ELECTRONIC COMMUNICATIONS

Let B be a two dimensional Brownian motion and let the frontier of B[0, 1] be defined as the set of all points in B[0, 1] that are in the closure of the unbounded connected component of its complement. We prove that the Hausdorff dimension of the frontier equals 2(1 − α) where α is an exponent for Brownian motion called the two-sided disconnection exponent. In particular, using an estimate on α due to Werner, the Hausdorff dimension is greater than 1.015.


Introduction
Let B(t) be a Brownian motion taking values in IR 2 which we also consider as I C. Let B[0, 1] be the image of [0, 1].For any compact A ⊂ I C we define the frontier of A, fr(A) to be the set of points in A connected to infinity.More precisely, fr(A) is the set of x ∈ A such that x is in the closure of the unbounded connected component of I C \ A. Take a typical point x ∈ fr(B[0, 1]).Then locally at x the Brownian motion looks like two independent Brownian motions starting at x with the condition that x is not disconnected from infinity.In this paper, we use this idea to prove that the Hausdorff dimension of fr(B[0, 1]) is given in terms of a certain exponent for Brownian motion.Let B1 , B 2 be independent Brownian motions starting on the unit circle, and let Let D n be the event that 0 is contained in the unbounded connected component of I where IP x 1 ,x 2 denotes probabilities assuming B 1 (0) = x 1 , B 2 (0) = x 2 , and the supremum is over all |x 1 | = |x 2 | = 1.It is easy to see from the strong Markov property and Brownian Electronic Communications in Probability scaling that for all n, m ≥ 1, q(nm) ≤ q(n)q(m).
Standard techniques, using the subadditivity of the function f(k) = log q(2 k ), can be used to show that there exists an α > 0 such that as n → ∞, where ≈ denotes that the logarithms of boths sides are asymptotic.Moreover, q(n) ≥ n −2α for all n.We call α the two-sided disconnection exponent.(Sometimes 2α is called the two-sided disconnection exponent.)The value of α is not known.One can give an easy bound of α ≤ 1/2, by noting that with probability at least cn −1/2 , a Brownian motion in I C starting at 1 reaches the circle of radius n without hitting the negative real axis (see (7)).If B 1 and B 2 both reach the circle of radius n without hitting the negative real axis, then 0 is connected to infinity in the complement of Werner [9,10] has recently shown that 2α ≤ .985.A lower bound, α ≥ (2π 2 +1)/(8π 2 ), was given by Burdzy and Lawler [2].There is an interesting conjecture due to Mandelbrot [7] that α = 1/3, and recent simulations of Puckette and Werner [8] are consistent with this conjecture.Mandelbrot's conjecture was really about the Hausdorff dimension of the frontier of Brownian motion, but as we see from the main theorem of this paper the two quantities are related.We let dim h (A) denote the Hausdorff dimension of the set A.
where α is the two-sided disconnection exponent.Using Werner's estimate, we get an immediate corollary, dim h (fr(B[0, 1])) > 1.015. (2) Bishop, Jones, Pemantle, and Peres [1] have recently given a different argument to show that but their methods are not sufficient to prove (2).In the course of our proof we will rederive the estimate α < 1/2, so (3) can be concluded from this paper alone.The main technical tool in proving the theorem is to improve the estimate in (1).We show that where denotes that both sides are bounded by a constant times the other side.Using this estimate (and some slight generalizations) we can compute the Hausdorff dimension by covering the set of "frontier times" by approximate "frontier intervals" of size 2 −n and then letting n → ∞.The method is similar to that in [6] where the Hausdorff dimension of the set of cut points of a Brownian path is computed.In the next section we give the main proof assuming the key estimate (4).The estimate and generalizations are derived in the last section.This work was done while the author was visiting the University of British Columbia.I would like to thank Chris Burdzy for the argument at the end of Section 2, and I would like to thank the referee for a careful reading and suggestions for improvement of this paper.Let L be the set of frontier times for B, It is well known [5] that with probability one Brownian motion doubles the Hausdorff dimension of sets.Hence it suffices to prove that with probability one Throughout this paper we use B(x, r) to denote the closed disc of radius r about x.
