The Exit Place of Brownian Motion in the Complement of a Horn

Consider the domain lying outside a horn. We determine asymptotics of the logarithm of the chance that Brownian motion in the domain has a large exit place. For a certain class of horns, the behavior is given explicitly in terms of the geometry of the domain. We show that for some horns the behavior depends on the dimension, whereas for other horns, it does not. Analytically, the result is equivalent to estimating the harmonic measure of the part of the domain lying outside a cylinder with large diameter.


Introduction
For an open set D ⊆ R d , let τ D be the first exit time of d-dimensional Brownian motion B t from D. It is well-known that the exit place of two-dimensional Brownian motion from the upper half plane U has a Cauchy distribution (Durrett (1984)). Thus the chance that the exit place is large decays as the reciprocal of the distance to the origin: P x (|B τ U | > r) ∼ C(x)r −1 as r → ∞.
In particular, |B τ U | has a moment of order p > 0 iff p < 1.
In addition to this, we showed there is an analogous result for more general cones. Since the result will be used below, we state it explicitly. An open set G ⊆ R d is a cone if for each c > 0, cG = G. If G is a cone and G ∩ S d−1 is C 3 , then we showed that where p(d, G) = −(d − 2) + (d − 2) 2 + 4γ 1 2 and γ 1 is the smallest positive Dirichlet eigenvalue of the Laplace-Beltrami operator ∆ S d−1 on G ∩ S d−1 .
It is natural to ask how the geometry of other unbounded domains will influence the chance of a large exit position. Note that an equivalent analytic formulation is to determine how the harmonic measure of the part of the domain outside a large ball depends on the geometry. For example, Bañuelos and Carroll (2005) used Carleman-type estimates and a conformal technique to study the parabolic-type domain where 0 < α < 1 and A > 0. (Note that α = 1 corresponds to the axially symmetric cone C described above). One of their principal results was that the chance of a large exit place is subexponential: where λ 1 is the smallest positive eigenvalue of the Dirichlet Laplacian on the unit ball of R d . Thus the chance depends on both the dimension and the geometry of the domain. Note when d = 1, √ λ 1 = π 2 . The authors actually derived more refined results; please see that paper for the details.
With this in mind, when d = 2, Ω is obtained by revolving the region {(ρ, z) : ρ > 1, |z| < Aρ α } about the z-axis. In contrast, the parabolic-type region studied by Bañuelos and Carroll would be obtained by revolving about the z-axis.
Since the (d + 1)-dimensional Laplacian in cylindrical coordinates is given by a further advantage of the representation (3) is that by symmetry, the chance of a large exit place |B(τ Ω D )| of (d + 1)-dimensional Brownian motion from Ω D is the same as the chance of a large exit place of the diffusion corresponding to the operator 1 2 from D. We will make use of this fact below.
The domain Ω is a special case of the following more general domain studied by several authors: For example, under some regularity conditions on the function a(·), Ioffe and Pinsky (1994) completely identified the positive harmonic functions in Ω a vanishing continuously at the boundary. Aikawa and Murata (1996), Murata (2002) and Murata (2005) weakened the conditions on a(·) and also extended the result to nonsymmetric domains where the condition |x d+1 | < a(|x|) is replaced by b(|x|) < |x d+1 | < a(|x|). Collet et al (2006) studied ratio limit theorems for the Dirichlet heat kernel in Ω a with a(t) = √ t; they also found the chance of a long lifetime of Brownian motion in that domain. Using another method, DeBlassie (2007) found the chance of a long lifetime of Brownian motion for a(t) = t α , 0 < α < 1. Although all these authors call Ω a a horn-shaped domain, as we saw above in three dimensions, it is better to call it the complement of a horn-shaped domain to conform to more common usage in the literature. Now we dispense with the case α = 1 in the following example.
where A > 0, and define Since we are interested in the chance of a large exit place |B(τ Ω D )| of (d + 1)-dimensional Brownian motion from Ω D , it is intuitively clear that modifying D within a ball will have little effect on the chance of a large exit place (this idea is made precise below in the proof of our main result for the case 0 < α < 1; we omit the details for the example). Thus Ω D is more or less the case of Ω from (2) with α = 1.
On the other hand, since cΩ D = Ω D for any c > 0, Ω D is a generalized cone in R d+1 and so the analogue of the formula (1) holds: where Below in section 6 we will show that p(d) is not independent of d. Thus the limit in (5) is not independent of the dimension.
Our main result is the following theorem.
In light of the example above, the fact that the limit is independent of the dimension is interesting and surprising. The reason for the independence is that there is a delicate balance of competing effects between the geometry of the domain and the transience of the process. More precisely, recall that the problem reduces to the study of the diffusion corresponding to the operator (4) in the region The Bessel-type drift d−1 ρ in (4) pushes the process away from the boundary and tends to increase the chance of a large exit place. Although the drift is small when the process is far away, when α = 1, it is not negligible as indicated by the dimensional dependence of the limit (5). As an aside, remember that a Bessel-type drift added to a one-dimensional Brownian motion cannot be neglected even when the process is is far away: the drift can completely change the recurrence/transience of the process.
On the other hand, Theorem 1.1 tells us that in the case when α < 1, the effect of the drift is negligible. The key is that the α < 1 domain expands more slowly than the α = 1 domain. Indeed, because of this difference in expansion, the vertical component of the process tends to force a quicker exit, hence a smaller exit place, from the α < 1 domain. The effect is to reduce the chance of a large exit place. Thus there are two competing effects on the chance of a large exit place. When α < 1 the tendency to reduce the chance of a large exit place completely overwhelms the tendency of the drift to increase the chance and so we get dimensional independence of the limit in Theorem 1.1. It is interesting to note there is a "phase transition" from power law decay to subexponential decay in the asymptotic behavior of the chance of a large exit place as the growth of the domain passes from linear to sublinear.
The organization of the article follows. In Section 2 we state some preliminaries and reduce the problem to two dimensions. A Feynman-Kac representation of an h-transform is given in Section 3. In Section 4 we derive an upper bound for d ≥ 3 and a lower bound for d = 2, both using an h-transform. In Section 5 we use results of Carroll and Hayman on a conformal transformation to a strip together with an h-transform to derive a lower bound for d ≥ 3 and an upper bound for d = 2. In Section 6 we prove the dimensional dependence of the limit in (5).

