Alpha-stable random walk has massive thorns
aa r X i v : . [ m a t h . P R ] F e b α -STABLE RANDOM WALK HAS MASSIVE THORNS ALEXANDER BENDIKOV AND WOJCIECH CYGAN
Abstract.
We introduce and study a class of random walks definedon the integer lattice Z d – a discrete space and time counterpart of thesymmetric α -stable process in R d . When < α < any coordinateaxis in Z d , d ≥ , is a non-massive set whereas any cone is massive. Weprovide a necessary and sufficient condition for the thorn to be a massiveset. Introduction
Motivating questions.
This paper is motivated by the following twoclosely related questions.1. Assuming that the probability φ on the group Z d is symmetric andits support generates the whole Z d , what is the possible decay of the Greenfunction G ( x ) = X n ≥ φ ( n ) ( x ) as x tend to infinity?2. If φ is as above, which sets are massive/recurrent with respect to therandom walk driven by φ ?Recall that the answer to the first question is known when φ is symmetric,has finite second moment and d ≥ . Indeed, it is proved in
Spitzer [19](see also
Saloff-Coste and Hebisch [11] for the treatment of generalfinitely generated groups) that G ( x ) ∼ c ( φ ) k x k − d at infinity. When thesecond moment of φ is infinite but φ belongs to the domain of attractionof the α -stable law with d/ < α < min { d, } , G ( x ) ∼ c ( φ ) k x k α − d l ( k x k ) Research of A. Bendikov was supported by National Science Centre, Poland, GrantDEC-2012/05/B/ST1/00613 and by SFB 701 of German Research Council.Research of W. Cygan was supported by: National Science Centre, Poland, GrantDEC-2012/05/B/ST1/00613 and DEC-2013/11/N/ST1/03605; German Academic Ex-change Service (DAAD); SFB 701 of German Research Council.2010
Mathematics Subject Classification : 31A15, 60J45, 05C81.
Key words and phrases : capacity, Green function, Lévy process, random walk, regularvariation, subordination. -STABLE RANDOM WALK HAS MASSIVE THORNS 2 at infinity, where l is an appropriately chosen slowly varying function, see Williamson [20]. However, there are many symmetric probabilities φ forwhich the behaviour of the Green function G at infinity is not known.In the present paper we use discrete subordination, a natural techniquedeveloped in Bendikov and Saloff-Coste [3] that produces interestingexamples of probabilities φ for which one can estimate the behaviour of theGreen function G at infinity. This in turn allows us to describe massivenessof some interesting classes of infinite sets. For instance, we give necessaryand sufficient conditions for the thorn to be a massive set, see Section 4.Massiveness of thorns for the simple random walk in Z d , d ≥ , was studiedin the celebrated paper of Itô and McKean [12].The main idea behind this technique is the well-known idea of subordi-nation in the context of continuous time Markov semigroups but the ap-plications we have in mind require some adjustments and variations. Theresults we obtain shed some light on the questions formulated above. Thepresent paper is concerned with examples when φ has neither finite supportnor finite second moment. Subordinated random walks.
In the case of continuous time Markovprocesses, subordination is a well-known and useful procedure of obtainingnew process from an original process. The new process may differ verymuch from the original process, but the properties of this new process canbe understood in terms of the original process. The best known applicationof this concept is obtaining the symmetric stable process from the Brownianmotion. See e.g.
Bendikov [1].From a probabilistic point of view, a new process ( Y t ) t> is obtainedfrom the original process ( X t ) t> by setting Y t = X ς t , where the "subordi-nator" ( ς s ) s> is a nondecreasing Lévy process taking values in (0 , ∞ ) andindependent of ( X t ) t> . See e.g.
Feller [8, Section X.7].From an analytical point of view, the transition function h ς ( t, x, B ) ofthe new process is obtained as a time average of the transition function h ( t, x, B ) of the original process, that is, h ς ( t, x, B ) = ∞ Z h ( s, x, B )d µ t ( s ) . -STABLE RANDOM WALK HAS MASSIVE THORNS 3 In this formula µ t ( s ) is the distribution of the random variable ς t . Subordi-nation was first introduced by Bochner in the context of semigroup theory.See [8, footnote, p. 347].Ignoring technical details, the minus infinitesimal generator B of theprocess ( Y t ) t> is a function of the minus infinitesimal generator A of theprocess ( X t ) t> , that is, B = ψ ( A ) . See
Jacob [13, Chapters 3 & 4] for adetailed discussion.A discrete time version of subordination in which the functional calculusequation B = ψ ( A ) serves as the defining starting point has been consideredby Bendikov and Saloff-Coste in [3]. Given a probability φ on Z d considerthe random walk X = { X ( n ) } n ≥ driven by φ . In its simplest form, discretesubordination is the consideration of a probability Φ defined as a convexlinear combination of the convolution powers φ ( n ) . That is, Φ = X n ≥ c n φ ( n ) , where c n ≥ and P n ≥ c n = 1 . We easily find that Φ ( n ) = X k ≥ n (cid:16) X k + ... + k n = k n Y i =1 c k i (cid:17) φ ( k ) . The probabilistic interpretation is as follows: let ( R i ) be a sequence of i.i.d.integer valued random variables, which are independent of X and such that P ( R i = k ) = c k . Set τ n = R + . . . + R n , then P ( τ n = k ) = X k + ... + k n = k n Y i =1 c k i and Φ ( n ) is the law of Y ( n ) = X ( τ n ) .The other way to introduce the notion of discrete subordination is touse Markov generators. Let P be the operator of convolution by φ . Theoperator L = I − P may be considered as minus the Markov generator ofthe associated random walk. For a proper function ψ we want to define a"subordinated" random walk with Markov generator − ψ ( L ) . The appro-priate class of functions is the class of Bernstein functions, see the book Schilling, Song and Vondraček [18].Recall that a function ψ ∈ C ∞ ( R + ) is called a Bernstein function if itis non-negative and ( − n − ψ ( n ) ( x ) ≥ , for all x > and all n ∈ N . Theset of all Bernstein functions we denote by BF . Each function ψ ∈ BF has -STABLE RANDOM WALK HAS MASSIVE THORNS 4 the following representation ψ ( θ ) = a + bθ + Z (0 , ∞ ) (cid:0) − e − θs (cid:1) d ν ( s ) , (1.1)for some constants a, b ≥ and some measure ν (the Lévy measure) suchthat Z (0 , ∞ ) min { , s } d ν ( s ) < ∞ . Proposition 1.1. [3, Proposition 2.3]
Assume that ψ is a Bernstein func-tion with its representation (1.1), such that ψ (0) = 0 , ψ (1) = 1 and set c ( ψ,
1) = b + Z (0 , ∞ ) te − t d ν ( t ) ,c ( ψ, n ) = 1 n ! Z (0 , ∞ ) t n e − t d ν ( t ) , n > . (1.2) Let φ be a probability on Z d . Let P be the operator of convolution by φ andset P ψ = I − ψ ( I − P ) . (1.3) Then P ψ is the convolution by a probability Φ defined as Φ = X n ≥ c ( ψ, n ) φ ( n ) . (1.4) Example 1.2.
