Yaglom limit for stable processes in cones
aa r X i v : . [ m a t h . P R ] D ec YAGLOM LIMIT FOR STABLE PROCESSES IN CONES
KRZYSZTOF BOGDAN, ZBIGNIEW PALMOWSKI, AND LONGMIN WANG
Abstract.
We give the asymptotics of the tail of the distribution of the first exit time of the isotropic α -stable Lévy process from the Lipschitz cone in R d . We obtain the Yaglom limit for the killed stable processin the cone. We construct and estimate entrance laws for the process from the vertex into the cone. For thesymmetric 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 us to express thetemporal asymptotics of the distribution of the process at infinity by means of the spatial asymptotics ofharmonic functions of the process at the vertex; on the representation of the probability of survival of theprocess in the cone as a Green potential; and on the approximate factorization of the heat kernel of thecone, which secures compactness and yields a limiting (Yaglom) measure by means of Prokhorov’s theorem. Keywords.
Yaglom limit ⋆ stable process ⋆ Lipschitz cone ⋆ quasi-stationary measure ⋆ Green function ⋆ Martin kernel ⋆ excursions. Contents
1. Introduction 22. Preliminaries 33. Full picture 64. Proofs 84.1. Proof of Lemma 7 84.2. Proof of (39) 94.3. Proof of Theorem 3 104.4. Proof of Theorem 5 104.5. Relatively uniform convergence 124.6. Proof of Theorem 1 124.7. Proof of Proposition 2 125. Symmetric Cauchy process on half-line 12References 14
Date : August 5, 2018.2000
Mathematics Subject Classification. K. Bogdan — Z. Palmowski — L. Wang Introduction
Let < α < , d = 1 , , . . . , and let X = { X t , t ≥ } be the isotropic α -stable Lévy process in R d . Wedenote by P x and E x the probability and expectation for the process starting from any x ∈ R d , see Section 2for details. Let Γ ⊂ R d be an arbitrary Lipschitz cone with vertex at the origin . We define(1) τ Γ = inf { t > 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 , (2) lim t →∞ P x (cid:18) X t t /α ∈ A (cid:12)(cid:12) τ Γ > t (cid:19) = µ ( A ) , x ∈ Γ . The above condition τ Γ > t means that X stays, or survives, in Γ for time longer than t . Theorem 1asserts that, given its survival, X t rescaled by t /α has a limiting distribution independent of the startingpoint. We note that rescaling is essential for the limit to be nontrivial. The Yaglom limit µ correspondswith 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 toconsider the effectively ultimate distribution (called a quasi-stationary distribution) [...]Namely, µ is a quasi-stationary distribution for ( t + 1) − /α X t in the following sense. Proposition 2.
Let P µ ( · ) = R Γ P y ( · ) µ (d y ) . For every Borel set A ⊂ R d , (3) P µ (cid:18) X t ( t + 1) /α ∈ A (cid:12)(cid:12) τ Γ > t (cid:19) = µ ( A ) , t ≥ . Note that Y t = ( t + 1) − /α X t is a time-inhomogenous Markov process and under P µ , the law of Y 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 one-dimensional cone Γ = (0 , ∞ ) , Yaglom limits similar to (2), and also using rescaling, were given by Haasand Rivero [45]. Their proofs rely on precise estimates for the tail distribution of exponential functionalsof non-increasing Lévy processes and are completely different from ours. As we will see below, the Yaglomlimit may be obtained from the asymptotics (i.e. limits) of the survival probability P x ( τ Γ > t ) . We note thatsuch asymptotics were studied for the multi-dimensional Brownian motion by DeBlassie [38]. Bañuelos andSmits [5] gave the asymptotics of the heat kernel of the cone in terms of the orthonormal eigenfunctions ofthe Laplace-Beltrami operator on the cone’s spherical cap. Denisov and Wachtel [41] derived a result similarto Theorem 1 for multidimensional random walks by using coupling with the Brownian motion. The taildistribution of τ Γ for the isotropic α -stable Lévy process and wedges Γ ⊂ R was estimated by DeBlassie[39]. Bañuelos and Bogdan [3] also provided estimates but not asymptotics for general cones in R d . Theyused the boundary Harnack principle (BHP), which turns out to be very useful also in our situation, becauseit, in fact, yields the asymptotics of the survival probability P x ( τ Γ > for x → , as we show below. Theseasymptotics are given in Theorem 3, and they lead to Theorem 1, to sharp estimates for the density