Yaglom limit for stable processes in cones

We give the asymptotics of the tail of the distribution of the first exit time of the isotropic $\alpha$-stable L\'evy process from the Lipschitz cone in $\mathbb{R}^d$. We obtain the Yaglom limit for the killed stable process for the cone. We construct and estimate entrance laws for the process from the vertex into the cone. For the symmetric Cauchy process and the positive half-line we give a spectral representation of the Yaglom limit. Our approach relies on the scalings of the stable process and the cone, which allow to express the temporal asymptotics of the distribution of the process at infinity by means of the spatial asymptotics of harmonic functions of the process at the vertex; on the representation of the probability of survival of the process in the cone as a Green potential; and on the approximate factorization of the heat kernel of the cone, which secures compactness and yields a limiting (Yaglom) measure by means of Prokhorov's theorem.


Introduction
Let 0 < α < 2, d = 1, 2, . . ., and let X = {X t , t ≥ 0} be the isotropic α-stable Lévy process in R d . We denote by P x and E x the probability and expectation for the process starting from any x ∈ R d , see Section 2 for details. Let Γ ⊂ R d be an arbitrary Lipschitz cone with vertex at the origin 0. We define (1) τ Γ = inf{t > 0 : X t / ∈ Γ} , the time of the first exit of X from Γ. The following measure µ will be called the Yaglom limit for X and Γ.
Theorem 1. There is a probability measure µ concentrated on Γ such that for every Borel set A ⊂ R d , The above condition τ Γ > t means that X stays, or survives, in Γ for time longer than t. Theorem 1 asserts that, given its survival, X t rescaled by t 1/α has a limiting distribution independent of the starting point. We note that rescaling is essential for the limit to be nontrivial. The Yaglom limit µ corresponds with the idea of 'quasi-stationarity', as expressed by Bartlett [6]: It still may happen that the time to extinction is so long that it is still of more relevance to consider the effectively ultimate distribution (called a quasi-stationary distribution) [...] Namely, µ is a quasi-stationary distribution for (t + 1) −1/α X t in the following sense.
Proposition 2. Let P µ (·) = Γ P y (·) µ(dy). For every Borel set A ⊂ R d , Note that Y t = (t + 1) −1/α X t is a time-inhomogenous Markov process and under P µ , the law of Y 0 is µ. This is the first paper where the Yaglom limit is identified for the multi-dimensional α-stable Lévy processe. For the one-dimensional self-similar processes, including the symmetric α-stable Lévy process in the onedimensional cone Γ = (0, ∞), Yaglom limits similar to (2), and also using rescaling, were given by Haas and Rivero [45]. Their proofs rely on precise estimates for the tail distribution of exponential functionals of non-increasing Lévy processes and are completely different from ours. As we will see below, the Yaglom limit may be obtained from the asymptotics (i.e. limits) of the survival probability P x (τ Γ > t). We note that such asymptotics were studied for the multi-dimensional Brownian motion by DeBlassie [38]. Bañuelos and Smits [5] gave the asymptotics of the heat kernel of the cone in terms of the orthonormal eigenfunctions of the Laplace-Beltrami operator on the cone's spherical cap. Denisov and Wachtel [41] derived a result similar to Theorem 1 for multidimensional random walks by using coupling with the Brownian motion. The tail distribution of τ Γ for the isotropic α-stable Lévy process and wedges Γ ⊂ R 2 was estimated by DeBlassie [39]. Bañuelos and Bogdan [3] also provided estimates but not asymptotics for general cones in R d . They used the boundary Harnack principle (BHP), which turns out to be very useful also in our situation, because it, in fact, yields the asymptotics of the survival probability P x (τ Γ > 1) for x → 0, as we show below. These asymptotics are given in Theorem 3, and they lead to Theorem 1, to sharp estimates for the density function of the Yaglom limit µ, and to the existence and estimates of laws of excursions of the stable process from the vertex into the cone, which we give in Theorem 5.
Information on quasi-stationary (QS) distributions for time-homogeneous Markov processes can be found in the classical works of Seneta and Vere-Jones [66], Tweedie [71], Jacka and Roberts [48]. The bibliographic database of Pollet [64] gives detailed history of QS distributions. In particular, Yaglom [73] was the first to explicitly identify QS distributions for the subcritical Bienaymé-Galton-Watson branching process. Part of the results on QS distributions concern Markov chains on positive integers with an absorbing state at the origin [36,72,42,44,66,76]. Other objects of study are the extinction probabilities for continuous-time branching process and the Fleming-Viot process [1,43,59]. A separate topic is the one-dimensional Lévy processes exiting from the positive half-line. Here the case of the Brownian motion with drift was resolved by Martinez and San Martin [61], complementing the result for random walks obtained by Iglehart [46]. The case of jump Lévy processes was studied by E. Kyprianou [58], A. Kyprianou and Palmowski [59] and Mandjes et al. [60]. These papers are based on the Wiener-Hopf factorization and Tauberian theorems. They are intrinsically one-dimensional and they do not use the boundary asymptotics of harmonic functions or rescaling to obtain the limiting distribution. We also note in passing that these results relate to the behavior of the one-dimensional Lévy processes and random walks conditioned to stay positive, for which we refer the reader to Bertoin [8], Bertoin and Doney [9], and Chaumont and Doney [32].
On a general level our development depends on a compactness argument based on recent sharp estimates of the heat kernel of cones and on a formula expressing the survival probability P x (τ Γ > t) as a Green potential. The latter allows us to obtain the spatial asymptotics of the survival probability at the vertex of the cone Γ in terms of the cone's Martin kernel with the pole at infinity, and is a consequence of BHP. By scaling we then obtain the asymptotics of the survival probability as t → ∞. The construction allows for the identification of the limiting boundary behavior of the heat kernel at the vertex of the cone. Such asymptotics are completely new, and may be regarded as a culmination of the study of the Dirichlet fractional Laplacian, which started with boundary estimates and asymptotics of harmonic functions, developed into estimates and asymptotics of the Green function, gave the Martin representation of harmonic functions, and resolved into sharp estimates of the heat kernel. The development was initiated by Bogdan [14] and Song and Wu [68] with proofs of (BHP) for the fractional Laplacian. Then Jakubowski [50] gave sharp estimates of the Green function. Bogdan et al. [23] gave the boundary limits of ratios of harmonic functions and Bogdan et al. [19] gave sharp estimates of the Dirichlet heat kernel. Related works on the Dirichlet problem in cones are given by DeBlassie [39], Kulczycki [54], Kulczycki and Burdzy [30], Méndez-Hernández [62], Bogdan and Jakubowski [21], Michalik [63], Kulczycki and Siudeja [56], and Bogdan and Grzywny [18]. For smooth domains we refer to the pioneering works by Kulczycki [53], Song and Chen [35] and Kim et al. [51,52]. Historically, the results for cones often preceded and informed generalizations to Lipschitz and arbitrary open sets. We expect similar generalizations for the asymptotics of heat kernels. The present paper only resolves the asymptotics of the heat kernel of the fractional Laplacian in the Lipschitz cone at the vertex, so there is much more work left to do.
The paper is organized as follows. In Section 2 we give basic notation and facts. In Section 3 we present our main results, which complement Theorem 1 and Proposition 2. Most of the proofs are given in Section 4. In Section 5 we discuss in detail the Cauchy process on the positive half-line and we give a spectral decomposition of its Yaglom limit.

