The Exit Place of Brownian Motion in an Unbounded Domain

For Brownian motion in an unbounded domain we study the inﬂuence of the “far away” behavior of the domain on the probability that the modulus of the Brownian motion is large when it exits the domain. Roughly speaking, if the domain expands at a sublinear rate, then the chance of a large exit place decays in a subexponential fashion. The decay rate can be explicitly given in terms of the sublinear expansion rate. Our results encompass and extend some known special cases.


Introduction
In this article we use a new approach to study the effect of the "far away" behavior of an unbounded domain on the probability that the modulus of the Brownian motion is large when it exits the domain. We study domains of the form and Ω = (ρ, z) ∈ 2 : ρ > 1 2 , |z| < a(ρ) × S n−1 , n ≥ 2 where a : 1 2 , ∞ → [0, ∞) is continuous and positive on 1 2 , ∞ . In the case of Ω, we understand it to be a subset of n+1 where the triple (ρ, z, θ ) denotes the cylindrical coordinates of a point (x, x n+1 ) ∈ n × withx = 0: One way to understand the difference between the domains D and Ω is to look at them in three dimensions. There D is obtained by revolving the set {(x, y): x > 1 2 , | y| < a(x)} about the x-axis, while Ω is obtained by revolving about the y-axis.
Roughly speaking, we will show that if the growth of a is sublinear and the oscillations far away are not too severe, then the probability the modulus of Brownian motion is large when it exits the domain is subexponentially small and we can explicitly identify the decay rate.
An equivalent analytic formulation of this problem is to ask how the harmonic measure behaves outside a compact set. A related problem is to determine the sharp order of integrability of the exit position.
Several authors have studied these questions in various domains, using different tools. For a cone of angle θ (which corresponds to D above with a(x) = c x), Burkholder (1977) used his L p -inequalities for Brownian motion to explicitly find p(θ ) > 0 for which the p > 0 moment of the exit place is finite iff p < p(θ ). By classical estimates for harmonic measure (Haliste (1984) and Essén and Haliste (1984)), there are positive C 1 and C 2 such that where B is Brownian motion and τ is the exit time from the cone. The analogue is also true for more general cones. Using the explicit form of the heat kernel for a cone (due to Bañuelos and Smits (1997)), Bañuelos and DeBlassie (2006) obtained a series expansion for d d r P x (|B(τ)| > r) which implies the behavior P x (|B(τ)| > r) ∼ C r −p(θ ) as r → ∞, and C was explicitly identified. Again, there is an analogous result for more general cones. Bañuelos and Carroll (2005) studied the domain D above with a(x) = Ax α , where A > 0 and 0 < α < 1. Denoting the exit time of Brownian motion from D by τ D , those authors showed that where λ 1 > 0 is the smallest eigenvalue for the Dirichlet Laplacian in the unit ball of d−1 (note when d = 2, λ 1 = π 2 ). They also showed that for d = 2, (1−α) . In dimension d ≥ 3, they proved the expectation is finite for b < λ 1 A(1−α) and infinite for b > λ 1 A(1−α) . The critical case was left open. Their method was to use a conformal mapping and a technique of Carleman to estimate harmonic measure.
For the domain Ω above with a(x) = Ax α , 0 < α < 1, DeBlassie (2008b) reduced the computation of the probability the modulus of Brownian motion is large upon exiting the domain to the two-dimensional case studied by Bañuelos and Carroll (2005). The method used conformal maps coupled with the Feynman-Kac formula and the Comparison Theorem for stochastic differential equations. The main result obtained was that It is interesting to note that in contrast with the domains considered by Bañuelos and Carroll, the limit is independent of the dimension. This is counter-intuitive; see DeBlassie (2008a) for more explanation.
It does not seem possible to use the conformal method mentioned above for boundary functions other than a(x) = Ax α for α ∈ (0, 1). This is because the method relies very much on delicate estimates, due to Carroll and Hayman (2004), of the derivative of a certain conformal map. The specific power law growth x α is crucial to their argument and it is not at all clear how to extend their estimates to more general functions.
Instead, we take a new approach that will permit extension of (3) and (5) to much more general functions a(x), and it will also resolve the critical case in dimension d ≥ 3 for finiteness of left open by Bañuelos and Carroll. The basic idea is to represent the density (with respect to surface measure on the boundary) of harmonic measure as the normal derivative of the Green function. Then we can use estimates of harmonic functions due to Cranston and Li (1997) to estimate the normal derivative. The representation of the density of harmonic measure as the normal derivative of the Green function is a classical fact for bounded smooth domains (Miranda (1970), Garabedian (1986), Gilbarg and Trudinger (1983)). But here our domains are unbounded and the classical proof must be modified. Basically, trouble arises because the proof requires the Divergence Theorem to hold, and since the domain is unbounded, integrability issues become significant.
Before stating our results, we present the basic assumptions on D from (1).