functionof the Yaglom limit µ , and to the existence and estimates of laws of excursions of the stable process fromthe vertex into the cone, which we give in Theorem 5.Information on quasi-stationary (QS) distributions for time-homogeneous Markov processes can be foundin the classical works of Seneta and Vere-Jones [66], Tweedie [71], Jacka and Roberts [48]. The bibliographicdatabase of Pollet [64] gives detailed history of QS distributions. In particular, Yaglom [73] was the first toexplicitly identify QS distributions for the subcritical Bienaymé-Galton-Watson branching process. Part of aglom limit for stable processes in cones 3 the results on QS distributions concern Markov chains on positive integers with an absorbing state at theorigin [36, 72, 42, 44, 66, 76]. Other objects of study are the extinction probabilities for continuous-timebranching process and the Fleming-Viot process [1, 43, 59]. A separate topic is the one-dimensional Lévyprocesses exiting from the positive half-line. Here the case of the Brownian motion with drift was resolvedby 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] andMandjes et al. [60]. These papers are based on the Wiener-Hopf factorization and Tauberian theorems. Theyare intrinsically one-dimensional and they do not use the boundary asymptotics of harmonic functions orrescaling to obtain the limiting distribution. We also note in passing that these results relate to the behaviorof the one-dimensional Lévy processes and random walks conditioned to stay positive, for which we refer thereader 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 estimatesof the heat kernel of cones and on a formula expressing the survival probability P x ( τ Γ > t ) as a Greenpotential. The latter allows us to obtain the spatial asymptotics of the survival probability at the vertex ofthe cone Γ in terms of the cone’s Martin kernel with the pole at infinity, and is a consequence of BHP. Byscaling we then obtain the asymptotics of the survival probability as t → ∞ . The construction allows for theidentification of the limiting boundary behavior of the heat kernel at the vertex of the cone. Such asymptoticsare 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 estimatesand asymptotics of the Green function, gave the Martin representation of harmonic functions, and resolvedinto 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 theGreen function. Bogdan et al. [23] gave the boundary limits of ratios of harmonic functions and Bogdan etal. [19] gave sharp estimates of the Dirichlet heat kernel. Related works on the Dirichlet problem in conesare given by DeBlassie [39], Kulczycki [54], Kulczycki and Burdzy [30], Méndez-Hernández [62], Bogdanand Jakubowski [21], Michalik [63], Kulczycki and Siudeja [56], and Bogdan and Grzywny [18]. For smoothdomains 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 arbitraryopen sets. We expect similar generalizations for the asymptotics of heat kernels. The present paper onlyresolves the asymptotics of the heat kernel of the fractional Laplacian in the Lipschitz cone at the vertex, sothere 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 wepresent our main results, which complement Theorem 1 and Proposition 2. Most of the proofs are givenin Section 4. In Section 5 we discuss in detail the Cauchy process on the positive half-line and we give aspectral decomposition of its Yaglom limit. 2. Preliminaries
As defined in the introduction, X = { X t , t > } is the isotropic α -stable Lévy process on the Euclideanspace R d . The process is determined by the jump measure with the density function(4) ν ( y ) = 2 α Γ(( d + α ) / π d/ | Γ( − α/ | | y | − d − α , y ∈ R d , where < α < , d = 1 , , . . . . The coefficient in (4) is chosen so that(5) Z R d [1 − cos( ξ · y )] ν ( y )d y = | ξ | α , ξ ∈ R d , K. Bogdan — Z. Palmowski — L. Wang for convenience. Here ξ · y is the Euclidean scalar product and | ξ | is the Euclidean norm. We always assumein this paper that the considered sets, measures and functions are all Borel. The process X is Markovianwith the following time-homogeneous transition probability P t ( x, A ) = Z A p t ( x − y )d y , t > , x ∈ R d , A ⊂ R d , and p t is the smooth real-valued function on R d with the Fourier transform:(6) Z R d p t ( x )e i x · ξ d x = e − t | ξ | α , ξ ∈ R d . In particular, if α = 1 , then X is a Cauchy process, and we have p t ( x ) = Γ(( d + 1) / π − ( d +1) / t (cid:0) | x | + t ) ( d +1) / , see [17, 69]. For every α ∈ (0 , , the infinitesimal generator of X is the fractional Laplacian,(7) ∆ α/ ϕ ( x ) = lim ε ↓ Z | y | >ε [ ϕ ( x + y ) − ϕ ( x )] ν ( y )d y , x ∈ R d , defined at least on smooth compactly supported functions φ ∈ C ∞ c ( R d ) , cf. [15, 17, 7, 28, 49, 65, 75, 57].The following scaling property is a consequence of (6),(8) p t ( x ) = t − d/α p ( t − /α x ) , x ∈ R d , t > . Furthermore,(9) c − (cid:18) t | x | d + α ∧ t − d/α (cid:19) ≤ p t ( x ) ≤ c (cid:18) t | x | d + α ∧ t − d/α (cid:19) , x ∈ R d , t > , see [11, 20, 28] for the explicit constant c . Below we will use the notation f ≈ g when functions f , g ≥ are comparable i.e. their ratio is bounded between two positive constants (uniformly on the whole domainof the functions). In particular we can rewrite (9) as follows:(10) p t ( x ) ≈ t − d/α ∧ t | x | d + α , x ∈ R d , t > . 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 , that is, an open Lipschitz set Γ ⊂ R d such that ∈ ∂ Γ , and if y ∈ Γ and r > then ry ∈ Γ . Recall that an open set D ⊂ R d is calledLipschitz if there exist R > and Λ > such that for every Q ∈ ∂D , there exist a Lipschitz function φ Q : R d − → R with Lipschitz constant not greater than Λ and an orthonormal coordinate system CS Q such thatif y = ( y , . . . , y d − , y d ) in CS Q coordinates, then D ∩ B ( Q, R ) = { y : y d > φ Q ( y , . . . , y d − ) } ∩ B ( Q, R ) . We note that the trivial cones
Γ = R d and Γ = ∅ are excluded from our considerations because we require ∈ ∂ Γ , and the Lipschitz condition excludes, e.g., R d \{ } . 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 − = { 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,(11) p Γ t ( x, y ) := p t ( x, y ) − E x [ τ Γ < t ; p t − τ Γ ( X τ Γ , y )] , x , y ∈ R d , t > , where E x [ τ Γ < t ; p t − τ Γ ( X τ Γ , y )] = R { τ Γ
Second, we use the following boundary Harnack principle (BHP) from Bogdan [14]: There is a constant C = C (Γ , α ) > such that if r > and functions u, v ≥ are regular harmonic in Γ r with respect to X and vanish on Γ c ∩ B r , then(BHP) u ( x ) v ( y ) ≤ Cu ( y ) v ( x ) , x , y ∈ Γ r . Generalizations of (BHP) can be found in [68, 23] for the fractional Laplacian and in [24, 51, 52] for more gen-eral jump Markov processes. By an oscillation-reduction argument, (BHP) implies that lim Γ ∋ x → u ( x ) /v ( x ) exists [3, 23]. Without essential loss of generality, in what follows we assume that := (0 , . . . , , ∈ Γ . The above directly implies the existence of(20) M ( y ) = lim Γ ∋ x, | x |→∞ G Γ ( x, y ) G Γ ( x, ) , y ∈ R d .M is called the Martin kernel with the pole at infinity for Γ . It is the unique nonnegative function on R d thatis regular harmonic with respect to X on every Γ r and such that M = 0 on Γ c and M ( ) = 1 [3, Theorem3.2]. The function is locally bounded on R d and homogeneous of degree β = β (Γ , α ) , that is,(21) M ( x ) = | x | β M ( x/ | x | ) , x = 0 . Furthermore, < β < α . The exponent β is decreasing in Γ and it delicately depends on the geometryof 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 shallsee below, using (BHP) and M we can capture the boundary asymptotics of harmonic functions and someGreen potentials.Following [3] we consider the Kelvin transformation K of M :(22) K ( x ) = | x | α − d M ( x/ | x | ) = | x | α − d − β K ( x/ | x | ) , x = 0 , see also Bogdan and Żak [29] for a more general discussion. The function K is called the Martin kernel at for Γ . Clearly, K ( ) = 1 and K = 0 on Γ c . By [3, Theorem 3.4], K ( x ) = E x [ τ B < ∞ ; K ( X τ B )] , x ∈ R d , for every open set B ⊂ Γ with dist(0 , B ) > . In particular K is α -harmonic in Γ and satisfies(23) K ( y ) = lim Γ ∋ x → G Γ ( x, y ) G Γ ( x, ) , y ∈ R d \ { } . Full picture
Theorem 1 and Proposition 2 are manifestations of phenomena which we present later in this section.By (BHP), a finite positive limit(24) C = lim Γ ∋ x → G Γ ( x, ) /M ( x ) exists. We denote(25) κ Γ ( z ) = Z Γ c ν ( z − y )d y , where ν is a jump measure defined in (4), and we define(26) C = C Z Γ Z Γ K ( y ) p Γ1 ( y, z ) κ Γ ( z )d z d y . aglom limit for stable processes in cones 7 Theorem 3.