The power function ψ α ( s ) = s α/ , α ∈ (0 , belongs to theclass BF . Its Lévy density ν α ( t ) is given by ν α ( t ) = α/ − α/ t − − α/ . The probabilities c ( ψ α , n ) are given by c ( ψ α , n ) = α/ − α/
2) Γ( n − α/ n + 1) ∼ α/ − α/ n − − α/ . Choosing ψ = ψ α in Proposition 1.1, we see that the Markov generators ofthe initial and new random walks are related by the equation I − P ψ α = ( I − P ) α/ . Definition 1.3.
Let X = { X ( n ) } n ≥ be the random walk driven by φ . Therandom walk with the transition operator P ψ defined at (1.3) will be calledthe ψ -subordinated random walk and will be denoted by X ψ = { X ψ ( n ) } n ≥ .When ψ = ψ α and X = S is the simple random walk, we call X ψ the α -stable random walk and denote it by S α . -STABLE RANDOM WALK HAS MASSIVE THORNS 5 It is straightforward to show that the increments of S α belong to thedomain of attraction of the α -stable law. This fact justifies the name " α -stable random walk" given in the Definition 1.3. Notation.
For any two non-negative functions f and g , f ( r ) ∼ g ( r ) at a means that lim r → a f ( r ) /g ( r ) = 1 , f ( x ) = O ( g ( x )) if f ( x ) ≤ Cg ( x ) , for someconstant C > , and f ( x ) ≍ g ( x ) if f ( x ) = O ( g ( x )) and g ( x ) = O ( f ( x )) .2. Green function asymptotic
Let S be the simple random walk and ψ ∈ BF . Assuming that thesubordinated random walk S ψ is transient we study asymptotic behaviourof its Green function G ψ .In the course of study we will use the following technical assumption:the function ψ satisfies ψ (0) = 0 , ψ (1) = 1 and(2.1) ψ ( λ ) = λ α/ /l (1 /λ ) , where < α < and l ( λ ) varies slowly at infinity.Recall that a function f defined in a neighbourhood of is said to varyregularly of index β at if for all λ > , lim x → f ( λx ) f ( x ) = λ β . When β = 0 , one says that f varies slowly at . Any regularly varyingfunction of index β is of the form f ( x ) = x β l ( x ) , where l is a slowly varyingfunction. For example, each of the following functions vary regularly at of index β : x β (log 1 /x ) δ , x β exp { (log 1 /x ) δ } , < δ < , etc.A function F defined in a neighbourhood of ∞ is said to vary regularlyof index β at ∞ if f ( x ) = F (1 /x ) varies regularly of index − β at . Let c ( ψ, k ) , k ∈ N , be the probabilities defined at (1.2). For k ≤ we set c ( ψ, k ) = 0 and consider τ = ( τ n ) n ≥ – random walk on Z whoseincrements τ n +1 − τ n have distribution c = { c ( ψ, k ) } k ∈ Z . The random walk τ has non-negative increments, in particular it is transient. Let C ( B ) = X k ≥ c ( k ) ( B ) , B ⊂ Z be its potential measure; here c ( k ) is the Dirac measure concentrated at when k = 0 and the k -fold convolution of the probability c when k ≥ . -STABLE RANDOM WALK HAS MASSIVE THORNS 6 Setting C ( n ) = C ( { n } ) we obtain C ( n ) = 0 for n < , C (0) = 1 and C ( n ) = n X k =1 c ( ψ, k ) C ( n − k ) , n ≥ . Recall that a function ψ ∈ BF is called a special Bernstein function,in short ψ ∈ SBF , if the function λ (cid:14) ψ ( λ ) is also a Bernstein function.Evidently ψ α ∈ SBF whereas ψ ( λ ) = 1 − e − λ does not belong to SBF . Inparticular,
SBF ⊂ BF is a proper inclusion.
Lemma 2.1.
