Exit times of Symmetric Α-stable Processes from Unbounded Convex Domains

Let X t be a d-dimensional symmetric stable process with parameter α ∈ (0, 2). Consider τ D the first exit time of X t from the domain D = (x, y) ∈ R × R d−1 : 0 < x, |y| < φ(x) , where φ is concave and lim x→∞ φ(x) = ∞. We obtain upper and lower bounds for P z {τ D > t} and for the harmonic measure of X t killed upon leaving D ∩ B(0, r). These estimates are, under some mild assumptions on φ, asymptotically sharp as t → ∞. In particular, we determine the critical exponents of integrability of τ D for domains given by φ(x) = x β [ ln(x + 1) ] γ , where 0 ≤ β < 1, and γ ∈ R. These results extend the work of R. Bañuelos and R. Bogdan (2).


Introduction
Let X t be a d-dimensional symmetric α-stable process of order α ∈ (0, 2].The process X t has stationary independent increments and its transition density p α (t, z, w) = f α t (z − w) is determined by its Fourier transform exp(−t|z| α ) = R d e iz•w f α t (w)dw.
When α = 2, the process X t is a d-dimensional Brownian motion running at twice the usual speed.
Let D be a domain in R d , and let X D t be the symmetric α-stable process killed upon leaving D. If α ∈ (0, 2), H α the self-adjoint positive operator associated to X D t is non-local.Analytically this operator is obtained by imposing Dirichlet boundary conditions on D to the pseudo-differential operator (−∆) α/2 , where ∆ is the Laplace operator in R d .The transition density of X D t is denoted by p α D (t, x, y) and is the first exit time of X t from D.
In the Brownian motion case, it is known there are domains such that the distribution of the exit time has sub exponential behavior.As a matter of fact, consider the domain D = D p given by where p < 1 and |x| is the euclidean norm in R d−1 .R.
for some c > 0. Similar results were obtained by M. van den Berg (22) for the asymptotic behavior of p 2 D (t, x, y).Notice D p is obtained by moving B(0, x p 1 ), the ball centered at the origin 0 ∈ R d−1 of radius x p 1 , along the straight line l x 1 = (x 1 , 0, . . ., 0).R. D. DeBlassie and R. Smits (15) extend (1.4) to domains generated in a similar way by a curve γ.We should also mentioned the work of Collet et al. ( (10), (11)) where the authors study domains of the form D = R d \ K, for K a compact subset of R d .In this case, there exits c > 0 such that It is then natural to ask if, for α ∈ (0, 2), there are domains in R d such that has subexponential behavior as t → ∞.
In this paper we will study the behavior of (1.6) and the behavior of the harmonic measure for unbounded domains of the form where φ is an increasing concave function, such that lim r→∞ φ(r) r = 0 , and As shown in §5, for φ(x) = [ ln(x + 1) ] µ , µ ∈ R, (1.6) has sub exponential behavior.
We will denote the ball of radius r centered at the origin, 0 of R n , by B(0, r).The following result, which we believe is of independent interest, will be fundamental in the study of (1.6).
Theorem 1.1.Let 0 < x, and D r = D ∩ B(0, r) where D is given by (1.7).Then there exists M > 0 and c α d > 0 such that for all r ≥ M where z = (x, 0, . . ., 0).This theorem can be combined with the results of (19) to obtain upper and lower bounds on (1.6).For instance one can show that there exist c α d > 0 and M > 0 for all z ∈ D and all t, r > M .Our bounds on (1.6) will imply the following result.
The paper is organized as follows.In §2 we setup more notation and give some preliminary lemmas.Theorem 1.1 is proved in §3.We obtained bounds on the asymptotic behavior of (1.6) in §4, and finish by proving Theorem 1.2 in §5.
Throughout the paper, the letters c,C, will be used to denote constants which may change from line to line but which do not depend on the variables x, y, z, etc.To indicate the dependence of c on α, or any other parameter, we will write c = c(α), c α or c α .

Preliminary results.
Throughout this paper the norm in the Euclidean space, regardless of dimension, will be denote by | • |, and φ : R + → R + will be an increasing concave function such that φ(0) = 0 and lim Notice that the concavity of φ implies Lemma 2.1.Let D be the domain given by (1.7).If u > 0 and z = (u, 0, . . ., 0).Then Proof.Let u > 0. A simple computation shows that there exists x 0 > 0 such that Then the monotonicity of φ and (2.3) imply On the other hand, thanks to (2.3) and the desired result immediately follows.
In the next section we will approximate certain integrals over D using spherical coordinates.For this we will need to study the behavior of the cross section angles.
For r > 0, let x r be the solution of and the angle between the x-axis and (x r , 0, . . ., 0, φ(x r )).
One easily sees that (2.1) and (2.6) imply lim and there exists M > 0 such that for all r ≥ M and 0 ≤ ϕ ≤ θ(r).