For any compact A ⊂ I C we will write Q(A) for the unbounded connected component of I C \ A and Q(A) for the closure of Q(A).Note that A 1 ⊂ A 2 implies Q(A 1 ) ⊃ Q(A 2 ).For every n, let I(j, n) = [(j − 1)/2 n , j/2 n ].We will say that I(j, n) is an approximate frontier interval if Let K j,n be the indicator of the event "I(j, n) is an approximate frontier interval" and let Proof: It is easy to see that with probability one there exists an n such that 0, 1 It is easy to see that this implies that But by continuity of B, for all n sufficiently large, where j = j t is chosen with j − 1 < t2 n ≤ j.Hence I(j, n) and I(j + 1, n) are not frontier intervals and hence t ∈ L n .
The next lemma will be proved in the next section (see Lemma 3.15).In this paper we use c, c 1 , c 2 for positive constants; the exact values of c, c 1 , c 2 may vary from line to line.
Lemma 2.2 There exist c 1 , c 2 such that for and Let µ n be the (random) measure on [0, 1] whose density with respect to Lebesgue measure is 2 αn on every interval I(j, n) with 2 n−2 < j ≤ 3 • 2 n−2 and K j,n = 1.On all other intervals the density is zero.Note that µ n is supported on L n ∩ [1/4, 3/4] and hence any weak limit of the and hence by the Borel-Cantelli Lemma, By standard arguments this implies that dim h (L ∩ [1/4, 3/4]) ≤ β with probability one and since this holds for all β > 1 − α, To get the lower bound we first note that Lemma 2.2 implies Hence by standard arguments, there exists a c 3 > 0 such that for all n, Let I β (µ) denote the β-energy of a measure µ on [0, 1], i.e., Here the sums are over all 2 n−2 < j, k ≤ 3 • 2 n−2 and u β is a positive constant, depending on β, whose value may change from line to line.In particular, Therefore, using (5), On the event above, let µ be any weak limit of the µ n .Then it is easy to verify that µ is supported on L ∩ [1/4, 3/4]; µ(L) ≥ c 3 ; and I β (µ) ≤ 2u β /c 3 .By standard arguments (see [4,Theorem 4.13]) this implies that with probability at least Since this is true for every β < 1 − α, we can conclude There is nothing special about the interval [1/4, 3/4].By a similar argument we can show that for every 0 ≤ a < b ≤ 1, there exists a u = u(a, b) > 0 such that We can adapt the proof above to prove a slightly different result.Let Kj,n be the indicator function of the event "I(j, n) is an approximate frontier interval; B(I(j, n)) ⊂ {1 ≤ (z) ≤ 2}; (B(1)) ≥ 3".Similarly to the proof of Lemma 2.2 we can show that for Hence if Jn = By using the same argument as above, we can then show that To prove the main theorem, we have to show that u(0, 1) = 1.When searching for cutpoints for Brownian motion [6], a similar problem arose.An argument similar to that above showed that for every interval, with positive probability the Hausdorff dimension of the set of "cut times" in that interval was given by a certain value, 1 − ζ.In order to show that this result holds with probability one, it was shown that there were cut times arbitrarily close to t = 0 and then a zero-one law could be used.This idea will not work in the case of the frontier of Brownian motion because B(0) is not a point on the frontier.However, by concentrating on a different point, the point at which the imaginary part of B t is a minimum, we can show that the dimension of L is 1 − α with probability one.We will only sketch the argument which is due to K. Burdzy.(It is not too difficult to give the complete details, but this is such an "obvious" fact that it should not take up too much of this paper.)Write B t = X t + iY t where X t , Y t are independent one-dimensional Brownian motions.Let It is easy to check that with probability one there is a unique (random) time σ ∈ (0, 1) with It suffices to prove that Let W, Z be independent h-processes in the upper half plane starting at 0, i.e., W, Z are independent Brownian motions "conditioned to have positive imaginary part for all positive times."It is intuitively clear that near B σ , the Brownian motion B looks like two independent h-processes conditioned to have imaginary part greater than R. Burdzy and San Martin made this notion precise (see [3, Lemma 2.1]) and showed that any "local" event at B σ has the same probability as the corresponding local event for W, Z.Let E n be the event that Then by the result of Burdzy and San Martin, It is not hard to show using 0-1 Laws that Let τ W n be the first time that the imaginary part of W is at least 2 −n and similarly define τ Z n .Using (6), one can show that there is a c > 0 such that for each n By the strong Markov property and the transience of the h-processes this implies that there is a c > 0 such that (We have extended the definition of frontier in a natural way to include some noncompact A.) In particular,

Brownian motion estimates
Let B 1 , B 2 be independent Brownian motions taking values in IR 2 = I C starting at For the first part of this section, we will also assume an initial configuration is given.An initial configuration will be an ordered pair Γ = (Γ 1 , Γ 2 ) of closed subsets of {z : 0 < |z| ≤ 1} such that Γ j ∩ {z : |z| = 1} = {x j }, j = 1, 2.
