 ), where G(t) is defined by 1 G(t) |W s | −2 ds = t. Similarly, (X t := β(π(W G(H(t)) )), t ∈ [0, M := H −1 (G −1 (ǫ))]) is a Brownian motion in R 2 , where M is random and H(t) is another time change, with |H ′ | going to 1 uniformly as r(D) → 0 and π(W G(H(t)) ) is in D.
With probability 1, W t is a single value, i.e., W t = W s for t = s, in which case for n sufficiently large, X s ∈ D n implies |W G(H(s)) | → |W t | and G(H(s)) → t. The sets β −1 (D n ) are contained in balls D ′ n with r(D ′ n )/r(D n ) → 1, so M 0 1 D ′ n (π(W G(H(s)) ) ds r(D ′ n ) 2 log 2 r(D ′ n ) → 2.