Let ψ ∈ BF satisfy (2.1). The strong renewal property (2.2) C ( n ) ∼ α/ n α/ − l ( n ) , n → ∞ , holds in the following two cases: (i) ψ ∈ BF and < α < , (ii) ψ ∈ SBF and < α < .Proof. Define an auxiliary function M ( x ) , x ∈ R , as M ( x ) = X k ≤ x C ( k ) . Observe that M is a right continuous step-function having jumps at integers.More precisely M ( x ) = 0 for x < , M ( x ) = C (0) for ≤ x < , M ( x ) = C (0) + C (1) for ≤ x < etc. We compute the Laplace-Stieltjes transform L ( M ) of the function M , L ( M )( λ ) = Z R e − λx d M ( x ) = ∞ X k =0 e − λk C ( k ) (2.3) = ∞ X k =0 ∞ X n =0 e − λk c ( n ) ( k ) = ∞ X n =0 ∞ X k =0 e − λk P ( τ n = k )= ∞ X n =0 (cid:16) E (cid:0) e − λτ (cid:1)(cid:17) n = 11 − E (cid:0) e − λτ (cid:1) . We claim that,(2.4) E (cid:0) e − λτ (cid:1) = 1 − ψ (cid:0) − e − λ (cid:1) . Indeed, by Proposition 1.2, E (cid:0) e − λτ (cid:1) = ∞ X k =1 e − λk c ( ψ, k ) . -STABLE RANDOM WALK HAS MASSIVE THORNS 7 Using (1.1) and the fact that ψ (1) = 1 we obtain − ψ (1 − e − λ ) = 1 − b (1 − e − λ ) − Z (0 , ∞ ) (cid:0) − e − t (1 − e − λ ) (cid:1) d ν ( t )= 1 − (cid:16) b + Z (0 , ∞ ) (1 − e − t ) d ν ( t ) (cid:17) + be − λ + Z (0 , ∞ ) e − t ∞ X n =1 t n e − nλ n ! d ν ( t )= be − λ + X n ≥ n ! (cid:16) Z (0 , ∞ ) e − t t n d ν ( t ) (cid:17) e − λn = X n ≥ c ( ψ, n ) e − λn , as desired. It follows that L M ( λ ) = 1 ψ (cid:0) − e − λ (cid:1) . Hence, by (2.1) we obtain L M ( λ ) ∼ λ − α/ l (1 /λ ) , as λ → + . By the Karamata’s Tauberian Theorem [4, Theorem 1.7.1],(2.5) M ( x ) ∼ (cid:0) α (cid:1) x α l ( x ) , as x → ∞ . By [4, Theorem 8.7.3], the equation (2.5) is equivalent to ∞ X k = n c ( ψ, k ) ∼ n − α/ l ( n )Γ (cid:0) − α (cid:1) , as n → ∞ . (2.6)Moreover, recall that C (0) = 1 , C ( n ) = n X k =1 c ( ψ, k ) C ( n − k ) , n > . (2.7)The celebrated Garsia-Lamperti theorem [9, Theorem 1.1] says that (2.7)and (2.6) imply that, when < α < , C ( n ) ∼ Γ (cid:16) − α (cid:17) sin (cid:0) πα/ (cid:1) π n α/ − l ( n ) , as n → ∞ . Using the Euler’s reflection formula Γ( z )Γ(1 − z ) = π sin( πz ) we obtain (2.2).Let us pass to the proof of (ii). Since ψ ∈ SBF , we have ψ ( λ ) = b + Z ∞ e − λt u ( t )d t (2.8) -STABLE RANDOM WALK HAS MASSIVE THORNS 8 for some b ≥ and some non-increasing function u : (0 , ∞ ) (0 , ∞ ) sat-isfying R u ( t )d t < ∞ , see [18, Theorem 11.3]. Set Φ( λ ) = 1 (cid:14) ψ ( λ ) andobserve that by (2.1), Φ( λ ) ∼ λ − α/ l (1 /λ ) , λ → . Applying both the Karamata Tauberian Theorem [4, Theorem 1.7.1] andthe Monotone Density Theorem we obtain u ( t ) ∼ α/ t α/ − l ( t ) , t → ∞ . (2.9)On the other hand L ( M )( λ ) = Φ(1 − e − λ ) = X k ≥ ( − k Φ ( k ) (1) k ! e − λk , whence by the uniqueness of the Laplace transform we obtain C ( k ) = 1 k ! Z ∞ t k e − t u ( t )d t, k ∈ N . We claim that C ( k ) = 1 k ! Z kk/ t k e − t u ( t )d t + O ((2 /e ) k ) . To prove the claim observe that the function t t k e − t is unimodal with max at the point t = k . Hence for a, b and k large enough we will have Z a t k e − t u ( t )d t ≤ a k e − a Z a u ( t )d t, a < k and Z ∞ b t k e − t u ( t )d t ≤ b k +1 e − b Z ∞ b u ( t ) t d t, b > k. In particular, choosing a = k/ , b = 2 k and applying (2.9) we obtain k ! (cid:16) Z k/ t k e − t u ( t )d t + Z ∞ k t k e − t u ( t )d t (cid:17) = O (cid:0) (2 /e ) k (cid:1) , which evidently proves the claim.Once again applying (2.9) we get k ! Z kk/ t k e − t u ( t )d t ∼ l ( k ) k !Γ( α/ Z kk/ t k + α/ − e − t d t. It is straightforward to show that k ! Z kk/ t k + α/ − e − t d t = 1 k ! Z ∞ t k + α/ − e − t d t + O (cid:0) (2 /e ) k (cid:1) . -STABLE RANDOM WALK HAS MASSIVE THORNS 9 At last, all the above show that C ( k ) ∼ l ( k )Γ( k + α/ α/ k + 1) ∼ α/ k α/ − l ( k ) . The proof of (ii) is finished. (cid:3)
Remark 2.2.
Remember that in the continuous time setting to each func-tion ψ ∈ BF is associated a unique convolution semigroup ( η t ) t> of mea-sures supported on [0 , ∞ ) such that L η t ( λ ) = e − tψ ( λ ) . A function ψ ∈ SBF is characterized by the fact that the potential measure U = R ∞ η t d t restricted to (0 , ∞ ) is absolutely continuous with respect tothe Lebesgue measure and its density u ( t ) is a decreasing function. Whetherthis is true in the discrete time setting, i.e. the sequence C ( k ) is decreasing,is an open question at the present writing .We present here some partial answer to this question. Recall that afunction ψ ∈ BF is called a complete Bernstein function, ψ ∈ CBF inshort, if its Lévy measure ν is absolutely continuous with respect to theLebesgue measure and its density ν ( s ) is completely monotone, i.e. ν ( s ) = Z [0 , ∞ ) e − st µ ( dt ) , Observe that in fact µ is supported on (0 , ∞ ) and satisfies Z (0 , ∞ ) min( t − , t − ) µ ( dt ) < ∞ . CBF ⊂ SBF is a proper inclusion. For all of this we refer to [18].
Theorem 2.3.
For ψ ∈ CBF the renewal sequence { C ( k ) } k ∈ N defined as C (0) = 1 and C ( k ) = k X n =0 c ( ψ, n ) C ( k − n ) , k ≥ , is decreasing.Proof. We give a proof of the statement in four steps.