We have < C < ∞ and (27) lim Γ ∋ x → P x ( τ Γ > M ( x ) = C . The proofs of Theorem 3 and the other results of this section are mostly deferred to Section 4, to allowfor a streamlined presentation.Theorem 3, the scaling property of X and the β -homogeneity of M , that is (21) and (8), yield the followingresult, which refines Lemma 4.2 of Bañuelos and Bogdan [3]. Corollary 4.
Uniformly as Γ ∋ t − /α x → we have (28) P x ( τ Γ > t ) ∼ C M ( x ) t − β/α . Another consequence of Theorem 3 is the following theorem, which is the main result of the paper.
Theorem 5.
The following limit exits, (29) n t ( y ) = lim Γ ∋ x → p Γ t ( x, y ) P x ( τ Γ > , ( t, y ) ∈ (0 , ∞ ) × Γ . It is a finite strictly positive jointly continuous function of t and y , and we have (30) µ ( A ) = Z A n ( z ) d z , A ⊂ Γ . Furthermore, for < s, t < ∞ , y ∈ Γ , n t ( y ) = t − ( d + β ) /α n ( t − /α y ) , (31) n ( y ) ≈ P y ( τ Γ > | y | ) d + α , (32) n t + s ( y ) = Z Γ n t ( z ) p Γ s ( z, y )d z . (33)In view of (33), n t ( y )d y defines an entrance law of excursions surviving at least time t from into Γ , cf.Rivero and Haas [45], Blumenthal [10, page 104] and Bañuelos et al. [2].We note that Yano [74] studies excursions of symmetric Lévy processes into R \ { } , a situation notdiscusses in this paper. We however note that in our situation Z Γ n t ( x )d x = t − β/α , t > , which nicely corresponds with [74, Example 1.1], because for Γ = R \ { } and α ∈ (1 , , β = α − , see [3]. Example . If d = 1 , Γ is the half-line (0 , ∞ ) , then M ( x ) = x α/ for x > [3, Example 3.2]. By [19,Theorem 2], P x ( τ Γ > ≈ x α/ ∧ , x > . By (32), n ( x ) ≈ x α/ ∧ x − d − α , x > . Therefore by (31),(34) n t ( x ) ≈ ( x α/ t − − /α ) ∧ ( x − − α t / ) , t, x > . 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. K. Bogdan — Z. Palmowski — L. Wang
Some of the other objects we study can also be expressed in terms of n . Namely we have(35) K ( y ) = lim Γ ∋ x → G Γ ( x, y ) P x ( τ Γ > P x ( τ Γ > M ( x ) M ( x ) G Γ ( x, )= C C lim Γ ∋ x → Z ∞ p Γ t ( x, y ) P x ( τ Γ >
1) d t = C C Z ∞ n t ( y )d t . Therefore,(36) C C = Z ∞ n t ( )d t may be interpreted as the expected amount of time spent at 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 ofBlumenthal and Getoor [11, 12] (see also [37]), who showed that p t ( x ) ∼ tν ( x ) as t | x | − α → . More resultsof 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(37) P x (cid:18) τ Γ > t, X t t /α ∈ A (cid:19) = P t − /α x ( τ Γ > , X ∈ A ) and(38) P x ( τ Γ > t ) = P t − /α x ( τ Γ > . Then the Ikeda-Watanabe formula (19) gives the representation(39) P x ( τ Γ >
1) = G Γ P Γ1 κ Γ ( x ) . 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 Γ1 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 . Lemma 7. If f is a measurable function on Γ , bounded on Γ and G Γ | f | ( ) < ∞ , then (40) lim Γ ∋ x → G Γ f ( x ) M ( x ) = C Z Γ K ( y ) f ( y )d y < ∞ . Proofs