Blanket Assumptions
The lim sup conditions as well as the condition a(t)a (t) → 0 as t → ∞ quantify the statement that the oscillations of a(·) far away are not too severe. ii) By the Mean Value Theorem, the requirement that a (t) → 0 as t → ∞ implies a(t)/t → 0 as t → ∞, which in turn implies a(t) is sublinear and iii) If a(·) is C 3 , then the domain D satisfies the blanket assumptions in the following cases: • a(t) = At α for large t, where A > 0 and α < 1 with α = 0; note if α = 1, then D is a cone.
• a(t) = Ae −γt p for large t, where A, γ and p are positive; • a(t) = At(log t) −p for large t, where A and p are positive.
For any > 0 and x ∈ d , write For any process Z t in d , we will write be the Bessel function of the first kind, we denote its first positive zero by j ν .
Remark 1.3. i) When d = 2, the problem is easily handled with conformal methods and the limiting behavior given holds upon replacing ν by |ν|. Note too that j ν = π/2 in this case. ii) Note that the first Dirichlet eigenvalue of ∆ d−1 on the unit ball in d−1 is in fact Thus we recover the result (3) of Bañuelos and Carroll for 0 < α < 1 and extend it to α < 0.
With some additional conditions on a(·), Theorem 1.2 can be extended with no further effort: Theorem 1.4. Let d ≥ 3. In addition to the Blanket Assumptions on D from (1), suppose Then for some positive C 1 and C 2 , for all large N , Remark 1.5. If A > 0 and 0 = α < 1, then a(t) = At α satisfies the hypotheses of Theorem 1.4. By Remark 1.3 iii), we can mimic the argument of Bañuelos and Carroll giving (4) for d = 2 and resolve the critical case for d ≥ 3 they left open: Next consider the domain Ω from (2). Then we can write where D is now given by (1) with d = 2, and we continue to make the Blanket Assumptions on D.
Theorem 1.6. Let n ≥ 2 and suppose B t is (n + 1)-dimensional Brownian motion. Then Remark 1.7. i) Analogous to Remark 1.3 iii), we recover our earlier result (5). ii) Theorem 1.6 can be sharpened much like Theorem 1.2 was sharpened by Theorem 1.4. iii) Note that in contrast with Theorem 1.2, there is dimensional independence in the limit. The explanation for this counter-intuitive result is the same as that given in DeBlassie (2008a) for the special case of a(x) = Ax α .
In addition to the results of the authors mentioned above, there are other studies involving the domains D and Ω. Ioffe and Pinsky (1994) identified the Martin boundary of Ω. This result was extended by Aikawa and Murata (1996) and Murata (2002Murata ( ), (2005 to asymmetric versions of Ω and they also found a series expansion for the Martin kernel. Related results were announced in Maz'ya (1977) and Kesten (1979). The growth of the Martin kernel at infinity for Ω and D was determined in DeBlassie (2008b) and (2009), respectively. Collet et al. (2006) proved a ratio limit theorem for the Dirichlet heat kernel in Ω for a(t) = t