Claim 1 . There exist a measure m on (0 , ∞ ) such that c ( ψ,
1) = b + Z (0 , ∞ ) e − r m (d r ) and c ( ψ, n ) = Z (0 , ∞ ) e − ( n +1) r m (d r ) , n > . -STABLE RANDOM WALK HAS MASSIVE THORNS 10 We consider the case n > . Since ψ ∈ CBF , c ( ψ, n ) = 1 n ! Z (0 , ∞ ) t n e − t ν ( t )d t = 1 n ! Z (0 , ∞ ) d t t n e − t Z (0 , ∞ ) e − st µ (d s )= Z (0 , ∞ ) µ (d s ) 1 n ! Z (0 , ∞ ) t n e − t (1+ s ) d t = Z (0 , ∞ ) µ (d s )(1 + s ) n +1 . Substitution log(1 + s ) = r gives c ( ψ, n ) = Z (0 , ∞ ) e − ( n +1) r m (d r ) , as desired. Claim 2 . { c ( ψ, n ) } n ∈ N satisfies c ( ψ, n − c ( ψ, n + 1) > c ( ψ, n ) , n > . It is enough to consider the case n > . We apply Claim 1, c ( ψ, n − c ( ψ, n + 1) = Z (0 , ∞ ) m (d s ) Z (0 , ∞ ) m (d t ) e −{ ns +( n +2) t } = Z (0 , ∞ ) m (d s ) Z (0 , ∞ ) m (d t ) e − ( n +1)( s + t ) e s − t = Z (0 , ∞ ) m (d s ) Z (0 , ∞ ) m (d t ) e − ( n +1)( s + t ) cosh( s − t ) > Z (0 , ∞ ) m (d s ) Z (0 , ∞ ) m (d t ) e − ( n +1)( s + t ) = c ( ψ, n ) . The strong inequality follows from the fact that, by (2.1), m is not a Diracmeasure. Claim 3 . { C ( n ) } n ∈ N satisfies C ( n − C ( n + 1) > C ( n ) , n > . Indeed, we have C ( n ) = n X k =1 c ( ψ, k ) C ( n − k ) , n ≥ and c ( ψ, n − c ( ψ, n + 1) > c ( ψ, n ) , n > . The remarkable de Bruijn-Erdös theorem [7, Theorem 1] yields the desiredresult. -STABLE RANDOM WALK HAS MASSIVE THORNS 11
Finally we prove that { C ( k ) } k ∈ N is a decreasing sequence. By Claim 3,the sequence C ( k + 1) /C ( k ) increases. Assume that C ( k + 1) /C ( k ) ≥ ,for some k ∈ N . Then there are some a > and N ∈ N such that C ( k + 1) /C ( k ) > a for all k ≥ N . It follows that C ( n ) ≥ a n − N C ( N ) , for all n ≥ N . Contradiction, because for all n ≥ , C ( n ) = P ( ∃ k : τ k = n ) ≤ . Thus C ( k ) decreases. (cid:3) Let p ( n, x ) be a transition function of the simple random walk S . By p ψ ( n, x ) we denote a transition function of the subordinated random walk S ψ and by G ψ its Green function, p ψ ( n, x ) = ∞ X k =1 p ( k, x ) P ( τ n = k ) and G ψ ( x ) = ∞ X n =1 p ψ ( n, x ) = ∞ X k =1 p ( k, x ) C ( k ) . Theorem 2.4.
Assume that ψ ∈ BF satisfies (2.1) with < α < d andthat (2.2) holds. Then G ψ ( x ) ∼ C d,α k x k d ψ (1 / k x k ) , x → ∞ , where C d,α = (cid:16) d (cid:17) α/ π − d/ Γ (cid:0) α (cid:1) Γ (cid:16) d − α (cid:17) . Proof.
Remember that p ( k, x ) is the k -step transition probability of thesimple random walk started at 0. Since p ( k, x ) = 0 for k < k x k√ d , we have G ψ ( x ) = X k ≥ k x k√ d C ( k ) p ( k, x )= X k> k x k A C ( k ) p ( k, x ) | {z } = I + X k x k√ d ≤ k ≤ k x k A C ( k ) p ( k, x ) | {z } = I , where A > is a constant which will be specified later.Our further analysis is based on the results of G.F. Lawler [14, Section1.2]. We write n ↔ x when n + x + ... + x d is even . Set q ( n, x ) = 2 (cid:16) d πn (cid:17) d e − d k x k n -STABLE RANDOM WALK HAS MASSIVE THORNS 12 and define the error function E ( n, x ) = (cid:26) p ( n, x ) − q ( n, x ) if n ↔ x, if n = x. By [14, Theorem 1.2.1],(2.10) | E ( k, x ) | ≤ c k x k − k − d/ , for some c > and all k ≥ .To study I we may assume that x ↔ , then p (2 k + 1 , x ) = 0 , for all k ≥ . Writing I in the form, I = X k> k x k A C (2 k ) q (2 k, x ) | {z } I + X k> k x k A C (2 k ) E (2 k, x ) | {z } I and using (2.2) and (2.10) we obtain I ≤ c X k> k x k A k α/ − l ( k ) k − d/ k x k ∼ c Z ∞ k x k A t α/ − d/ − l ( t ) d t as x → ∞ , (2.11)for some constant c > . By [4, Proposition 1.5.10], Z ∞ k x k A t α/ − d/ − l ( t ) d t ∼ d − α A d − α k x k α − d l ( k x k ) as x → ∞ . It follows that lim k x k→∞ k x k d − α l ( k x k ) I = 0 . Similarly, when k x k → ∞ , I ∼ (cid:16) d π (cid:17) d/ Γ (cid:0) α (cid:1) X k> k x k A (2 k ) α/ − l (2 k ) (2 k ) − d/ exp (cid:26) − d k x k k (cid:27) ∼ (cid:16) d π (cid:17) d/ Γ (cid:0) α (cid:1) Z ∞ k x k A t α/ − d/ − exp (cid:26) − d k x k t (cid:27) l ( t ) d t. Applying [4, Proposition 4.1.2] we obtain I ∼ (cid:16) d (cid:17) α/ π − d/ Γ (cid:0) α (cid:1) k x k α − d l ( k x k ) Z Ad/ s d/ − α/ − e − s d s. -STABLE RANDOM WALK HAS MASSIVE THORNS 13 It follows that lim k x k→∞ k x k d − α l ( k x k ) I = (cid:16) d (cid:17) α/ π − d/ Γ (cid:0) α (cid:1) Z dA/ s d/ − α/ − e − s d s := C ( A ) . To estimate I we use the Gaussian upper bound from [11, Theorem 2.1], I ≤ c X k x k√ d ≤ k ≤ k x k A k α/ − l ( k ) k − d/ exp (cid:26) −k x k c k (cid:27) ∼ c Z k x k A k x k√ d t α/ − d/ − exp (cid:26) −k x k c t (cid:27) l ( t ) d t = c c α/ − d/ k x k α − d Z √ d k x k c Ac s d/ − α/ − e − s l (cid:16) k x k c s (cid:17) d s ≤ c c α/ − d/ k x k α − d Z ∞ Ac s d/ − α/ − e − s l (cid:16) k x k c s (cid:17) d s, for some constants c , c > . Next we apply [4, Theorem 1.5.6] and theDominated Convergence Theorem lim sup k x k→∞ k x k d − α l ( k x k ) I ≤ c c α/ − d/ Γ (cid:0) α (cid:1) Z ∞ Ac s d/ − α/ − e − s d s := C ( A ) . All the above show that, for any fixed
A > , lim sup k x k→∞ k x k d − α l ( k x k ) G ψ ( x ) ≤ C ( A ) + C ( A ) and lim inf k x k→∞ k x k d − α l ( k x k ) G ψ ( x ) ≥ C ( A ) . At last, we have lim A →∞ C ( A ) = 0 and lim A →∞ C ( A ) = (cid:16) d (cid:17) α/ π − d/ Γ (cid:0) α (cid:1) Γ (cid:16) d − α (cid:17) . The proof is finished. (cid:3)
Remark 2.5.