Proof of Lemma 7. If G Γ | f | ( x ) < ∞ for some x ∈ Γ , then by [16, Lemma 5.1], G Γ | f | ( x ) < ∞ foralmost all x ∈ R d . Let < δ < . Choose x ∈ Γ δ/ so that G Γ | f | ( x ) < ∞ . By (BHP),(41) G Γ ( x, y ) G Γ ( x, ) ≤ c G Γ ( x , y ) G Γ ( x , ) , x , y ∈ Γ , | x | < δ/ , | y | ≥ δ for some constant c . By (23), (41), and Fatou’s lemma we see that the right-hand side of (41) is finite.Observe that we also have(42) Z Γ \ Γ δ G Γ ( x , y ) | f ( y ) | d y ≤ Z Γ G Γ ( x , y ) | f ( y ) | d y < ∞ . By (23), (41), (42) and the dominated convergence theorem,(43) lim Γ ∋ x → Z Γ \ Γ δ G Γ ( x, y ) G Γ ( x, ) f ( y )d y = Z Γ \ Γ δ K ( y ) f ( y )d y . aglom limit for stable processes in cones 9 Next we consider the integral over Γ δ . By our assumptions on f , a change of variables, and the scalingproperty G Γ ( δx, δy ) = δ − d + α G Γ ( x, y ) we can conclude that for some constant c , Z Γ δ G Γ ( x, y ) | f ( y ) | d y ≤ c δ d Z Γ G Γ ( x, δz )d z = c δ α Z Γ G Γ ( δ − x, z )d z , x ∈ Γ δ . (44)Now, by (BHP), for some contant c ,(45) G Γ ( v, y ) M ( y ) ≤ c G Γ ( v, ) M ( ) = c G Γ ( v, ) , v ∈ Γ \ Γ , y ∈ Γ . Indeed, by the symmetry of G Γ , the regular harmonicity of y G Γ ( v, y ) on Γ follows from (18). Further-more, the continuity of α -harmonic functions allows us to use in (45). By (18) and (45) we have G Γ ( x, y ) = G Γ ( x, y ) + E x (cid:2) G Γ ( X τ Γ2 , y ) (cid:3) ≤ G Γ ( x, y ) + c E x (cid:2) G Γ ( X τ Γ2 , ) (cid:3) M ( y ) ≤ G Γ ( x, y ) + c G Γ ( x, ) M ( y ) , x , y ∈ Γ . (46)By identities (44), (46) and the local boundedness of M , we have(47) Z Γ δ G Γ ( x, y ) | f ( y ) | d y ≤ c δ α Z Γ G Γ ( δ − x, z )d z + c δ α G Γ ( δ − x, ) for every x ∈ Γ δ . Let c = inf z ∈ Γ Z Γ \ Γ ν ( z − w )d w . Clearly, c > . By the Ikeda–Watanabe formula (19) and (BHP), for x ∈ Γ δ , Z Γ G Γ ( δ − x, z )d z ≤ c − Z Γ \ Γ Z Γ G Γ ( δ − x, z ) A d,α | w − z | d + α d z d w ≤ c − P δ − x (cid:0) X τ Γ2 ∈ Γ (cid:1) ≤ c G Γ ( δ − x, ) . Again by (BHP) we have G Γ ( δ − x, ) ≈ M ( δ − x ) = δ − β M ( x ) ≈ δ − β G Γ ( x, ) for x ∈ Γ δ/ . In view of (47),(48) Z Γ δ G Γ ( x, y ) G Γ ( x, ) | f ( y ) | d y ≤ c δ α − β , x ∈ Γ δ/ . From (43), Fatou’s lemma and (48) it follows that Z Γ \ Γ δ K ( y ) f ( y )d y ≤ lim inf Γ ∋ x → G Γ f ( x ) G Γ ( x, ) ≤ lim sup Γ ∋ x → G Γ f ( x ) G Γ ( x, ) ≤ Z Γ \ Γ δ K ( y ) f ( y )d y + c δ α − β . Taking the limit in the above identity as δ → and using the fact that α > β , we establish that(49) lim Γ ∋ x → G Γ f ( x ) G Γ ( x, ) = Z Γ K ( y ) f ( y )d y . We then apply (24), which completes the proof. (cid:3)
Proof of (39).
We observe that by the Lipschitz condition and Sztonyk [70, Theorem 1] we have P x ( X τ Γ ∈ ∂ Γ) = 0 for every x ∈ Γ . Thus,(50) P x ( X τ Γ − = X τ Γ ) = 0 . K. Bogdan — Z. Palmowski — L. Wang
Now for x ∈ Γ c we have τ Γ = 0 P x -a.s. and (39) holds true. To prove (39) for x ∈ Γ , observe that by (19)and (50), P x ( τ Γ >
1) = Z ∞ Z Γ p Γ s ( x, z ) κ Γ ( z )d z d s = Z ∞ Z Γ Z Γ p Γ s ( x, w ) p Γ1 ( w, z )d w κ Γ ( z )d z d s = G Γ P Γ1 κ Γ ( x ) . (cid:3) Proof of Theorem 3.