in two dimensions. They used their theorem to determine the probability that Brownian motion remains in this particular domain for a long time. Using different methods, DeBlassie (2007) extended the latter result to all dimensions for the functions a(t) = t α , where 0 < α < 1. Pinsky (2009) determined spectral properties of the Neumann Laplacian in Ω (and more general domains) as well as a transience/recurrence dichotomy for Brownian motion in Ω with normal reflection at the boundary.
In Bañuelos et al. (2001), Li (2003) and Lifshits and Shi (2002), the probability that Brownian motion remains in D for a long time was derived in the case when a(t) = t α , 0 < α < 1. For this particular domain, van den Berg (2003) found long-time asymptotics for the corresponding Dirichlet heat kernel.
In the case of the domain D with a(t) → 0 as t → ∞, bounds on the Dirichlet eigenfunctions of the Laplacian in D were obtained by Bañuelos and Davis (1992) and (1994), Bañuelos and van den Berg (1996), Cranston and Li (1997) and Lindemann et al. (1997).
Here is the organization of the article. In section 2 we study the domain D from (1). A representation theorem is stated, giving the density of harmonic measure as the normal derivative of the Green function. In subsection 2.1, the representation theorem is used in conjunction with a result of Cranston and Li on the asymptotics of harmonic functions to prove Theorems 1.2 and 1.4. Then in subsections 2.2-2.4, the proof of the representation theorem is given.
In section 3 we shift attention to the domain Ω from (2). The problem is reduced to two-dimensions, where now the relevant operator is the Laplacian plus a first order term, with corresponding process X . In subsection 3.1, by suitably conditioning the process, we eliminate the drift and state a representation theorem for the exit place density of X in terms of the conditioned process. Then we proceed analogously to the case of D considered in section 2.
Please note that throughout the article, c will be a scalar whose exact value can change from line to line.
Acknowledgement. I thank the Associate Editor for a detailed list of comments and suggestions that improved the exposition of the article. I am especially grateful for the suggested way to prove Lemma 3.3. It is much simpler and more elegant than my original cumbersome argument. (1) In this section we will prove Theorems 1.2 and 1.4. Since the domain D has a non-polar component, the Brownian motion-killed upon exiting D-is transient in D. Hence it has a Green function we denote by G D (x, y). Note that analytically one says ( 1 2 ∆ d , D) is subcritical and in fact G D (x, y) is the (minimal) Green function for ( 1 2 ∆ d , D). As indicated in the introduction, the next result describes the normal derivative of G D (x, ·) as more or less being the density of the harmonic measure based at x.

