On the most visited sites of planar Brownian motion

: Let ( B t : t ≥ 0) be a planar Brownian motion and deﬁne gauge functions φ α ( s ) = log(1 /s ) − α for α > 0. If α < 1 we show that almost surely there exists a point x in the plane such that H φ α ( { t ≥ 0: B t = x } ) > 0, but if α > 1 almost surely H φ α ( { t ≥ 0: B t = x } ) = 0 simultaneously for all x ∈ R 2 . This resolves a longstanding open problem posed by S. J. Taylor in 1986.


Introduction and statement of main results
Let (B t : t ≥ 0) be a standard planar Brownian motion.Dvoretzky, Erdős and Kakutani (1958) first showed that, almost surely, there exist points x in the plane such that {t ≥ 0 : B t = x}, the set of times where the Brownian path visits x, is uncountably infinite.Modern proofs of this fact are given in Le Gall (1987) and Mörters and Peres (2010).The result naturally raises the question: How large can the sets {t ≥ 0 : B t = x} be?Kaufman's famous dimension-doubling theorem implies that, almost surely, for all points x in the plane this set has Hausdorff dimension zero.S. James Taylor in his influential survey paper 'The measure theory of random fractals' of 1986 raises the problem in terms of Hausdorff gauge functions.Letting φ : (0, ε) → [0, 1] be a right continuous, increasing function with φ(0+) = 0, we denote the φ-Hausdorff measure of the set E ⊂ R. Problem 5 in Taylor (1986) is the following question: Which gauge functions φ are such that, almost surely, we have H φ ({t ≥ 0 : B t = x}) = 0 for all x ∈ R 2 ?
In the paragraph following this question Taylor focuses on the gauge functions of the form φ α (s) = log(1/s) −α for α > 0, moving to a weaker but still challenging form of the problem.He notes that the results of Perkins and Taylor (1987) imply that these functions satisfy the above condition for α > 2 and states that It is not much more than a guess, but my hunch is that φ α satisfies the condition for α > 1, but not for 0 < α < 1.
The weaker form of the problem is reiterated in Perkins and Taylor (1987) as Problem 3.11.More than fifteen years later Xiao (2004) notes that the problem is still open even in the weaker form.The problem has recently been reiterated as Problem 5 in the open problems section of Mörters and Peres (2010).It is the aim of the present paper to solve the weaker form of the problem and to confirm Taylor's conjecture.
Although the solution involves both a new lower and upper bound, only the lower bound involves substantial work.It relies on the construction of an intersection local time for points of infinite multiplicity, which is due to Bass, Burdzy and Khoshnevisan (1994).Denoting by B(x, ε) ⊂ R 2 the open disc with centre x and radius ε and by B the open disc B(0, 1) we let N x ε be the number of times the Brownian path (B t : t ≥ 0) travels from the point x ∈ B to the circle ∂B(x, ε) before it leaves B for the first time.Naturally for most points x we have N x ε = 0, but some points satisfy for some a > 0. If (1.1) holds we say that Brownian motion 'spends a units of local time at x'.Note that the points x are by definition points of infinite multiplicity.Bass, Burdzy and Khoshnevisan (1994) in their Theorem 1.1 show that, for 0 < a < 1 2 there exists a (random) measure β a on the plane such that (1.1) holds for β a -almost every x.Moreover the measure is nonzero and has carrying Hausdorff dimension 2 − a almost surely.
Our main result shows that the points selected according to β a also have a large inverse image under the Brownian motion, thus providing a lower bound for Taylor's Problem 5.
Then, almost surely, The proof of Theorem 1.1 is given in Section 2. To solve the weaker form of Taylor's Problem 5 we also need an upper bound confirming that the bound above has the right power of the logarithm.
Theorem 1.2.For every gauge function φ with φ(ε) log(1/ε) → 0, almost surely, The proof of Theorem 1.2, which uses a simple first moment estimate, is given in Section 3. Combining Theorems 1.1 and 1.2 confirms Taylor's conjecture on the weaker form of Problem 5 as stated in the abstract.The strong form of the problem remains open at this point, see Section 4 for a new conjecture and some further remarks on this.