We set T j 1 = 0; for n > 1 we set, as before, and we let For each n ≥ 1, let Θ j (n) be the argument of B j (T j n ).We will consider arguments modulo 2π, i.e., 0 and 2π are the same argument.Let where the supremum is over all |x 1 | = |x 2 | = 1 and the initial configuration is given by Γ j = {x j }.By Brownian scaling and the strong Markov property, q(nm) ≤ q(n)q(m), and hence by standard subadditivity arguments there exists an α such that as n → ∞, Here ≈ denotes that the logarithms of both sides are asymptotic.Moreover, q(n) ≥ n −2α .We know (see Section 1) that α ∈ (0, 1).
Let Y n = Y n ( Γ) be the supremum of all > 0 such that Note that {Y n > 0} = D n .In the proofs we will use a standard estimate about Brownian motion.Let The θ = π/2 result is the standard "gambler's ruin" estimate for Brownian motion, and the estimate for other θ can be derived by conformal mapping.
Lemma 3.1 There exists a constant c 1 > 0 such that if Γ is any initial configuration with Y 1 > 0, then Proof: If Y 1 > 1/10 then it is easy to prove the lemma by direct construction of an event so we assume Y 1 = r ≤ 1/10.Without loss of generality we will assume that 0 = Θ 1 (1) ≤ Θ 2 (1) = θ ≤ π.Consider the events . By standard estimates (using conformal mapping) it can be shown that It is easy to see that U 1 ∩ U 2 ⊂ {Y 2 ≥ 1/4}, and hence we get the result.
The next lemma is a very important technical lemma.It states in a uniform way that two Brownian motions conditioned not to disconnect the origin from infinity have a reasonable chance of being "not too close to disconnecting."Let F s be the σ-algebra generated by {B j (t) : 0 ≤ t ≤ T j s , j = 1, 2}.
Lemma 3.2 There exists a constant c 1 > 0 such that if Γ is any initial configuration with Proof: For every 3/2 ≤ ρ ≤ 2, let V (ρ) be the event For any > 0, it is easy to see by direct construction of an event (see, e.g., the proof of Lemma 3.1) that there is a u > 0 such that for any initial configuration with Y 1 ≥ , )} ≥ u .
Choose integer N sufficiently large so that exp[ Let h N = 3/2 and for n > N, let where the infimum is over all initial configurations with Y 1 ≥ 2 −n .By the comment above, r(n) > 0 for each fixed n.We will show below that there is a c 2 > 0 such that for all n > N, and hence there exists a c 1 > 0 such that r(n) ≥ c 1 for all n.This clearly gives the lemma.By the strong Markov property, Brownian scaling, and the definition of r(n), we can see that Choose n > N, and let Γ be an initial configuration with Y 1 ≥ 2 −n .By Lemma 3.1, Let s j = s j,n = exp{j2 −n } and let and hence If we can show that we will have derived (8).