By [53] we have G Γ ( x, w ) > for all x, w ∈ Γ . Thus P Γ1 κ Γ is finite almosteverywhere. Furthermore, the following estimate holds:(51) P Γ1 κ Γ ( x ) ≈ P x ( τ Γ > for x ∈ Γ . Indeed, if | x | ≤ , then by (10),(52) p ( x, y ) ≈ ∧ | x − y | − d − α ≈ (1 + | y | ) − d − α . Furthermore, by [18] or [19, Theorem 2], the following approximate factorization holds(53) p Γ1 ( x, y ) ≈ P x ( τ Γ > P y ( τ Γ > p ( x, y ) , x , y ∈ Γ . Thus,(54) P Γ1 κ Γ ( x ) ≈ P x ( τ Γ > Z Γ P y ( τ Γ >
1) (1 + | y | ) − d − α κ Γ ( y )d y, x ∈ Γ , which finishes the proof of (51). Note also that the integral on the right-hand side of (54) is finite. Thisallows us to establish the asymptotic behavior of P x ( τ Γ > . The estimate (51) ensures that f ( x ) = P Γ1 κ Γ ( x ) satisfies the assumptions of Lemma 7. Thus (27) is a direct consequence of the representation (39) and theidentity (40). This completes the proof. (cid:3) Proof of Theorem 5.
Consider the family of measures { µ x : x ∈ Γ } defined by(55) µ x ( A ) = R A p Γ1 ( x, y )d y P x ( τ Γ > , x ∈ Γ , A ⊂ R d . We start by proving that the above family of measures is tight. Indeed, by (53) and (52) we can bound thedensity of µ x by the integrable function:(56) p Γ1 ( x, y ) P x ( τ Γ > ≈ P y ( τ Γ > | y | ) − d − α , x ∈ Γ , y ∈ R d . We will prove now that the measures µ x converge weakly to a probability measure µ on Γ as Γ ∋ x → :(57) µ x ⇒ µ as Γ ∋ x → . To prove (57), we consider an arbitrary sequence { x n } such that Γ ∋ x n → . By the tightness of the familyof measures { µ x : x ∈ Γ } there exists a subsequence { x n k } such that µ x nk converge weakly to a probabilitymeasure µ as k → ∞ .Let φ ∈ C ∞ c (Γ) and u φ = − ∆ α/ φ . The function u φ is bounded and continuous and G Γ u φ ( x ) = φ ( x ) for x ∈ R d , see [22, Eq. (19)] and [23, Eq. (11)] for more details. By (7) we have | u φ ( x ) | ≤ c p ( x, . Then,(58) P Γ1 | u φ | ( x ) ≤ c p ( x ) , aglom limit for stable processes in cones 11 and for every x ∈ Γ , G Γ P Γ1 | u φ | ( x ) ≤ Z R d Z R d G Γ ( x, y ) p ( y, z ) | u φ ( z ) | d z d y ≤ c Z R d G Γ ( x, y ) p ( y, y < ∞ , see [23, Eq. (74)]. By Fubini’s theorem,(59) G Γ P Γ1 u φ ( x ) = P Γ1 G Γ u φ ( x ) = P Γ1 φ ( x ) . It follows from Lemma 7 and (58) that lim Γ ∋ x → P Γ1 φ ( x ) M ( x ) = lim Γ ∋ x → G Γ P Γ1 u φ ( x ) M ( x ) = C Z Γ K ( y ) P Γ1 u φ ( y )d y . Denoting µ x ( φ ) = R Γ φ ( y ) µ x (d y ) and applying Theorem 3 we get a finite limit lim Γ ∋ x → µ x ( φ ) = lim Γ ∋ x → P Γ1 φ ( x ) P x ( τ Γ >
1) = R Γ K ( y ) P Γ1 u φ ( y )d y R Γ K ( y ) P Γ1 κ Γ ( y )d y . In particular, µ ( φ ) = lim k →∞ µ x nk ( φ ) does not depend on the choice of the subsequence { x n k } . Thus, µ x weakly converges to this µ as Γ ∋ x → .We are now in a position to prove that n 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 Γ1 ( · , y ) ∈ C ( R d ) ,(60) p Γ2 ( x, y ) = Z Γ p Γ1 ( x, z ) p Γ1 ( z, y )d z = P Γ1 φ y ( x ) , x , y ∈ Γ . Thus, for all y ∈ Γ ,(61) p Γ2 ( x, y ) P x ( τ Γ >
1) = P Γ1 φ y ( x ) P x ( τ Γ >