Proof of the lower bound
Let P z and E z be the distribution and the corresponding expectation of planar Brownian motion started at z ∈ R 2 , and let τ (A) be the first hitting time of a Borel set A ⊂ B. In particular we let τ = τ (∂B) be the first hitting time of the unit disc and denote by p B t (• , • ) the transition (sub-)density for the Brownian motion killed at τ .Let B = (B t : 0 ≤ t ≤ τ ) be the Brownian motion started at the origin and killed upon leaving B. The idea of the proof of Theorem 1.1 is to use the representation of B as seen from a typical point chosen accordingly to β a , which is given in Section 5 of Bass et al. (1994).The main ingredient of this representation is the construction of the process (Z x a (t) : 0 ≤ t ≤ τ ) which we now recall.
Fix some x ∈ B and let h be a strictly positive harmonic function in B \ {x} with zero boundary values on ∂B and a pole at x such that The h-transform of B is a Markov process in B \ {x} with transition density p h t given by The distribution of this process started in y ∈ B \ {x} is denoted P y h .Let C * [0, ∞) be the set of all paths e : [0, ∞) → B ∪ {∆} which are continuous on some interval [0, σ) and then jump to the isolated coffin state ∆.The canonical process on C * [0, ∞) will be denoted by X, i.e.X t (e) = e t for all e ∈ C * [0, ∞) and t ≥ 0. There is an, up to a constant factor unique, positive and σ-finite measure • H x is strong Markov for the transition densities p h t (• , • ), The excursion law H x will be normalised so that The trajectories of Z x a are assembled from three parts, ) is an h-process in B \ {x} which starts from 0 and is stopped when it approaches x at time t 1 .
By (2.4), T (u) is well defined for all u < ∞ almost surely.Note that, almost surely, T (u) < ∞ for each u and, for almost all u, there is a point (s, e s ) ∈ Y such that s = T (u).For such u we define and for the remaining u we define Z 2 (u) = x.Let t 2 = s<1 σ(e s ) and observe that this defines a continuous process (Z 2 (t) : 0 ≤ t ≤ t 2 ).
(iii) (Z 3 (t) : 0 ≤ t ≤ t 3 ) is a Brownian motion starting from x and killed at the first exit from the unit ball B.
We assume that Z 1 , Z 2 and Z 3 are independent.The distribution of the process (Z x a (t) : 0 ≤ t ≤ τ ) will be denoted by Q x a .The process Z x a may be thought of as a Brownian motion conditioned to spend a units of local time at x.It is possible to interpret Q x a as the distribution of B conditioned by the event that x is in the support of β a .This follows from the 'Palm measure' decomposition of β a stated below and proved in Bass et al. (1994), Theorem 5.2.
Lemma 2.1.For every a ∈ (0, 1 2 ) and every nonnegative measurable function Let L x a be the right-continuous generalised inverse of the sum of the excursion lengths of Z x a from x, L x a (t) := sup θ : L x a is the local time (in the sense of excursion theory) of Z x a at x.The following proposition is the main ingredient of the proof of the lower bound.
Before proving Proposition 2.2, we show how this implies Theorem 1.1.
First, the natural link between the Hausdorff measure H ϕ {t ≥ 0 : B t = x} and the law of the iterated logarithm in (2.5) is the Rogers-Taylor theorem stated below, see Mörters and Peres (2010), Proposition 6.44, for a proof.
Lemma 2.3.Let µ be a Borel measure on R and let φ be a Hausdorff gauge function.If Λ ⊂ R is a closed set and In our case we consider the closed set Λ x := {t ≥ 0 : Z x a (t) = x}, the (probability) measure ℓ x a given by ℓ x a [t, t + ε] := L x a (t + ε) − L x a (t), the gauge function ϕ and the sets Then, by Lemma 2.3, we have ), and we now show that, Q x a -almost surely, ℓ x a (A α ) = 1 for suitable α > 0.
For 0 ≤ t ≤ 1 let T t := inf{s > 0 : L x a (s) ≥ t} < ∞.By construction the process (Z x a (T t + s) : 0 ≤ s ≤ T 1 − T t ) has the same law as (Z x a (t 1 + s) : 0 ≤ s ≤ T 1−t ).Hence it follows from Proposition 2.2 that, for all t ∈ (0, 1), By applying Fubini's theorem we get Leb t ∈ (0, 1) : lim sup Hence ℓ x a (A α ) = 1 for all α > a −1 2/π, and so To complete the proof of the lower bound we apply Lemma 2.1 to the nonnegative, mea- and this implies that that is H ϕ (Λ x ) ≥ a π/2 for β a -almost every x, P 0 -almost surely, as required to prove Theorem 1.1 subject to the proof of Proposition 2.2.
Lemma 2.5.We have where ς(ǫ) is the first hitting time of ∂B(x, ǫ).By the strong Markov property of the excursion measure, and in view of (2.1) and (2.3), we have Analogously, we get The following lemma, see Port and Stone (1978), Proposition 4.1, will be used to replace p B θ by the Brownian transition kernel in the main part of the integral.Lemma 2.6.Let U be an open set and x ∈ U .Then there is a δ 0 > 0 such that lim t↓0 p U t (ζ, ξ)/p t (ζ, ξ) = 1 uniformly for all ζ, ξ ∈ B(x, δ 0 ).
Proof.For any small δ > 0 we have since the integral over the complement of B(x, δ) is bounded.In view of (2.1) and Lemma 2.6, for every ε > 0, we find a small δ > 0 and d = d(δ) > 0 such that, for θ < d, and, changing variables s 2 = r 2 /θ, we get Similarly, for the lower bound we have which completes the proof.
We can now use Lemma 2.4 and get (2.5) for the gauge function