One useful observation is that if we assume that the function ψ satisfies ψ ( θ ) ≍ θ α/ /l (1 /θ ) at -STABLE RANDOM WALK HAS MASSIVE THORNS 14 and belongs to the class SBF , then following the line of reasons of Lemma2.1 and Theorem 2.4, we obtain that C ( k ) ≍ k α/ − l ( k ) at ∞ and G ψ ( x ) ≍ k x k α − d l ( k x k ) at ∞ . Whether this is true when ψ ∈ BF \ SBF is an open question at the presentwriting. In the closely related paper [2] some partial results in this directionare obtained. 3. Massive sets
Basic definitions.
Let X = { X ( n ) } n ≥ be a transient random walk on Z d .Let B be a proper subset of Z d and p B the hitting probability of B . The set B is called massive / recurrent if p B ( x ) = 1 for all x ∈ Z d and non-massive otherwise.Let π B ( x ) be the probability that the random walk X starting from x visits the set B infinitely many times. The set B is massive if and only if π B ≡ ; for non-massive B , π B is identically .Let G ( x, y ) be the Green function of X . In general, the function p B isexcessive, whence it can be written in the form p B = G̺ B + π B . When B is a non-massive set, i.e. π B ≡ , p B is a potential. It is calledthe equilibrium potential of B , respectively ̺ B - the equilibrium distribution .When B is non-massive, the capacity of B is defined as Cap ( B ) = X y ∈ B ̺ B ( y ) . The quantity
Cap ( B ) can be also computed as Cap ( B ) = sup { X y ∈ B ̺ ( y ) : ̺ ∈ Ξ B } , where Ξ B = { ̺ ≥ supp ̺ ⊂ B and G̺ ≤ } . For all of this we refer to spitzer [19, Chapter VI]. -STABLE RANDOM WALK HAS MASSIVE THORNS 15
Test of massiveness.
Assume that the Green function G ( x ) is of the form: G ( x ) = a ( x ) χ ( k x k ) , x = 0 , (3.1)where χ is a non-decreasing function satisfying the doubling condition χ (2 θ ) ≤ Cχ ( θ ) , for all θ > and some C > , (3.2)and c ≤ a ( x ) ≤ c for some c , c > uniformly in x .For a set B define the following sequence of sets B k = { x ∈ B : 2 k ≤ k x k < k +1 } , k = 0 , , . . . . Theorem 3.1.
A set B is non-massive if and only if ∞ X k =0 Cap ( B k ) χ (2 k ) < ∞ . To prove this statement, crucial in fact in our study, we use the assump-tions (3.1) and (3.2) and follow step by step the classical proof by Spitzer[19, Section 26, T1].
Example 3.2.
Let S be the simple random walk in Z . The set B = Z + × { } × { } is S -massive. Moreover, its proper subset P × { } × { } ,where P is the set of primes, is massive, see [12], [15].Let < α < and S α be the α -stable random walk in Z . We claim thatthe set B = Z + × { } × { } is not massive. To prove the claim we applyTheorem 3.1 with χ ( θ ) = θ − α . Let | B k | be the cardinality of B k . Since Cap ( B k ) ≤ | B k | , we have ∞ X k =0 Cap ( B k ) χ (2 k +1 ) ≤ ∞ X k =0 | B k | ( k +1)(3 − α ) ≤ ∞ X k =0 X n : ( n, , ∈ B k n − α = ∞ X n =1 n − α < ∞ . Example 3.3.
Let B be the hyperplane { x ∈ Z d : x = 0 } , d ≥ . Weclaim that(i) If < α < , then B is a non-massive set with respect to S α ;(ii) If ≤ α ≤ , then B is a massive set with respect to S α .Let s α ( n ) be the projection of S α ( n ) on the x -axis. Evidently the set B is S α -massive if and only if the random walk { s α ( n ) } is reccurent. The -STABLE RANDOM WALK HAS MASSIVE THORNS 16 characteristic function of the random variable S α (1) is H α ( θ ) = 1 − (1 − d d X j =1 cos θ j ) α/ , θ ∈ R d . It follows that the characteristic function h α ( ξ ) of s α (1) is h α ( ξ ) = 1 − d − α/ (1 − cos ξ ) α/ , ξ ∈ R . Let p ( n ) be the probability of return to in n steps defined by the randomwalk { s α ( n ) } , then taking the inverse Fourier transform we obtain p ( n ) = 12 π Z π − π (cid:0) h α ( ξ ) (cid:1) n d ξ. It follows that X n ≥ p ( n ) = 12 π Z π − π d ξ − h α ( ξ ) ≍ Z d ξξ α < ∞ if and only if < α < . By the well known criterion of transience, s α ( n ) istransient. 4. Thorns
In this section we assume that the dimension d of the lattice Z d satisfies d ≥ . For x = ( x , . . . , x d − , x d ) we set x ′ = ( x , . . . , x d − ) and write x = ( x ′ , x d ) . The thorn T is defined as T = { ( x ′ , x d ) ∈ Z d : k x ′ k ≤ t ( x d ) , x d ≥ } , where t ( n ) is a non-decreasing sequence of positive numbers. We study S α -massiveness of T .The problem of massiveness of thorns with respect to the simple randomwalk was studied in Itô and McKean [12]. When d = 3 the thorn T is S -massive, because the straight line is S -massive. Whence for the simplerandom walk one assumes that d ≥ .By Cap α ( B ) we denote the S α -capacity of the set B ⊂ Z d , whereas ] Cap α ( A ) stands for the capacity of the set A ⊂ R d , associated with therotationally invariant α -stable process. Proposition 4.1.
Assume that lim sup n →∞ t ( n ) /n = δ > , then the thorn T is S α -massive for any < α < and d ≥ . -STABLE RANDOM WALK HAS MASSIVE THORNS 17 Proof.