1) = µ x ( φ y ) → µ ( φ y ) < ∞ , as Γ ∋ x → , 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 Γ ∋ x → p Γ t ( x, y ) P x ( τ Γ > t/ − d/α lim Γ ∋ x → p Γ2 (cid:0) ( t/ − /α x, ( t/ − /α y (cid:1) P ( t/ − /α x ( τ Γ >
1) lim Γ ∋ x → P x ( τ Γ > t/ P x ( τ Γ > t/ − ( d + β ) /α n (( t/ − /α y ) . This proves the existence of the limit n t ( y ) for general t > , 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 everybounded continuous function φ on Γ we have µ ( φ ) = lim Γ ∋ x → P Γ1 φ ( x ) P x ( τ Γ >
1) = lim Γ ∋ x → Z Γ p Γ1 ( x, y ) P x ( τ Γ > φ ( y )d y = Z Γ n ( y ) φ ( y )d y . This completes the proof of the fact that the limit n ( y ) from (5) is well-defined and gives the densityfunction of the quasi-stationary measure µ . Note that (33) follows directly from the Chapman-Kolmogorovequation and the dominated convergence theorem: n t + s ( y ) = lim Γ ∋ x → Z Γ p Γ t ( x, z ) P x ( τ Γ > p Γ s ( z, y )d z = P Γ s n t ( y ) . To end the proof we show that n t ( y ) is jointly continuous on (0 , ∞ ) × Γ . Indeed, p Γ1 ( z, y ) /p Γ1 ( z, y ) ≈ for z ∈ Γ , if y, y ∈ Γ are close to each other. The continuity of n ( y ) follows from the dominated convergencetheorem and the continuity of p Γ1 . The joint continuity of n t ( y ) follows from the scaling property. (cid:3) K. Bogdan — Z. Palmowski — L. Wang
Relatively uniform convergence.Lemma 8. If ≤ c − f n ≤ f m ≤ cf n for all m, n , and f = lim f n , then lim R f n d η = R f d η . This is true because if the integral R f d η is finite, then the dominated convergence theorem applies.4.6. Proof of Theorem 1.
By the scaling property of X t we have P x (cid:18) X t t /α ∈ A | τ Γ > t (cid:19) = P x (cid:0) τ Γ > t, X t t /α ∈ A (cid:1) P x ( τ Γ > t ) = P t − /α x ( τ Γ > , X ∈ A ) P t − /α x ( τ Γ > R A p Γ1 ( t − /α x, y )d y P t − /α x ( τ Γ > , x ∈ Γ , t > . We have shown in the proof of Theorem 5 that p Γ1 ( t − /α x, y ) / P t − /α x ( τ Γ > converges relatively uniformlyto n ( y ) as t → ∞ , in the sense of the condition in Lemma 8, see (56). This yields the Yaglom limit µ (d x ) = n ( x )d x . (cid:3) Proof of Proposition 2.
Recall that by (30) we have µ (d z ) = n ( z ) d z . By using (33) and (31), for A ⊂ R d we get P µ (cid:18) X t ( t + 1) /α ∈ A, τ Γ > t (cid:19) = Z Γ Z ( t +1) /α A n ( x ) p Γ t ( x, y )d y d x = Z ( t +1) /α A n t +1 ( y )d y = Z ( t +1) /α A ( t + 1) − ( d + β ) /α n (cid:16) ( t + 1) − /α y (cid:17) d y =( t + 1) − β/α Z A n ( y )d y = ( t + 1) − β/α µ ( A ) . In particular, P µ ( τ Γ > t ) = ( t + 1) − β/α , which ends the proof. (cid:3) Symmetric Cauchy process on half-line
Let d = α = 1 and Γ = (0 , ∞ ) . Then X t is the symmetric Cauchy process on R and τ Γ = inf { t ≥ X t < } , which is sometimes called the ruin time. For this particular situation we can add specific spectral informationon the Yaglom limit µ . Following [55], for x > , we let(63) r ( x ) = √ π Z ∞ t (1 + t ) / exp (cid:18) π Z t log s s d s (cid:19) e − tx d t and(64) ψ ( x ) = sin (cid:16) x + π (cid:17) − r ( x ) . Theorem 9. If X t is the symmetric Cauchy process on R and Γ = (0 , ∞ ) , then µ has the density function (65) n ( y ) = lim x → p Γ1 ( x, y ) P x ( τ Γ >
1) = r π Z ∞ λ / ψ ( λy )e − λ d λ , y > . Proof.