Preliminaries
Suppose G ⊆ R n is open and b : G → R n is bounded and measurable. When G = R n , by the diffusion associated with 1 2 ∆ R n + b · ∇ on G we will mean the diffusion associated with the corresponding martingale problem, which is well-posed in this context (Stroock and Varadhan (1979)). When G = R n , extend b to be bounded and measurable on all of R n and let X be the diffusion associated with the extended operator on R n . The diffusion associated with 1 2 ∆ R n +b·∇ on G is then defined to be X killed upon exiting G.
denote the cylindrical coordinates of x and in these coordinates, and denote by X the diffusion associated with L on D.
By symmetry, for x = (x, x d+1 ) ∈ Ω, the function P x (|B(τ Ω )| > N ) is independent of the angular coordinate θ =x/|x|. Consequently if Thus Theorem 1.1 is an immediate consequence of the following theorem.
The next reduction is analogous to a reduction in Bañuelos and Carroll (2005). It shows that the heart of the matter is in the behavior of the first coordinate of the exit place.
Lemma 2.2. For large N , Thus As for the other inequality, choose is the first coordinate of the intersection of the curve z = Aρ α with the circle z 2 + ρ 2 = N 2 and so it is easy to see In particular, if |X(τ D )| > N , then X τ D is outside the circle ρ 2 + z 2 = N 2 and so Combining Lemma 2.2 with the next result, we immediately obtain Theorem 2.1.
The rest of the article, except for the last section, will be devoted to proving Theorem 2.3.
We will need the following theorem due to Bañuelos and Carroll (2005).
We close this section with a result needed to obtain the lim sup behavior in the case d = 2. Roughly speaking, it says that the part of D inside a bounded set does not have too much influence on the exit place being far away.
Let Γ be a non-intersecting Lipschitz curve, lying in D except for its endpoints, which lie on ∂D. Then Γ divides D into two components; let D Γ be the unbounded one.
Let γ be a curve lying in D, except for its endpoints, which lie on ∂D. We also assume x ∈ γ, (by the strong Markov property) Now if sup y∈γ P y (σ Γ < τ D ) = 1, then by continuity and that P z (σ Γ < τ D ) = 0 when z ∈ ∂D, the L-harmonic function P • (σ Γ < τ D ) would take on an interior maximum on D Γ , contrary to the maximum principle. Thus Observe that the function is L-harmonic on D and vanishes continuously on Using this in (7) we end up with The functions are positive and L-harmonic on a neighborhood in D of γ ∩ D and both vanish continuously on a neighborhood in ∂D of the endpoints of γ. Hence by the boundary Harnack principle (Pinsky (1995), Theorem 8.0.1 on page 333), for some Thus for some c 2 > 0 depending on x, since w ≤ 1. Using the definition of v, when combined with (8), this gives Remark 2.6. Let B = (B 1 , B 2 ) be two-dimensional Brownian motion and set S = (r, w) : |w| < π 2 and S M = (r, w) : r > M, |w| < π 2 , M > 0. Then the same proof as in Lemma 2.5 gives that if x ∈ S M then for some c > 0, for all large N .