Preliminaries
As defined in the introduction, X = {X t , t > 0} is the isotropic α-stable Lévy process on the Euclidean space R d . The process is determined by the jump measure with the density function where 0 < α < 2, d = 1, 2, . . .. The coefficient in (4) is chosen so that for convenience. Here ξ · y is the Euclidean scalar product and |ξ| is the Euclidean norm. We always assume in this paper that the considered sets, measures and functions are all Borel. The process X is Markovian with the following time-homogeneous transition probability and p t is the smooth real-valued function on R d with the Fourier transform: In particular, if α = 1, then X is a Cauchy process, and we have [17,69]. For every α ∈ (0, 2), the infinitesimal generator of X is the fractional Laplacian, [15,17,7,28,49,65,75,57]. The following scaling property is a consequence of (6), Furthermore, [11,20,28] for the explicit constant c. Below we will use the notation f ≈ g when functions f , g ≥ 0 are comparable i.e. their ratio is bounded between two positive constants (uniformly on the whole domain of the functions). In particular we can rewrite (9) as follows: We will also write lim f (x)/g(x) = 1 as f (x) ∼ g(x).
As stated, Γ denotes a generalized Lipschitz cone in R d with vertex 0, that is, an open Lipschitz set Γ ⊂ R d such that 0 ∈ ∂Γ, and if y ∈ Γ and r > 0 then ry ∈ Γ.
Recall that an open set D ⊂ R d is called Lipschitz if there exist R > 0 and Λ > 0 such that for every Q ∈ ∂D, there exist a Lipschitz function φ Q : We note that the trivial cones Γ = R d and Γ = ∅ are excluded from our considerations because we require 0 ∈ ∂Γ, and the Lipschitz condition excludes, e.g., R d \{0}. In particular for d = 1, Γ is necessarily a half-line. We note that the cone Γ is characterized by its intersection with the unit sphere S d−1 = {x ∈ R d : |x| = 1}. The first exit time from Γ, as defined in (1), yields the heat kernel p Γ t (x, y) of the cone, For bounded or nonnegative functions f we have We also note that , and p Γ satisfies the Chapman-Kolmogorov equations: see, e.g., [15,27,33]. Since Γ is Lipschitz, by the exterior cone condition and Blumenthal 0-1 law, From (8), the following scaling property follows: By (14), a similar scaling holds for the survival probability: We define the Green function of Γ: and the Green operator: for integrable or nonnegative functions f . Clearly, by (12), We should note that G Γ is always locally integrable because Γ = R and Γ = R \ {0}, cf. [29].
For r ∈ (0, ∞) we let B r = {|x| < r} and we define the truncated cone By the strong Markov property, for t > 0 and x, y ∈ R d , Integrating the above identity against dt (see [23, eq. (15)]), we obtain where we assume that the expectation is finite. Two more facts are crucial in our development.
The above directly implies the existence of M is called the Martin kernel with the pole at infinity for Γ. It is the unique nonnegative function on R d that is regular harmonic with respect to X on every Γ r and such that M = 0 on Γ c and M (1) = 1 [3, Theorem 3.2]. The function is locally bounded on R d and homogeneous of degree β = β(Γ, α), that is, Furthermore, 0 < β < α. The exponent β is decreasing in Γ and it delicately depends on the geometry of Γ. When Γ is a right-circular cone, a rather explicit estimate for M is available [63, Theorem 3.13], expressed in terms of β. More information on β for narrow right-circular cones is given in [26]. As we shall see below, using (BHP) and M we can capture the boundary asymptotics of harmonic functions and some Green potentials.
Following [3] we consider the Kelvin transformation K of M : see also Bogdan and Żak [29] for a more general discussion. The function K is called the Martin kernel at 0 for Γ. Clearly, K(1) = 1 and K = 0 on Γ c . By [3, Theorem 3.4], for every open set B ⊂ Γ with dist(0, B) > 0. In particular K is α-harmonic in Γ and satisfies

