Ergodic aspects of some Ornstein-Uhlenbeck type processes related to Lévy processes
aa r X i v : . [ m a t h . P R ] S e p Ergodic aspects of someOrnstein-Uhlenbeck type processesrelated to L´evy processes
Jean Bertoin ∗ Abstract
This work concerns the Ornstein-Uhlenbeck type process associated toa positive self-similar Markov process ( X ( t )) t > which drifts to ∞ , namely U ( t ) := e − t X (e t − . We point out that U is always a (topologically) re-current Markov process and identify its invariant measure in terms of thelaw of the exponential functional ˆ I := R ∞ exp(ˆ ξ s )d s , where ˆ ξ is the dualof the real-valued L´evy process ξ related to X by the Lamperti transfor-mation. This invariant measure is infinite (i.e. U is null-recurrent) if andonly if ξ L ( P ) . In that case, we determine the family of L´evy processes ξ for which U fulfills the conclusions of the Darling-Kac theorem. Our ap-proach relies crucially on a remarkable connection due to Patie [25] withanother generalized Ornstein-Uhlenbeck process that can be associated tothe L´evy process ξ , and properties of time-substitutions based on additivefunctionals. Keywords:
Ornstein-Uhlenbeck type process, Stationarity, Self-similar Markovprocess, L´evy process, Exponential functional, Darling-Kac theorem.
AMS subject classifications:
Let ( ξ t ) t > be a real-valued L´evy process which drifts to ∞ , that is lim t →∞ ξ t = ∞ a.s. The so-called exponential functional I ( t ) := Z t exp( ξ s )d s defines a random bijection I : R + → R + , and we denote its inverse by τ . Awell-known transformation due to Lamperti [19], X ( t ) := exp( ξ τ ( t ) ) , yields a Markov process ( X ( t )) t > on (0 , ∞ ) that enjoys the scaling property(with index ), in the sense that for every x > , ( xX ( t/x )) t > is a version of ∗ Institut f¨ur Mathematik, Universit¨at Z¨urich, Winterthurerstrasse 190, CH-8057 Z¨urich,Switzerland.Email: [email protected] started from x . Conversely, any Markov process X on (0 , ∞ ) that fulfills thescaling property (with index ) and drifts to ∞ can be constructed in this way.We refer to the survey by Pardo and Rivero [24] and references therein for adetailed presentation of the topic.The question of the existence of a truly self-similar version ( ˜ X t ) t > , that is, ˜ X is a Markov process with the same transition probabilities as X and furtherthere is the identity in distribution ( c ˜ X ( t/c )) t > d ) = ( ˜ X ( t )) t > for every c > , is equivalent to the question of whether is an entrance boundary for theMarkov process X . This was raised by Lamperti, and settled in the presentsetting in [5]: the answer is positive if and only if ξ ∈ L ( P ) (recall thatthen E ( ξ ) > , since the test of Chung and Fuchs ensures that in dimension ,centered L´evy processes are recurrent and therefore oscillate), and further thestationary law can then be expressed in terms of the exponential functional ˆ I of the dual L´evy process.On the other hand, there is another well-known transformation `a la Ornstein-Uhlenbeck, also due to Lamperti [18], that yields a bijection between self-similarprocesses and stationary processes. In the present setting, assuming again that ξ ∈ L ( P ) and writing ˜ X for the self-similar version of X , ˜ U ( t ) := e − t ˜ X (e t ) , t ∈ R is a stationary process on (0 , ∞ ) . Furthermore, the scaling property ensuresthat U ( t ) := e − t X (e t − , t > (1.1)is Markovian, and the Markov processes U and ˜ U have the same semigroup.The initial motivation for this work is to analyze the situation when ξ L ( P ) . We shall show that the Ornstein-Uhlenbeck type process U still possessesa stationary version ˜ U , but now under an infinite measure Q which is absolutelycontinuous with respect to P . More precisely, U is a (null) recurrent Markovprocess and its invariant measure ν can be expressed similarly as in the positiverecurrent case in terms of the dual exponential functional ˆ I . When E ( ξ ) = ∞ ,the claim that U is recurrent might look surprising at first sight, since theL´evy process may grow faster than any given polynomial (think for instance ofstable subordinators). One could expect that the same might hold for X ( t ) =exp( ξ τ ( t ) ) , which would then impede the recurrence of U . However the time-substitution by τ has a slowing down effect when X gets larger, and actually X only grows linearly fast.Our main result is related to the celebrated Darling-Kac theorem, which canbe thought of as a version of Birkhoff’s ergodic theorem in infinite invariantmeasure; see e.g. Theorem 3.6.4 in [1] and Theorem 8.11.3 in [7]. We show thatif a : (0 , ∞ ) → (0 , ∞ ) is regularly varying at ∞ with index α ∈ (0 , , then forevery nonnegative f ∈ L ( ν ) , a ( t ) − R t f ( U ( s ))d s converges in distribution as t → ∞ towards a Mittag-Leffler distribution with parameter α if and only if This question makes also sense when ξ oscillates, that is lim sup t →∞ ξ t = ∞ and lim inf t →∞ ξ t = −∞ a.s. It was proved in [8] and [11] that the answer is