Feyman-Kac representation of an h-transform
In this section we will prove a result used repeatedly in the sequel. In essence, it allows us to convert the study of the exit distribution of a process into that of an h-transform of the process.
We are going to apply some results of Pinsky (1995). In order to do so, we review some definitions and facts found there. In what follows, β ∈ (0, 1).
Assumption H H H. The operator Assumption H H H. The operator Let U ⊆ R 2 be a bounded open set with C 2,β boundary and suppose satisfies Assumption H on U . Then by Theorem 3.6.1 in Pinsky (1995), the principal eigenvalue of L on U is given by where X t is the diffusion associated with L − V = 1 2 ∆ + b · ∇ on U . An operator L on a domain G is said to be subcritical if it possesses a Green function. Probabilistically, this corresponds to transience of the diffusion associated with L on G. If H ∈ C 2,β (G) with H > 0 on G, then define the H-transform of L to be the operator L H given by Theorem 3.1. Let G ⊆ R 2 be Lipschitz domain with piecewise C 2,β boundary and denote the C 2,β part of ∂G by T . Suppose H, b ∈ C ∞ (G) and H > 0 on G. Let X t and Y t be the diffusions associated with the operators Proof. It suffices to show for each nonnegative smooth function f on R 2 with compact support and supp f ∩ ∂G ⊆ T , To this end, let U ⊆ G be a bounded open set with C 2,β boundary such that Notice then that By our hypotheses, L satisfies Assumption H on U , hence we can apply an existence/uniqueness theorem for the Dirichlet problem (Pinsky (1995), Theorem 3.3.1) to get a unique solution g ∈ C 2,β (U ) to Then by the martingale property and optional stopping, for any open U ′ ⊆ U ′ ⊆ U , Now b is bounded and continuous on U , hence (L, U ) is subcritical. Since H ∈ C 2,β (U ) and H > 0 on U , and since L satisfies H loc on U , by Theorem 4.3.3 (iv) in Pinsky (1995) the generalized principal eigenvalues λ C (L, U ) and λ C (L H , U ) of L and L H , respectively, on U coincide. But L and L H satisfy Assumption H on U , hence by Theorem 4.3.2 in Pinsky (1995), λ C (L, U ) = λ 0 (L, U ) and λ C (L H , U ) = λ 0 (L H , U ). Thus we conclude λ 0 (L H , U ) = λ 0 (L, U ) < 0, by hypothesis. As a consequence, because we also have f H ∈ C 2,β (U ) by positivity of H on G, we can apply another existence/uniqueness theorem (Theorem 3.6.5 in Pinsky (1995) Applying a version of the Feynman-Kac formula (Theorem 3.6.6 (iii) in Pinsky (1995)) to L H , in our notation we get (there is a misprint in the statement of that Theorem:L there should be L 0 ). On the other hand, by (11) the functionṽ = g H ∈ C 2,β (U ) also solves (13). Uniqueness then forces g H =ṽ = v; that is, by (12), Since τ G (X) and τ G (Y ) are a.s. finite, Hence by Fatou's lemma and dominated convergence in (14), By (10), Since LH has constant sign, and each of the latter is in L 1 (for LH ≤ 0 use that f /H is bounded and for LH ≥ 0 use (15)). Thus by boundedness and continuity of f , f H and LH H on G, we can apply dominated convergence to (14), letting U ↑ G to end up with This completes the proof of Theorem 3.1.