Full picture
Theorem 1 and Proposition 2 are manifestations of phenomena which we present later in this section. By (BHP), a finite positive limit exists. We denote where ν is a jump measure defined in (4), and we define Theorem 3. We have 0 < C 1 < ∞ and The proofs of Theorem 3 and the other results of this section are mostly deferred to Section 4, to allow for a streamlined presentation.
Theorem 3, the scaling property of X and the β-homogeneity of M , that is (21) and (8), yield the following result, which refines Lemma 4.2 of Bañuelos and Bogdan [3].
Another consequence of Theorem 3 is the following theorem, which is the main result of the paper. Theorem 5. The following limit exits, It is a finite strictly positive jointly continuous function of t and y, and we have Furthermore, for 0 < s, t < ∞, y ∈ Γ, n t+s (y) = Γ n t (z)p Γ s (z, y)dz .
The first expression gives the minimum if x α ≤ t (small space), and the second -if x α > t (short time). Estimates of n t (x) for half-spaces in R d may be obtained in a similar way.
Some of the other objects we study can also be expressed in terms of n. Namely we have Therefore, may be interpreted as the expected amount of time spent at 1 by the excursion from the vertex into Γ.
We note in passing that the spatial asymptotics of the heat kernel at infinity were given in the works of Blumenthal and Getoor [11,12] (see also [37]), who showed that p t (x) ∼ tν(x) as t|x| −α → 0. More results of this type for unimodal Lévy processes can be found in recent works of Tomasz Grzywny et al., including [37], however, the above papers only concern Γ = R d .
Our approach to Theorem 5 depends on three properties. First, the scaling (8) yields Then the Ikeda-Watanabe formula (19) gives the representation . Recall that κ Γ (x) may be considered as the killing intensity because it is the intensity of jumps of X from x to Γ c . Similarly, P Γ 1 κ Γ (x) may be interpreted as the intensity of killing precisely one unit of time from now. To actually prove the existence of n t in (29), we use the asymptotics of Green potentials at the vertex 0.
Observe that we also have By (23), (41), (42) and the dominated convergence theorem, Next we consider the integral over Γ δ . By our assumptions on f , a change of variables, and the scaling property G Γ (δx, δy) = δ −d+α G Γ (x, y) we can conclude that for some constant c 2 , Now, by (BHP), for some contant c 3 , Indeed, by the symmetry of G Γ , the regular harmonicity of y → G Γ (v, y) on Γ 1 follows from (18). Furthermore, the continuity of α-harmonic functions allows us to use 1 in (45). By (18) and (45) we have By identities (44), (46) and the local boundedness of M , we have for every x ∈ Γ δ . Let Clearly, c 4 > 0. By the Ikeda-Watanabe formula (19) and (BHP), for x ∈ Γ δ ,