The sequence t ( n ) is non-decreasing, whence by the assumption, lim sup n →∞ t (2 n )2 n ≥ δ/ . Hence t (2 n ) / n > δ/ for infinitely many n . Forsuch n consider the following sets T n = T ∩ { x ∈ Z d : 2 n ≤ k x k < n +1 } . (4.1)Let B n be the ball of radius δ n − centred at (0 , . . . , , · n − ) , see Figure1. Since t (2 n ) > δ/ · n > δ n − , we have B n ⊂ T n , whence Cap α ( T n ) ≥ Cap α ( B n ) . By the inequality (5.7), Section 5, for some c > , Cap α ( B n ) ≥ c ( n − d − α ) . It follows that X n ≥ Cap α ( T n )2 n ( d − α ) = ∞ . By Theorem 3.1, the thorn T is massive. (cid:3) Remark 4.2.
Using capacity bounds given in Section 5 and following thesame line of reasons as in the proof of Proposition 4.1, we show that thethorn T satisfying lim sup t ( n ) /n > is S ψ -massive, for any special Bern-stein function ψ which satisfy the assumptions in Theorem 2.4. When lim t ( n ) /n = 0 , S ψ -massiveness of the thorn T is a delicate question. Insuch a generality this question is opened at present. n n +132 n t (2 n ) t (2 n +1 ) Set T n Ball B n Figure 1.
Ball inscribed in the thorn. -STABLE RANDOM WALK HAS MASSIVE THORNS 18
Next we study the case lim n →∞ t ( n ) n = 0 . (4.2)Our reasons are based on the criterion of massiveness given in Theorem 3.1but require more andvanced tools than those in the proof of Proposition4.1. More precisely, we need upper and lower bounds of the α -capacity ofnon-spherically symmetric sets, long cylinders for instance.Let F L be a cylinder of height L with the unit disc as its base, F L = { ( x ′ , x d ) ∈ R d : k x ′ k ≤ , < x d ≤ L } . Proposition 4.3.
There exist constants c , c > which depend only on d and α such that the following inequality holds c L ≤ ] Cap α ( F L ) ≤ c L, L ≥ . Proof.
Indeed, for the upper bound we write L = k + m , where k = [ L ] and m = L − [ L ] . Then, for some c > , ] Cap α ( F L ) ≤ k ] Cap α ( F ) + ] Cap α ( F m ) ≤ c L. To obtain the lower bound we define the following sets D i = { ( x ′ , x d ) ∈ R d : k x ′ k ≤ , i − ≤ x d ≤ i } , i ≥ . Let µ i be the equilibrium measure of D i , i.e. µ i ( D i ) = ] Cap α ( D i ) . We have f G α µ i +1 ( x ) = f G α µ ( x − ie d ) , where f G α is the Green function associated with the symmetric α -stableprocess in R d and e d = (0 , , . . . , . Without loss of generality we canassume that L is an integer number. Define the following measure σ = µ + . . . + µ L . Clearly σ ( R d ) = L ] Cap α ( D ) . We claim that f G α σ ≤ K < ∞ . (4.3)Indeed, we have G α µ ( x ) ≤ , for all x , and lim k x k→∞ k x k d − α f G α µ ( x ) < C, for some constant C > . It follows that X i> f G α µ ( x − ie d ) ≤ C X i> k x − ie d k α − d ∧ . -STABLE RANDOM WALK HAS MASSIVE THORNS 19 Observe that the series above converges uniformly in x which proves theclaim. The inequality (4.3) in turn implies the lower bound ] Cap α ( F L ) ≥ σ ( F L ) /K = LK ] Cap α ( D ) . The proof is finished. (cid:3)
Define the following sets F − n = { ( x ′ , x d ) ∈ R d : k x ′ k ≤ t (2 n ) ,
43 2 n ≤ x d <
34 2 n +1 } ; F + n = { ( x ′ , x d ) ∈ R d : k x ′ k ≤ t (2 n +1 ) ,
34 2 n ≤ x d <
43 2 n +1 } ; F − n = F − n ∩ Z d and F + n = F + n ∩ Z d . Let Q ( b ) be the cube [0 , d centered at b . For any set B ⊂ Z d , we denoteby e B the subset of R d defined as e B = [ b ∈ B Q ( b ) . (4.4) Theorem 4.4.
Under the assumption (4.2), the thorn T is S α -massive ifand only if the series X n> (cid:16) t (2 n )2 n (cid:17) d − α − (4.5) diverges. Before embarking on the proof of Theorem 4.4 we illustrate the statementby the following example. Consider the thorn T with t ( n ) = n/ (log(1 + n )) β , β > . Then T is S α -massive if and only if β ≤ / ( d − α − . Proof.
Assume that the series (4.5) is convergent. Show that the set T isnon-massive. For any compact set A ⊂ R d and for any s > the followingscaling property holds ] Cap α ( sA ) = s d − α ] Cap α ( A ) , (4.6)see e.g. Sato [17, Example 42.17]. Using Proposition 4.3, the assumption(4.2) and the equation (4.6), for enough large n we have ] Cap α ( F + n ) = ] Cap α (cid:0) t (2 n +1 ) ·F + n /t (2 n +1 ) (cid:1) = t (2 n +1 ) d − α · ] Cap α (cid:0) F + n /t (2 n +1 ) (cid:1) ≤ c t (2 n +1 ) d − α · t (2 n +1 ) − · (cid:16)
43 2 n +1 −
34 2 n (cid:17) ≤ c t (2 n +1 ) d − α − · n +1 , -STABLE RANDOM WALK HAS MASSIVE THORNS 20 for some c , c > . Let T n be as in (4.1). Since T n ⊂ F + n , see Figure 2, Cap α ( T n ) ≤ Cap α ( F + n ) . By Theorem 5.2, Section 5, c ] Cap α ( e F + n ) ≤ Cap α ( F + n ) ≤ c ] Cap α ( e F + n ) , (4.7)for some c , c > . Using again Proposition 4.3 we obtain c ] Cap α ( F + n ) ≤ ] Cap α ( e F + n ) ≤ c ] Cap α ( F + n ) , (4.8)for some c , c > . All the above show that X n> Cap α ( T n )2 n ( d − α ) ≤ c X n> (cid:16) t (2 n +1 )2 n +1 (cid:17) d − α − < ∞ , as desired.Conversely, assume that the series (4.5) is divergent. Show that theset T is massive. Applying Proposition 4.3, the assumption (4.2) and theequation (4.6) we have ] Cap α ( F − n ) = ] Cap α (cid:0) t (2 n ) ·F − n /t (2 n ) (cid:1) = t (2 n ) d − α · ] Cap α (cid:0) F − n /t (2 n ) (cid:1) ≥ c ′ t (2 n ) d − α · t (2 n ) − · (cid:16)
34 2 n +1 −
43 2 n (cid:17) ≥ c ′ t (2 n ) d − α − · n , for some c ′ , c ′ > . Since F − n ⊂ T n , see Figure 2, Cap α ( T n ) ≥ Cap α ( F − n ) . Similarly to (4.7) and (4.8) we get c ′ ] Cap α ( e F − n ) ≤ Cap α ( F − n ) ≤ c ′ ] Cap α ( e F − n ) and c ′ ] Cap α ( F − n ) ≤ ] Cap α ( e F − n ) ≤ c ′ ] Cap α ( F − n ) , for some constants c ′ , c ′ , c ′ , c ′ > . Thus, at last, X n> Cap α ( T n )2 n ( d − α ) ≥ c ′ X n> (cid:16) t (2 n )2 n (cid:17) d − α − = ∞ , as desired. (cid:3) -STABLE RANDOM WALK HAS MASSIVE THORNS 21 n n +1 34 n +143 n Set F + n Set T n Set F − n Figure 2.