Harmonic measure estimates
In this section we study the harmonic measure of the domain for D given by (1.7).Our arguments follow the ideas of T. Kulczycki in (18).As a matter of fact, we are interested in the behavior of as r → ∞, where z ∈ D, and B is a borelian subset of D. For m ∈ Z, we define and B(A m ) to be the Borel subsets of A m .
To simplify the notation we set If x ∈ A m the probability that X jumps directly to B, B\D m+1 = ∅, when leaving the subdomain where p m is the Poisson kernel of X t killed upon leaving the domain D m+1 .
However the process X t , starting at x ∈ A m , could also jump out of D m+1 and reach B ⊂ A n after precisely k successive jumps to Thus we are interested in the behavior of where i 0 = m.The Markov property implies that where i 0 = m.Notice that the event is not empty if and only if k ≤ l − 2. Thus for m, n ∈ Z and k ∈ N with m < n.
Therefore the probability that X starts at x ∈ D and goes to B ∩ D after k jumps, of the type (3.3), is T. Kulzcycki prove that if x ∈ D −1 and B 1 ⊂ A 1 , then Thus to estimate the harmonic measure it is enough to have good estimates of σ(x, •).We will start by estimating the function q m (x, •). where In particular If n − m ≥ 2, and y ∈ B n we have Besides, if n > m + 2, then On the other hand, if n = m + 2, we have Then |y| d (y − 2 m+1 r) α/2 dy.
Finally if B n = A n .Using spherical coordinates we obtain from (2.9) and (3.10) follows.
The following corollary is an immediate consequence of the definition of q i 1 ,...,i k (x, •).
Then there exists c α d > 0 such that where for all l, k ∈ Z and all borelian sets B k contained in A k .
In order to estimate σ(•, •), we will need the following monotonicity result.
Proof.Recall that the function φ(x)/x is decreasing.Following the arguments of Lemma 3.1, we obtained a constant c d α such that On the other hand, using spherical coordinates and (2.9) and the result follows.
Let (i 1 , . . ., i k ) ∈ J(m, n), and 1 ≤ s < k − 1.By the definition of J(m, n) we have In addition, if i k−1 < n − 2, then for all ρ ≤ 2 n r Thus Since this inequality also holds when i k−1 = n − 2, we conclude that In order to obtain and upper bound on σ(x, B), we need to estimate Then there exists c α d > 0 such that for k ≥ 2 Proof.Thanks to Corollary 3.2 it is enough to prove that We will prove (3.21) by induction in k.Notice that Then (3.14) and the result follows for k = 2.
On the other hand, Lemma 3.3, and (3.18) imply and the result follows.
We finally obtain an upper bound on σ(x, B).
Then there exists a constant c α d such that Proof.The previous result implies that On the other hand one easily sees that (3.22) implies the converges of this series.
The proof of Lemma 3.8, Lemma 3.9 and Lemma 3.10 of (18) can be followed step by step to obtain the following result, which is the upper bound on Theorem 1.1.
Proposition 3.6.Let x ∈ D r/2 and B a Borelian subset of D \ D r .Then there exists c α d > 0 such that In particular We shall now obtain the lower bound in (1.10) of Theorem 1.1.
Notice that D r is a bounded domain that satisfies the exterior cone condition.It is well know that, see (17), We will estimate the integral on z using polar coordinates.Thanks to (2.9) there exists M ∈ R such that for all r ≥ M , Combining (3.28) and (3.29) we obtain the desired inequality.

Exit time estimates
T. Kulczycki proved the semigroup associated to the killed symmetric α-stable process on any bounded domain is intrinsic ultracontractive.Thus there exists c α d > 0 such that for all t > 1, where λ d is the principal eigenvalue of X t killed upon leaving B(0, 1) ⊂ R d .
We now use the results of §4 to obtained estimates for the distribution of the exit time.
for all z ∈ D and all t > 1.
Proof.Notice D r is a convex domain in R d .Let r(D r ) be the inradius of D r and I r = ( −r(D r ), r(D r ) ).
Then Theorem 5.1 in ( 19) and (4.1) imply We now obtained our lower bound on the asymptotic behavior of P z ( τ D > t ).
Proof.Let η < 1.The strong Markov property implies Let w ∈ D \ D r .Then Lemma 2.1 implies that there exists M > 0 such that for all r ≥ M Proof.Let 0 < r 1 < r 2 , then The result follows from Lemma 4.1 and Proposition 3.6.

Applications and examples
In this section we will apply the results of the previous section to the function A straight forward computation shows that φ satisfies the assumptions of Theorem 1.1 and §4, if either 0 ≤ β < 1, and µ ∈ R, or β = 1, and µ < −1.
As mention in (23) the asymptotic behavior of P z ( τ D > t ) is not known for α = 2, β = 1 and µ < −1.However using the well known behavior of the exit times from cones and a suitable approximation of the domain D we can prove that there exists M > 0, c 1 > 0 and c 2 > 0 such that ≤ c 1 exp −c 2 (ln t) 1−µ , for all t ≥ M .
Case II: We now study the case β = 0 and µ ∈ R.That is we now consider φ(x) = ( ln[x + 1] ) µ , where µ ∈ R. In this case we will have subexponential behavior of (1.6). Let

3 )
For any domain D ⊂ R d , we denote by d D (z) to the distance from z to the boundary ∂D.