By [3, Example 3.2 and 3.4], we have M ( x ) = ( x ∨ / , K ( x ) = ( x ∨ − / , x ∈ R . aglom limit for stable processes in cones 13 The Green function G Γ ( x, y ) is given by the well-known explicit Riesz’s formula:(66) G Γ ( x, y ) = 1 π arcsin s xy ( x − y ) , x , y > see [13] or [31, Theorem 3.3] with m = 0 . Thus the constant C defined in (24) is given by(67) C = lim x → G Γ ( x, M ( x ) = 2 π . For t > we define ξ ( t ) = 1 π t (1 + t ) / exp (cid:18) − π Z t log s s d s (cid:19) . Note that R ∞ log s/ (1 + s )d s = 0 . Thus(68) ξ ( t ) ∼ π t − / as t → ∞ . It follows from [55, Theorem 5] that(69) P x ( τ Γ ∈ d t ) = 1 t ξ (cid:18) tx (cid:19) d t . Therefore(70) P x ( τ Γ >
1) = Z ∞ t ξ (cid:18) tx (cid:19) d t = Z ∞ x − ξ ( t ) t d t ∼ π x / = 2 π M ( x ) as x → and hence C = C , where C is as defined in (26).By [55, (5.8)] we have that ≤ r ( x ) ≤ r (0) = sin( π/ and | ψ ( x ) | ≤ for x ≥ , where r ( x ) and ψ ( x ) are as defined in (63) and (64), respectively. Further, by [55, Theorem 2] the function ψ λ ( x ) = ψ ( λx ) is aneigenfunction of the semigroup P Γ t acting on C (Γ) , with the eigenvalue e − λt . Thus(71) p Γ t ( x, y ) = 2 π Z ∞ ψ λ ( x ) ψ λ ( y )e − λt d λ ; see [55, (7.4)]. Note that r ′ ( x ) = − √ π Z ∞ t (1 + t ) / exp (cid:18) π Z t log s s d s (cid:19) e − tx d t . Since − Z t log s s d s = Z ∞ t log s s d s is positive for all t > and it is regularly varying at ∞ of index − , the following estimates hold for x > : Z t (1 + t ) / exp (cid:18) π Z t log s s d s (cid:19) e − tx d t ≤ , (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ t (1 + t ) / exp (cid:18) π Z t log s s d s (cid:19) e − tx d t − Z ∞ t − / e − tx d t (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z ∞ (cid:12)(cid:12)(cid:12)(cid:12) t (1 + t ) / − t − / (cid:12)(cid:12)(cid:12)(cid:12) d t + Z ∞ t − / (cid:12)(cid:12)(cid:12)(cid:12) − exp (cid:18) π Z t log s s d s (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) d t ≤ π Z ∞ t − / Z ∞ t log s s d s d t < ∞ and (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ t − / e − tx d t − √ πx − / (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z t − / e − tx d t ≤ . K. Bogdan — Z. Palmowski — L. Wang
Thus there exists a constant c > such that for x > ,(72) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r ( x ) − r (0) − r π x / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z x (cid:12)(cid:12)(cid:12)(cid:12) r ′ ( s ) + 1 √ π s − / (cid:12)(cid:12)(cid:12)(cid:12) d s ≤ cx and(73) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ ( x ) − r π x / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) sin (cid:16) x + π (cid:17) − sin π (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r ( x ) − r (0) − r π x / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ cx . The above inequalities and (52) imply that(74) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p Γ1 ( x, y ) − r π x / Z ∞ λ / ψ ( λy )e − λ d λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ cx Z ∞ λψ ( λy )e − λ d λ . The identity (65) now follows from (70) and (74). Since we have (30), the proof is complete. (cid:3)
Acknowledgements.
We thank Victor Rivero for discussions on quasi-stationary distributions. Wethank Gavin Armstrong for comments on the paper.
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