Two cylinders inscribed in and circumscribedaround the thorn. 5.
Two comparisons.
Let ψ be a special Bernstein function (see Remark 4.2). Let B ψ be aLévy process in R d obtained by subordination of the Brownian motion B .Let S ψ be the random walk obtained by subordination of the simple randomwalk S . Let f G ψ ( x ) (resp. G ψ ) be the Green function of B ψ (resp. S ψ ).In what follows we assume that ψ ∈ BF satisfies the conditions of The-orem 2.4. Proposition 5.1.
The function f G ψ ( x ) has the following asymptotic f G ψ ( x ) ∼ A d,α k x k d ψ (1 / k x k ) , x → ∞ , where A d,α = Γ(( d − α ) / α · π d/ · Γ( α/ . In particular, f G ψ ( x ) ∼ (2 /d ) α/ G ψ ( x ) , x → ∞ . Proof. As ψ is a special Bernstein function, the potential measure U associ-ated with the corresponding (continuous time) subordinator has a monotonedensity u ( t ) , see e.g. [6, Chapter V, Theorem 5.1]. Since L ( U )( λ ) = 1 /ψ ( λ ) ,the Karamata theorem implies that the density function u ( t ) satisfies u ( t ) ∼ α/ t α/ − l ( t ) at ∞ . -STABLE RANDOM WALK HAS MASSIVE THORNS 22 Recall that, by definition, f G ψ ( x ) = Z ∞ (4 πt ) − d/ exp (cid:26) − k x k t (cid:27) u ( t ) d t, whence, as k x k → ∞ f G ψ ( x ) = 4 − π − d/ k x k − d Z ∞ s d/ − e − s u (cid:16) k x k s (cid:17) d s ∼ − π − d/ k x k − d Z ∞ s d/ − e − s u ( k x k ) (cid:16) s (cid:17) α/ − d s = 2 − α π − d/ k x k − d u ( k x k ) Z ∞ s d/ − α/ − e − s d s ∼ Γ( d − α )2 α ·· π d/ · Γ( α/ k x k α − d l ( k x k ) . Combining this result with that of Theorem 2.4 we obtain the claimedcomparison of Green functions f G ψ and G ψ . (cid:3) Let ] Cap ψ ( A ) be the capacity of a set A ⊂ R d associated with the process B ψ . Recall that by definition (see e.g. [5]) ] Cap ψ ( A ) = sup { µ ( A ) : µ ∈ K A } , where K A is the class of measures supported by A and such that f G ψ µ ( ξ ) = Z A f G ψ ( ξ − η ) µ (d η ) ≤ , for all ξ ∈ R d . Let
Cap ψ ( B ) be the capacity of a set B ⊂ Z d associated with the process S ψ . Similarly Cap ψ ( B ) = sup { X y ∈ B φ ( y ) : φ ∈ Ξ B } , where Ξ B = { φ ≥ supp φ ⊂ B and G ψ φ ≤ } . Theorem 5.2.
Let B be a bounded subset of Z d . Let e B be defined at (4.4).There exist constants c , c > , which depend only on d and ψ , and suchthat c ] Cap ψ ( e B ) ≤ Cap ψ ( B ) ≤ c ] Cap ψ ( e B ) . Proof.
Take a, b ∈ B . Let Q ( a ) be the cube [0 , d centered at a ∈ B . Let d η be the Lebesgue measure in R d . By Proposition 5.1 and radial monotonicity -STABLE RANDOM WALK HAS MASSIVE THORNS 23 of f G ψ , we can find a constant c > which does not depend on a and b ,and such that for ξ ∈ Q ( a ) and η ∈ Q ( b ) , Z Q ( b ) f G ψ ( ξ − η )d η ≤ c G ψ ( a − b ) . (5.1)Let E be the equilibrium distribution of B associated with the random walk S ψ . We define a new measure d ν ( η ) = X b ∈ B E ( b ) Q ( b ) ( η )d η. Using (5.1) we compute the potential f G ψ ν Z e B f G ψ ( ξ − η ) d ν ( η ) = X b ∈ B Z Q ( b ) f G ψ ( ξ − η ) E ( b ) d η ≤ c X b ∈ B G ψ ( a − b ) E ( b ) = c G ψ E ( a ) ≤ c . Thus, the measure c − ν belongs to the class K e B , therefore ] Cap ψ ( e B ) ≥ c ν ( e B ) . (5.2)On the other hand ν ( e B ) = Z e B d ν ( η ) = X b ∈ B Z Q ( b ) E ( b ) d η = X b ∈ B E ( b ) = Cap ψ ( B ) . (5.3)Combining (5.2) and (5.3) we obtain Cap ψ ( B ) = ν ( e B ) ≤ c ] Cap ψ ( e B ) . For the converse we use again Proposition 5.1 and radial monotonicityof f G ψ . Let a, b ∈ B . Choose c > , which does not depend on a and b ,such that for ξ ∈ Q ( a ) , η ∈ Q ( b ) , c G ψ ( a − b ) ≤ f G ψ ( ξ − η ) . (5.4)Let e E be the equilibrium measure of e B , i.e. ] Cap ψ ( e B ) = e E ( e B ) . Define adistribution ̺ supported by the set B as ̺ ( b ) = e E ( Q ( b )) , b ∈ B. Let p = c G ψ ̺. Using (5.4) we get p ( a ) ≤ X b ∈ B f G ψ ( ξ − η ) ̺ ( b ) ≤ Z e B f G ψ ( ξ − η ) d e E ( η ) ≤ . -STABLE RANDOM WALK HAS MASSIVE THORNS 24 It follows that c ̺ ∈ Ξ B , whence Cap ψ ( B ) ≥ c ̺ ( B ) . (5.5)Computing ̺ ( B ) we obtain ̺ ( B ) = X b ∈ B ̺ ( b ) = X b ∈ B Z Q ( b ) d e E ( η ) = Z e B d e E ( η ) = e E ( e B ) . (5.6)From (5.5) and (5.6) we deduce that Cap ψ ( B ) ≥ c ̺ ( B ) = c e E ( e B ) = c ] Cap ψ ( e B ) . The proof is finished. (cid:3)
Corollary 5.3.