Upper bound for
Throughout this section we will use the notation of Section 2. In particular, is the nonangular part of 1 2 ∆ R d+1 expressed in cylindrical coordinates. We will prove the following theorem.
We make an H-transform to eliminate the drift in L and use Theorem 3.1 to convert the problem into a question about two-dimensional Brownian motion in D. To this end, let Then a simple computation yields that Notice then that the diffusion associated with L H − LH H on D is just two-dimensional Brownian motion B = (B 1 , B 2 ). It is known that P x (τ D (B) < ∞) = 1 (Bañuelos et al. (2001), Li (2003 or Lifshits and Shi (2000)). Since the exit time of X from D is the same as the exit time of (d + 1)-dimensional Brownian motion W from Ω, and the latter is known to be a.s. finite (DeBlassie (2007)), we get P x (τ D (X) < ∞) = 1.
Moreover, if U ⊆ D is a bounded open set with C 2,β boundary, then the exit time τ U (X) is the same as τ e U (W ), where But U is bounded, so for some λ( U ) > 0, Since L satisfies Assumption H on U , It is easy to verify all the other hypotheses of Theorem 3.1 hold for G there replaced by D and Y replaced by B. Thus we can conclude that for large N , Proof of Theorem 4.1 a). Since d ≥ 3, we have p(p − 1) ≥ 0 and (17) yields that Consider any η, ξ > 1 with 1 ξ + 1 η = 1. Then where C is independent of N . Another application of Theorem 2.4 yields lim sup .

Lower bound for
We continue using the notation of Section 2. To complete the proof of Theorem 2.3, in light of Theorem 4.1 it suffices to prove the following theorem.
Unlike in Section 4, we do not have the confluence of events where the exponential in the Feynman-Kac representation of the h-transform does not come into play. Here it does and the analysis is much more delicate. So instead of h-transforming first, we conformally map D into a strip and then h-transform. Using explicit properties of the conformal map we can analyze the exponential in the Feynman-Kac representation.
In the first subsection below we will describe properties of the conformal map, make the appropriate h-transform and apply Theorem 3.1 to obtain a Feynman-Kac representation. We prove Theorem 5.1 in the subsequent two subsections.