Theorem 2.1. For any Borel set
where ∂ ∂ n y is the inward normal derivative at y ∈ ∂ D and σ(d y) is the surface measure on ∂ D induced by the usual Riemannian structure on d .
The unboundedness of D complicates the proof of Theorem 2.1. We will break up the proof into several pieces, but before that, we now use it to prove Theorems 1.2 and 1.4.

Proof of Theorems 1.2 and 1.4
For d ≥ 3, let Ω d−1 be the unit ball in d−1 and set ν = d−3 2 . As pointed out earlier, the first where j ν is the first positive zero of the Bessel function J ν . Furthermore, the corresponding eigenfunction is The next theorem is due to Cranston and Li (1997)-see the two paragraphs just after the proof of their Theorem 2.1. Note that although they make the blanket assumption a(t) → 0 as t → ∞, this is not used to prove the version of their theorem that we use. Set

and suppose H is bounded and Hölder continuous on D M with
For y ∈ ∂ D, recall n y is the inward unit normal to ∂ D at y. Lemma 2.3. Let y = ( y 1 ,ỹ) ∈ ∂ D with y 1 > 1 2 . Then for z = (z 1 ,z) = y + hn y , Proof of Lemma 2.3. For y = ( y 1 ,ỹ) ∈ ∂ D with y 1 > 1 2 , it is a simple matter to show that Proof of Theorem 1.2. Define x 1 (N ) to be the first coordinate of the intersection of the circle ρ 2 + z 2 = N 2 with the curve z = a(ρ) in the ρz-plane: Fix x ∈ D and suppose M > |x| and δ ∈ (0, j ν ). Combined with the fact that G D (x, y) goes to 0 as the modulus of y ∈ D goes to infinity, Theorem 2.2 applied to u(·) = G D (x, ·) and H ≡ 0 on D M shows that we can choose M 1 > M and C > 0 such that h .
Using this and Lemma 2.3 in (9), we get that for some positive C 1 and C 2 for y = ( y 1 ,ỹ) ∈ ∂ D with y 1 > M 1 . Then by Theorem 2.1, also using that (8)), we get that for N > M 1 , The integrands depend only on y 1 , so this reduces to Since By l'Hôpital's rule and the fact that a (x) → 0 as x → ∞, for any γ > 0 that is close to Combining this with (10), we get that for some positive C 3 and C 4 , for large N , Take the natural logarithm, multiply by [ Let δ → 0 and use the fact that To complete the proof of Theorem 1.2, we show Indeed, recalling the definition of x 1 (N ) from (8) and writing x 1 for x 1 (N ), we have Then by the mean value theorem, there isx ∈ (x 1 , N ) such that Upon differentiating (8) with respect to N , To finish, use these limits and l'Hôpital's rule to get Proof of Theorem 1.4. Under the additional hypotheses of Theorem 1.2, the conclusion of Theorem 2.2 holds with δ = 0. Thus the argument leading to (12) goes through with δ = 0 there and we get the conclusion of Theorem 1.4.

Proof of Theorem 2.1: Preliminaries and a Reduction
To prove Theorem 2.1, it suffices to show that for each x ∈ D, for any nonnegative f ∈ C 3 ( d ) with compact support in d \{x}, we have To this end, write By the strong Markov property, u is harmonic in D, hence C ∞ there. Given > 0, by uniform But since |x − y| < δ 2 , for any t > 0 we can apply Theorem 2.2.2 (ii) in Pinsky (1995) to get Choosing t > 0 so large that the first term is less than , we get Now let x → y ∈ ∂ D and apply Corollary 2.3.4 in Pinsky (1995) to get Since > 0 was arbitrary, we get that u( For an unbounded set G ⊆ d , we define C 2,α (G) to be the set of all functions in C 2 (G) whose second order partials are uniformly Hölder continuous on any compact subset of G. Note we do not require G to be open.
Since f ∈ C 3 (D), by the Elliptic Regularity Theorem (Lemma 6.18 in Gilbarg and Trudinger (1983)), u ∈ C 2,α (D). Thus (14) becomes and so (13) amounts to the classical Green Representation of the solution to (15). As indicated in the introduction, since D is unbounded, complications arise.
We will write Now we fix x = (x 1 ,x) ∈ D and prove (13). For simplicity we will assume x = x 0 . Notice where the over bar denotes Euclidean closure. Given M > 0 so large that Then by Green's Second Identity, Writing where now the ∂ ∂ n y Once we prove and we can let M → ∞ and → 0 in (19) to get which is exactly (13), as desired.
Formulas (20)-(21) will be proved in the next two subsections.

Proof of (20)
The following result is a consequence of the proof of Lemma 6.5 in Gilbarg and Trudinger (1983) combined with the comments subsequent to the proof.

Lemma 2.5.
Let Ω be a domain in d with a C 2,α boundary portion T . Suppose u ∈ C 2,α (Ω ∪ T ) is a solution of Then for any z ∈ T there is δ > 0 such that for |u|.
Here δ depends only on the diameter of the domain of the C 2,α diffeomorphism ψ that straightens the boundary near z and C depends only on d, α, Λ and the C 2,α bounds on ψ.
We now apply Lemma 2.5.
If v ∈ C 2,α (D\{x}) is a solution of Proof. Define Then for Notice H M is obtained from D via translating by −M x 0 and then scaling by 1/a(M ). For any function g on D, we define g M (z) = g(a(M )z + M x 0 ), z ∈ H M .