From (43), Fatou's lemma and (48) it follows that
Taking the limit in the above identity as δ → 0 and using the fact that α > β, we establish that We then apply (24), which completes the proof.

4.3.
Proof of Theorem 3. By [53] we have G Γ (x, w) > 0 for all x, w ∈ Γ. Thus P Γ 1 κ Γ is finite almost everywhere. Furthermore, the following estimate holds: (10), Furthermore, by [18] or [19,Theorem 2], the following approximate factorization holds which finishes the proof of (51). Note also that the integral on the right-hand side of (54) is finite. This allows us to establish the asymptotic behavior of P x (τ Γ > 1). The estimate (51) ensures that f (x) = P Γ 1 κ Γ (x) satisfies the assumptions of Lemma 7. Thus (27) is a direct consequence of the representation (39) and the identity (40). This completes the proof.

Proof of Theorem 5. Consider the family of measures {µ
We start by proving that the above family of measures is tight. Indeed, by (53) and (52) we can bound the density of µ x by the integrable function: We will prove now that the measures µ x converge weakly to a probability measure µ on Γ as Γ ∋ x → 0: To prove (57), we consider an arbitrary sequence {x n } such that Γ 1 ∋ x n → 0. By the tightness of the family of measures {µ x : x ∈ Γ 1 } there exists a subsequence {x n k } such that µ xn k converge weakly to a probability measure µ as k → ∞.
We are now in a position to prove that n 1 is the density of the Yaglom limit µ appearing in (57) and that n t is well-defined. By the Chapman-Kolmogorov equation applied to φ y (·) = p Γ 1 (·, y) ∈ C 0 (R d ), Thus, for all y ∈ Γ, as Γ ∋ x → 0, see Theorem 3 and (57). This proves the existence of the limit n t defined in (29) for t = 2. Using the existence of this limit, the scaling property and Theorem 3 we can conclude now that for any (t, y) ∈ (0, ∞) × Γ the following holds true (62) n t (y) = lim This proves the existence of the limit n t (y) for general t > 0, and the equation (31). By (56) we get (32). By the weak convergence (57), Theorem 3, and the dominated convergence theorem, we get that for every bounded continuous function φ on Γ we have This completes the proof of the fact that the limit n 1 (y) from (5) is well-defined and gives the density function of the quasi-stationary measure µ. Note that (33) follows directly from the Chapman-Kolmogorov equation and the dominated convergence theorem: To end the proof we show that n t (y) is jointly continuous on (0, ∞) × Γ. Indeed, p Γ 1 (z, y)/p Γ 1 (z, y 1 ) ≈ 1 for z ∈ Γ, if y, y 1 ∈ Γ are close to each other. The continuity of n 2 (y) follows from the dominated convergence theorem and the continuity of p Γ 1 . The joint continuity of n t (y) follows from the scaling property. This is true because if the integral f dη is finite, then the dominated convergence theorem applies.
4.6. Proof of Theorem 1. By the scaling property of X t we have We have shown in the proof of Theorem 5 that p Γ 1 (t −1/α x, y)/P t −1/α x (τ Γ > 1) converges relatively uniformly to n 1 (y) as t → ∞, in the sense of the condition in Lemma 8, see (56). This yields the Yaglom limit µ(dx) = n 1 (x)dx.

Symmetric Cauchy process on half-line
Let d = α = 1 and Γ = (0, ∞). Then X t is the symmetric Cauchy process on R and which is sometimes called the ruin time. For this particular situation we can add specific spectral information on the Yaglom limit µ. Following [55], for x > 0, we let and (64) ψ(x) = sin x + π 8 − r(x) .