Preliminaries
Let f be the conformal mapping of P = {(ρ, z) : ρ > 0, |z| < Aρ α } onto the strip S = (r, w) : |w| < π 2 such that f is real on the real axis and f ′ is positive there. Write g : S → P for the inverse map and set h = Re(g). Bañuelos and Carroll (2005) used this map to derive their results on the distribution of the exit place of Brownian motion from a parabolic-type domain. We now state some properties of the map f .
Theorem 5.2. (1) The inverse map g extends to a homeomorphism of S onto P\{0} and the derivative g ′ has a continuous nonzero extension to S.
A simple modification of the proof of Proposition 4 in Bañuelos and Carroll (2005) shows g ′ has a continuous nonzero extension to S. Then the extension of g to a homeomorphism on S follows. By symmetry, the extension maps −∞ to 0. This proves part (1).
To prove part (3), note by part (1), f extends to a homeomorphism of P\{0} onto S. Then given N > 0 we can choose r N such that in other words, for r ≥ r N and |w| < π 2 , h(r, w) = Re(g(r, w)) > N.
In what follows, for T > 0 we will write S T = S ∩ {(r, w) : r > T }.
Let 0 < ε < 1 and by Corollary 5.3, choose large M > 1 so that and Making M larger (as we will below) does not change the validity of (20)-(21).
Let N > 1 and set Then for some R(N ), Lemma 5.5. As N → ∞, Proof. By Theorem 5.2 (2), Now let us convert our problem to the strip S. Define and let Y be the diffusion corresponding to L in S. Notice by the Cauchy-Riemann equations the operator |∇h| −2 L is just the operator L from Section 2 expressed in the coordinates (r, w) = f (ρ, z). In particular, Y and f (X) in f (D) have the same exit distribution on ∂S and so we can write Moreover, for any open U ⊆ D, Y and f (X) have the same exit distribution on ∂f (U ) and so Next, let us h-transform and get a Feynman-Kac representation.
Lemma 5.6. Let B = (B 1 , B 2 ) be two-dimensional Brownian motion. Let R > M and y ∈ S M .
(2) If d = 2 then Proof. We are going to verify the hypotheses of Theorem 3.1 for Since h is harmonic on S, and LH has constant sign. By Theorem 5.2 (1), inf ∂r 2 + ∂ 2 ∂w 2 , respectively, on G. Notice X = Y from the statements prior to Lemma 5.6 and Y is two-dimensional Brownian motion. We need to show τ G ( X) and τ G ( Y ) are a.s. finite. Write I = − π 2 , π 2 and note τ G ( Y ) ≤ τ I ( Y 2 ). But Y 2 is one-dimensional Brownian motion and it is known that for some positive constants c 1 an c 2 , P y 2 (τ I ( Y 2 ) > t) ≤ c 1 e −c 2 t .
Also, since X is two-dimensional Brownian motion with bounded drift, by the Cameron-Martin-Girsanov Theorem, P y (τ G ( X) > t) satisfies a similar bound. In particular both τ G ( X) and τ G ( Y ) a.s. finite. Moreover, if U ⊆ G is any bounded open set with C 2,β boundary then τ U ( X) ≤ τ G ( X), and so by the exponential decay of the tail distribution of the latter, λ 0 (L, U ) < 0. Thus all the hypotheses of Theorem 3.1 hold and we can conclude that for R > M

ds .
Recalling that X = Y and Y is two-dimensional Brownian motion, this translates into By (21), and so (25) leads to This gives part (1) of the lemma. Now assume d = 2. Then p = 1 2 and (25) yields part (2).
Lemma 5.7. For two-dimensional Brownian motion B and y ∈ S M , Proof. Choose x ∈ D such that f (x) = y. By conformal invariance of two-dimensional Brownian motion and (21)-(22), for large N and Then by Remark 2.6, for some c > 0, if N is large we have Applying Theorem 2.4, this yields Using Lemma 5.5, we get Thus lim T →∞ 1 T log P y (B 1 (τ S M ) > T ) = −1 and the conclusion of the lemma holds.