Then for
Thus for large M , Since v ∈ C 2,α (D\{x}) and v = 0 on ∂ D outside a compact set, by making M larger if necessary, we have that We are going to apply Lemma 2.5 to L M and v M on This is legitimate because by our hypotheses on b, there exists Λ > 0 such that for large M , By symmetry, compactness and our Blanket Assumptions on a(·), for each z ∈ ∂ H M ∩{− 1 2 ≤ z 1 ≤ 1 2 }, the C 2,α bounds on the diffeomorphism straightening the boundary near z are independent of z and large M . (Roughly speaking, for large M , the set H M ∩ {−1 < z 1 < 1} looks like the set {(z 1 ,z): − 1 < z 1 < 1, |z| < 1}). Combined with (23), it follows that the constant C appearing in Lemma 2.5 is independent of such z and large M . Likewise, the δ in the lemma is also independent of z and large M . The net effect is that for some δ > 0 and C > 0, for whenever z ∈ ∂ H M ∩ {− 1 2 ≤ z 1 ≤ 1 2 } and M is large.
In particular, given y ∈ π 0 ∩ H M ∩ {|ỹ| > 1 − δ 2 }, choose z ∈ π 0 ∩ ∂ H M such that d( y, π 0 ∩ ∂ H M ) = d( y, z). Then y ∈ B δ (z) and by (24), for M large, Since y ∈ π 0 , d( y, ∂ B(z) − T (z)) ≥ δ 2 and we get that for some C 1 > 0, for large M , On the other hand, by the Schauder interior estimates (Gilbarg and Trudinger (1983), Theorem 6.2), sup where C 2 > 0 is independent of large M . Combined with (25), we get that for some Converting back to v and using (22), for all large M we have which is to say as desired.
Now we can prove (20). Taking b = 0 in Lemma 2.6, v = u or G D (x, ·) satisfies the required hypotheses, so for some C > 0, for all large M , Since u is bounded, we end up with as desired.

Proof of (21)
Since Thus to prove (21), we need only check Now for p(t, y, z) = 1 (2πt) d/2 e −| y−z| 2 /2t , the transition density of Brownian motion killed upon exiting D is given by Analytically, p D is the Dirichlet heat kernel for 1 2 ∆ d on D. Thus where Consequently, recalling that ∂ ∂ n y is differentiation along the inward unit normal to ∂ B (x), To justify the differentiation under the expectation, bound the difference quotients using the Mean Value Theorem. Then dominated convergence applies because We have, as → 0, Thus to prove (26), we need only show But this is an immediate consequence of the continuity and boundedness of u, combined with the . (2) Recall n ≥ 2 and the horn Ω in n+1 is represented in cylindrical coordinates (ρ, z, θ ) by Ω = D × S n−1 , where

The Domain Ω Ω Ω from
The Laplacian expressed in the coordinates (ρ, z, θ ) is where ∆ S n−1 is the Laplace-Beltrami operator on S n−1 . Write L for the nonangular part of 1 2 ∆ n+1 : and let X t be the diffusion associated with L in {(ρ, z): ρ > 0}. Then by symmetry, for x = (ρ, z, θ ), Thus to prove Theorem 1.6, it suffices to show Using the relation derived in the proof of Theorem 1.2, we see the proof of Theorem 1.6 comes down to showing

The Analogue of Theorem 2.1 for X X X
Since n ≥ 2, starting at (ρ, z) with ρ > 0, the process X t stays in {(ρ, z): ρ > 0} forever. In fact, the first component of X t is an n-dimensional Bessel process and the second component is an independent one-dimensional Brownian motion. Thus the transition density p(t, y, w) of X t (with respect to Lebesgue measure) is the product of the transition densities of the components: for y = ( y 1 , y 2 ) and w = (w 1 , w 2 ), where is the modified Bessel function (see Ikeda and Watanabe (1981) for the transition density of the Bessel process).
Since D ⊆ {(ρ, z): ρ > 0}, by Lemma 3.1 (L, D) is subcritical and the corresponding Green function G D is associated with X t killed upon exiting D. Because L is not self-adjoint with respect to Lebesgue measure, the analogue of Theorem 2.1 takes on a slightly different form. We h-transform L, converting it into a self-adjoint operator that is easier to analyze. Here, if h ∈ C 2,α (D) is positive, then the h-transform of L is the operator L h given by We will take Then Since (L, D) is subcritical, so is (L h , D) (Pinsky (1995) Proposition 4.2.2) and its Green function is Now we can state the analogue of Theorem 2.1.