Let β > 0 be the probability that a Brownian motion starting at the origin, stopped upon reaching the sphere of radius 2, disconnects the ball of radius 1 from the sphere of radius 2.
Then, for n sufficiently large, we can see that if Γ is any initial configuration with Likewise, by scaling we can see that for all j < n 2 , Iterating this, we can conclude that Combining this with (10), we can conclude (11).
By applying this lemma to the configuration Γ n−1 , we see (using Brownian scaling) that for n ≥ 2, and any initial configuration, and hence Since paths with Y n ≥ 1/4 can be extended with positive probabilty to distance 2n without producing a disconnection, we get the following corollary.
For the remainder of this paper, we will not need to consider any initial configurations.Let B 1 , B 2 be independent Brownian motions starting on the unit circle.For any t 1 , t 2 (perhaps random), let J(t 1 , t 2 ) denote the set of θ such that there exists a continuous curve γ : [0, ∞) → I C satisfying: We consider J(t 1 , t 2 ) as a subset of the unit circle.It is easy to verify that J(t where l denotes Lebesgue measure on the unit circle.Note that D n = {X n > 0}.It seems intuitively clear that if D n holds then there should be a reasonable chance that J(T 1 n , T 2 n ) is not very small.The next lemma gives a rigorous statement of this.Lemma 3.4 There exists a δ > 0 such that for all n ≥ 1, where the supremum is over all By the strong Markov property, we get (for n > 2), if δ is chosen sufficiently small.But it easy to verify that Lemma 3.5 Let δ be as in Lemma 3.4.Let V j n = V j n ( ) be the event Then there exists a c > 0 such that for each n > 1, > 0 where the supremum is over all Proof: We will first show that there exists a c such that for each n, there exist It is easy to check that there is a u > 0 (independent of ) such that If a Brownian motion is at distance 1 − , there is a probability of at least c that it will hit the circle of radius 1/2 before hitting the circle of radius 1. Starting on the ball of radius 1/2 there is a positive probability that the Brownian motion will make a closed loop about the disc of radius 1/2 before leaving the circle of radius 1.From all this we see that the probability that B 1 [0, τ] disconnects 0 from infinity given that ρ < T 1 n is greater than c .If B 1 [0, τ] does not disconnect 0 from infinity, the strong Markov property says that the probability that X n ≥ δ is at most p(n).Therefore and hence {V 1 n ; X n ≥ δ} and do the same argument on B 2 .
Let δ be as in Lemma 3.4 and fix = δ/20.For any λ > 0, |x| = 1, let W j (λ, x) be the event that B j [0, T j 2 ] disconnects B(x, λ) from the sphere of radius 2. Let where |x 1 | = |x 2 | = 1.Note that the probability is indepedent of the x 1 , x 2 chosen and p(λ) → 1 as λ → 0. By the strong Markov property and Corollary 3.3, , by choosing λ sufficiently small we can conclude the following from Lemma 3.5.
Corollary 3.6 Let δ, be as above and V j n = V j n ( ) as in Lemma 3.5.Let W j (x) = W j (λ, x) be as above.There exist λ > 0 and c > 0 such that where the supremum is over all |x 1 | = |x 2 | = 1.
We now fix a λ that satisfies Corollary 3.6.Suppose we start the Brownian motions at y j with |y j − x j | ≤ λ/2.To determine whether or not the event occurs, we need only view the paths after the first time they hit the sphere of radius λ about x j .Hence, using either the exact form of the Poisson kernel or the Harnack inequality for harmonic functions, we can see for any |y 1 | = |y 2 | = 1 with |y j − x j | ≤ λ/2, Hence we can conclude the following.
Corollary 3.7 Let δ, be as above and V j n = V j n ( ) as in Lemma 3.5.Let W j (x j ) be as above.There exist λ > 0 and c > 0 such that sup

2
Main proof Let B(t) be a Brownian motion taking values in I C. Let Λ = fr(B[0, 1]).
1 , t 2 ) is one of: empty, an open interval of the unit circle, or the union of two disjoint open invervals.Moreover, if J(t 1 , t 2