Let B (0 , r ) ⊂ Z d be a ball of radius r > centered at .The following inequalities hold cr d ψ (1 /r ) ≤ Cap ψ ( B (0 , r )) ≤ Cr d ψ (1 /r ) , for some constants c, C > and all r > . In particular, cr d − α ≤ Cap α ( B (0 , r )) ≤ Cr d − α . (5.7) Proof.
Assume d ≥ . Let φ be the Lévy exponent of B ψ , that is E e iξB ψ ( t ) = e − tφ ( ξ ) , ξ ∈ R d . Since B ψ is a subordinated Brownian motion, we have φ ( ξ ) = ψ ( k ξ k ) , ξ ∈ R d . The function φ ( s ) is increasing whence [10, Proposition 3] applies in theform ] Cap ψ ( B (0 , r )) ≍ ψ ( r − ) r d . At last, Theorem 5.2 yields the desired result.When d ≤ we proceed as follows. We use [6, Proposition 5.55], ] Cap ψ ( B (0 , r )) ≍ r d R B (0 ,r ) f G ψ ( x )d x , and [6, Proposition 5.56], Z B (0 ,r ) f G ψ ( x ) ≍ E τ B (0 ,r ) , where τ B (0 ,r ) is B ψ –first exit time from the ball B (0 , r ) . We use [16, Theorem1 and p. 954], E τ B (0 ,r ) ≍ h ( r ) , -STABLE RANDOM WALK HAS MASSIVE THORNS 25 where h ( r ) = Z R d (cid:16) k x k r ∧ (cid:17) d ν ( x ) and ν is the Lévy measure associated with the Lévy exponent φ , see [16,Section 3]. By [10, Corollary 1], h ( r ) ≍ ψ ( r − ) . The proof is finished. (cid:3)
Acknowledgements.
This paper was started at Wrocław University andfinished at Bielefeld University (SFB-701). We thank A. Grigor’yan, W. Hansen,S. Molchanov and Z. Vondraček for fruitful discussions. We also thank theanonymous referee for valuable remarks.
References [1]
A. Bendikov,
Asymptotic formulas for symmetric stable semigroups ,Expo. Math. 12 (1994), 381–384. [2]
A. Bendikov and S. Molchanov,
On the strong renewal property in thesub-critical region , 2014, preprint. [3]
A. Bendikov and L. Saloff-Coste,
Random walks on groups and discretesubordination , Math. Nachr., 285 (2012), 580-605. [4]
N.H. Bingham, C.M. Goldie and J.L. Teugels,
Regular Variation , Cam-bridge University Press, Cambridge, 1987. [5]
R.M. Blumenthal and R.K. Getoor,
Markov processes and potential the-ory , Springer-Verlag, New York, 1968. [6]
K. Bogdan, T. Byczkowski, T. Kulczycki, M. Ryznar, R. Song andZ. Vondraček,
Potential analysis of stable processes and its extensions ,Lecture Notes in Mathematics, 1980. Springer-Verlag, Berlin, 2009. [7]
N.G. de Bruijn and P. Erdös,
On a Recursion Formula and on SomeTauberian Theorems , Journal of Research of the National Bureau ofStandards, 50(3), 1953, 161-164. [8]
W. Feller,
An introduction to probability theory and its applications , vol.II, John Wiley & Sons, New York, 1965. [9]
A. Garsia and J. Lamperti ,
A Discrete Renewal Theorem with InfiniteMean , Commentarii Mathematici Helvetici , 37 (1963), 221-234. [10]
T. Grzywny,
On Harnack inequality and Hölder regularity for isotropicunimodal Lévy processes , Potential Analysis, 41, 2014. [11]
W. Hebisch and L. Saloff-Coste,
Gaussian estimates for Markov chainsand random walks on groups , Ann. Probab., 21(2) (1993), 673-709. [12]
K. Itô and H.P. McKean,
Potentials and the random walk , Illinois J.Math. Volume 4, Issue 1 (1960), 119-132. [13]
N. Jacob,
Pseudo differential operators and Markov processes. Fourieranalysis and semigroups , vol. I, Imperial College Press, London, 2001. [14]
G.F. Lawler,
Intersections of random walks , Birkhäuser., Boston, 1996. [15]
H.P. McKean,
A problem about prime numbers and the random walk ,Illinois J. Math. Volume 5, Issue 2 (1961), 351. -STABLE RANDOM WALK HAS MASSIVE THORNS 26 [16]
W.E. Pruitt,
The Growth of Random Walks and Levy Processes , Ann.Probab. Volume 9(6), 1981, 948-956. [17]
K. Sato,
Lévy processes and infinitely divisible distributions , CambridgeUniversity Press, 1999. [18]
R. Schilling, R. Song and Z. Vondraček,
Bernstein functions , de GruyterStudies in Mathematics 37 , Walter de Gruyter & Co., Berlin, 2010. [19] F. Spitzer,
Principles of random walk , D. Van Nostrand Company, Inc.,New Jersey, 1964. [20]
J. Williamson,
Random walks and Riesz kernels , Pacific Journal ofMathematics, Vol. 25, No. 2, 1968.
A. Bendikov, Institute of Mathematics, Wrocław University, 50-384Wrocław, Pl. Grunwaldzki 2/4, Poland
E-mail address : [email protected] W. Cygan, Institute of Mathematics, Wrocław University, 50-384 Wrocław,Pl. Grunwaldzki 2/4, Poland
E-mail address ::