Theorem 3.2. For any Borel set
Before proving this theorem, we show how it yields (27), hence Theorem 1.6. Indeed, Theorem 3.2 implies Fix y ∈ D and let M > | y|, δ ∈ (0, j ν ). Since and so we can apply Theorem 2.2. Then we can repeat the proof of Theorem 1.2 almost word-forword to end up with the analogue of (12), except that now d = 2 and the upper and lower bounds have an extra factor of x 1 (N ) −p -this is due to the extra factor 1 h(w) = w −p 1 in the integrand of the expression above for P y (|X τ D | > N ). The rest of the argument after (12) still goes through because lim N →∞ log x 1 (N )

Proof of Theorem 3.2
Now L h is formally self-adjoint and it satisfies Hypothesis H loc in Pinsky (1995). Then by his Theorems 4.2.5, 4.2.8, and 8.1.1, • G h D is positive and jointly continuous off the diagonal; • L h G h D ( y, ·) = 0 on D\{ y}; • for each y ∈ D, there exist positive C 1 and C 2 along with r 0 ∈ (0, 1) such that • G h D ( y, ·) is continuous on D\{ y} with boundary value 0.
Fix y ∈ D. Exactly as in the proof of Theorem 2.1, it suffices to show for any nonnegative f ∈ C 3 ( 2 ) with compact support in 2 \{ y}, The argument giving (15) also gives that Proof. It is expedient to convert back to Brownian motion B in Ω. For x = (x, x n+1 ) ∈ n × , define To this end, choose K so large that supp(g) ⊆ B K (0). For a Borel set E, we will use η E to denote the first hitting time of E by B t . If we set G = supp(g) ∩ ∂ Ω and take M large, then for x ∈ Ω with It is well-known that (2007)), and so we have as desired.
Let ∆ w denote the two-dimensional Laplacian in the variable (w 1 , w 2 ). Now since by Green's second identity, where, analogous to (16), Then exactly as in §2 (cf. (17) and what follows), we will have that once we prove the analogues of (20)-(21): and Translating (37), we would then have It remains to verify (38) and (39).
Proof of (38). All the hard work was done in section 2. Taking b(ρ, z) = − p(p−1) Now we can apply Lemma 2.6 to ∆ + b = 2L h and v = u h or G h ( y, ·) to get that for some C > 0, for all large M , for w ∈ π M ∩ D, By Lemma 3.1, for large w 1 , G( y, w) is bounded (recall y is fixed), and so Then using that Combining this with (41), for large M , we get Since |u h (w)| ≤ M p sup z 1 ≥M |u(z)| for w ∈ π M ∩ D, the inequalities (40), (42) and (43) yield that for all large M , Thus as M → ∞, by Lemma 3.3. This completes the proof of (38).
Proof of (39). Recall the ∂ ∂ n w appearing in (39) is the inward normal derivative at ∂ B ( y). Since is bounded on a neighborhood ∂ B ( y); then by (31), Thus to prove (39), it suffices to show The transition density p D (t, w, z) of X t killed upon exiting D is given by where p(t, w, z) is from (28). Then where G is from Lemma 3.1. Writing Lemma 3.4. a) For > 0 so small that B 2 ( y) ⊆ D, denoting inward normal differentiation at the boundary of B ( y), We defer the technical proof to the next subsection. Differentiating (45) and using Lemma 3.4a to justify differentiation inside the expectation, Moreover, since u h is bounded near y and since h is bounded on D, part a) of Lemma 3.4 also implies that as → 0, Thus by (45) |u h (w) − u h ( y)|, which converges to 0 as → 0. Moreover, the limit in that part of the lemma also implies that the second integral converges to 2u h ( y) as → 0. This completes the proof of (39).
Then by the series representation for the hypergeometric function F from Lemma 3.1, it follows that H(γ 2 ) and H (γ 2 ) are bounded for z ∈ ∂ D and w ∈ B 2 ( y). Together with (50) and (47), this implies that ∇ w G h (z, w) is bounded for z ∈ ∂ D and w ∈ B 2 ( y), as desired.