Proof of Theorem 5.1 a)
In this subsection we will assume d ≥ 3. Consider any η, ξ > 1 with 1 η + 1 ξ = 1. Define (recall ε > 0 is from Section 5.1-see just before (20)). This is real because d ≥ 3 forces p−1 ≥ 0. Notice q is chosen so that Let Z be the diffusion associated with the operator Lemma 5.8. For two-dimensional Brownian motion B and R > M , Proof. For B(0) = y, we can write Since q > 2, Z 1 never hits 0. By the Comparison Theorem (Ikeda and Watanabe (1989), Since Z 2 (t) = B 2 (t) for all t ≥ 0, we can conclude that either and we have as desired.
We are going to compare P y (Y 1 (τ f (D) ) > R) and P y (B 1 (τ S M ) > R) and in order to do so, we must have y ∈ S M . In general though, it is possible that y / ∈ S M for P y (Y 1 (τ f (D) ) > R). The next lemma lets us get around this difficulty.
Lemma 5.9. Suppose y 1 ≤ M . Then for some c > 0, Proof. Write G = f (D)\S 2M . By the strong Markov property, for R > 2M wherec is independent of R > 2M .
Now we can complete the proof of Theorem 5.1 a). Recall we are writing y = f (x) for x = (ρ, z) ∈ D. Depending on the choice of ε made just before (20), it is possible that y 1 < M . Since we can make M larger without affecting the validity of (20)- (21), it is no loss to assume that we have y 1 < M . For large N , by (23) and Lemma 5.9, for some c > 0 independent of N , ds (by Lemma 5.6 (1)) (by (20) and that p − 1 ≥ 0).
We are going to apply Theorem 3.1 to

Noting that
we can repeat the argument in the proof of Lemma 5.6 to show the hypotheses of Theorem 3.1 are met. Thus that theorem with (X, Y ) there replaced by (Z, B) yields (by Lemma 5.7 and (27)), where c is independent of large N . Using this in (28), we get for somẽ c > 0 independent of N , .
Let ξ → 1 + to get part a) of Theorem 5.1.

Proof of Theorem 5.1 b)
We continue using the notation of Section 2. It will be necessary to make M larger so that f (x) / ∈ S M . We will compare X 1 (τ D ) with X 1 (τ D Γ ) where D Γ = g(S M ) and this requires the starting point x to be in D Γ or equivalently, f (x) ∈ S M . The corollary following the next lemma will let us get around this problem.
Lemma 5.10. Suppose z = (z 1 , z 2 ) ∈ D. Then for some constant c > 0, Proof. Let R 1 < z 1 < R 2 < 2z 1 and set Notice z ∈ G. The functions are positive and L-harmonic on G. Since they vanish continuously at ∂D∩{(ρ, z) : R 1 < ρ < R 2 }, by the boundary Harnack principle there is c > 0, independent of N , such that wherec is independent of N > 2z 1 .
Corollary 5.11. Let x ∈ D with f (x) / ∈ S M . Then for some c > 0, Since f (x) / ∈ S M , x lies to the left of the curve Γ. Then for N > 2K (by the strong Markov property).
But Re(X(σ K )) = K, so by Lemma 5.10 this gives as desired.
The grand conclusion is that 1 2 γ 1 (d) is the smallest positive Dirichlet eigenvalue of L on the interval Since the coefficients of L are smooth onĪ, by Theorem 4.3.2 in Pinsky (1995), and the definition after its proof, − 1 2 γ 1 (d) is the generalized principal eigenvalue of L on I. Let h(ϕ) = (sin ϕ) (1−d)/2 .
Then h is strictly positive and C 3 on I; hence by Theorem 4.3.3(iv) in Pinsky (1995), − 1 2 γ 1 (d) is also the generalized principal eigenvalue of the h-transformed operator L h on I. It is easy to compute that where V d (ϕ) = (d − 1) 2 1 + (d + 1) 2 cot 2 ϕ .
Since the operator L h is well-behaved onĪ, application of Theorems 4.3.2 and 3.6.1 in Pinsky (1995) yields that if B t is one-dimensional Brownian motion,then where τ I is the first exit time of B t from I. If d 1 < d 2 , then we have V d 1 < V d 2 , and the last limit implies that γ 1 (d 2 ) ≤ γ 1 (d 1 ), as desired.