Proof of the upper bound
Fix a ball B(0, R), R > 1, and stop Brownian motion at the first exit time τ from this ball.Given a cube Q of side length r inside this ball, we define recursively Lemma 3.1.There exists 0 < θ < 1 such that for any z ∈ B(0, R) and 0 < r < 1 2 , , where x is the centre of Q.The second factor can be bounded from below by a constant not depending on r.The first factor is bounded from below by the probability that planar Brownian motion started at any point in ∂B(0, 2 √ r) hits ∂B(0, 2R) before ∂B(0, r).This probability is given by log which is bounded from zero by a positive constant independent of r.
Lemma 3.2.Let C m be the set of dyadic cubes of side length 2 −m inside a fixed unit cube U ⊂ B(0, R).Almost surely there exists a (random) integer C so that for all m ≥ 1 and cubes Q ∈ C m and r = 2 −m we have τ Q mC > τ .
Proof.From Lemma 3.1 we get that, for any positive integer c, Now choose c so large that 4θ c < 1.Then, by the Borel-Cantelli lemma, for all but finitely many m we have τ Q cm ≥ τ for all Q ∈ C m .Finally, we can choose a random C > c to handle the finitely many exceptional cubes.
To complete the proof we note that, on the event in the lemma, for a given m we can cover any set {0 < t < τ : B t ∈ Q}, Q ∈ C m , with no more than Cm intervals of length r = 2 −m .This implies that, for any z ∈ U , H φ {0 < t < τ : B t = z} ≤ lim m→∞ Cmφ(2 −m ) = 0, under the assumption on φ.Theorem 1.2 follows as U ⊂ B(0, R) and R > 1 were arbitrary.

Outlook
As mentioned in the introduction, the strong form of Taylor's Problem 5 remains unresolved in this paper.Our work however allows us to make a conjecture.We believe that our lower bound is sharp, or in other words that for every gauge function φ with φ(ε)/ϕ(ε) → 0 where ϕ is as in Theorem 1.2, almost surely, H φ {t ≥ 0 : B t = x} = 0 for every x ∈ R 2 .
The reason for this belief is that our upper bound uses coverings of the level sets by intervals of fixed size.If we were able to adapt the size of the covering intervals to the fluctuations of the local time, one would expect a gain of order log log log(1/ε), similar to our lower bound.However, the technical difficulties in carrying out such an estimate appear to be considerable and we therefore defer verification of our conjecture to future work.