positive if andonly if the so-called ascending ladder height of ξ has a finite expectation. ( t ) − ξ t converges in distribution as t → ∞ to a positive stable random variablewith exponent α , where b denotes an asymptotic inverse of a .At the heart of our approach lies the fact that one can associate to the L´evyprocess ξ another generalized Ornstein-Uhlenbeck process, namely V ( t ) := exp( − ξ t ) ( I ( t ) + V (0)) , t > . Lindner and Maller [21] have shown that, since ξ drifts to ∞ , V always pos-sesses a stationary version ˜ V , no matter whether ξ is integrable or not. Patie[25] pointed at a remarkable connection between U and V via a simple timesubstitution, and this provides a powerful tool for the analysis of U .The rest of this paper is organized as follows. We start in Section 2 byproviding background on the generalized Ornstein-Uhlenbeck process V . Then,in Section 3, we construct a stationary version ˜ U of U under a possibly infiniteequivalent measure, and point at the topological recurrence of U . Finally, inSection 4, we address the Darling-Kac theorem for the occupation measure of ˜ U . We start by recalling the basic time-reversal property of L´evy processes, alsoknown as the duality identity, which plays an important role in this subject. Ifwe denote ˆ ξ the so-called dual L´evy process which has the same law as − ξ , thenfor every t > , there is the identity in distribution between c`adl`ag processes ( − ξ t + ξ ( t − s ) − ) s t ( d ) = ( ˆ ξ s ) s t . Following Carmona et al. [10] and Lindner and Maller [21], as well asother authors, we associate to the L´evy process ξ another generalized Ornstein-Uhlenbeck process ( V ( t )) t > , V ( t ) := exp( − ξ t ) ( I ( t ) + V (0)) = Z t exp( ξ s − ξ t )d s + V (0) exp( − ξ t ) , where the initial value V (0) is arbitrary and may be random. It was observed in[10] and [21] that the time-reversal property and the a.s. finiteness of the dualexponential functional ˆ I := ˆ I ( ∞ ) = Z ∞ exp( ˆ ξ s )d s (which is known to follow from our assumption that ξ drifts to ∞ , see Theorem1 in [6], or Theorem 2 in [14]), immediately implies that lim t →∞ V ( t ) = ˆ I in distribution, (2.1)independently of the initial value V (0) . The distribution of ˆ I , µ (d x ) := P ( ˆ I ∈ d x ) , x ∈ (0 , ∞ ) , V (0) hasthe same law as ˆ I and is independent of ξ , then the process ( V t ) t > is stationary.It will be convenient for us to rather work with a two-sided version ( ˜ V t ) t ∈ R whichcan easily be constructed as follows.Assume henceforth that ( ˆ ξ t ) t > is an independent copy of ( − ξ t ) t > , andwrite ( ˜ ξ t ) t ∈ R for the two-sided L´evy process given by ˜ ξ t = (cid:26) ξ t if t > , ˆ ξ | t |− if t < . We then set for every t ∈ R ˜ I ( t ) := Z t −∞ exp( ˜ ξ s )d s and ˜ V ( t ) = exp( − ˜ ξ t ) ˜ I ( t ) . Note that ˜ V (0) = ˜ I (0) = ˆ I , so the process ( ˜ V t ) t > is a version of V startedfrom its stationary distribution. The next statement records some importantproperties of ˜ V that will be useful for this study. Theorem 2.1. (i) The process ( ˜ V t ) t ∈ R is a stationary and strongly mixingFeller process, with stationary one-dimensional distribution µ .(ii) For every f ∈ L ( µ ) , we have lim t →∞ t Z t f ( ˜ V ( s ))d s = h µ, f i a.s.Proof. (i) By the time-reversal property, the two-sided process ˜ ξ has stationaryincrements, in the sense that for every t ∈ R , ( ˜ ξ t + s − ˜ ξ t ) s ∈ R has the same lawas ( ˜ ξ s ) s ∈ R . This readily entails the stationarity of ˜ V . The Feller property hasalready been pointed at in Theorem 3.1 in [2], so it only remains to justify thestrong mixing assertion. Unsurprisingly , this follows from (2.1) by a monotoneclass argument that we recall for completeness.Let L ∞ denote the space of bounded measurable functions g : (0 , ∞ ) → R and C b the subspace of continuous bounded functions. Introduce the vectorspace H := { g ∈ L ∞ : lim t →∞ E ( f ( ˜ V (0)) g ( ˜ V ( t ))) = h µ, f ih µ, g i for every f ∈ L ∞ } . We easily deduce from (2.1) that C b ⊆ H . Then consider a non-decreasingsequence ( g n ) n ∈ N in H with sup n ∈ N k g n k ∞ < ∞ and let g = lim n →∞ g n . Forevery f ∈ L ∞ , we have by stationarity E ( f ( ˜ V (0)) g ( ˜ V ( t ))) = E ( f ( ˜ V ( − t )) g ( ˜ V (0))) . So assuming for simplicity that k f k ∞ , the absolute difference | E ( f ( ˜ V (0)) g ( ˜ V ( t ))) − h µ, f ih µ, g i| If the Markov process V is µ -irreducible, then one can directly apply well-known factsabout stochastic stability; see Part III in Meyn and Tweedie [23]. However, establishingirreducibility for arbitrary generalized Ornstein-Uhlenbeck processes seems to be a challengingtask; see Section 2.3 in Lee [20] for a partial result. E ( g ( ˜ V (0))) − g n ( ˜ V (0))) + h µ, g − g n i + | E ( f ( ˜ V (0)) g n ( ˜ V ( t ))) − h µ, f ih µ, g n i| . The first two terms in the sum above coincide and can be made as small as wewish by choosing n large enough. Since g n ∈ H , this entails lim t →∞ | E ( f ( ˜ V (0)) g ( ˜ V ( t ))) − h µ, f ih µ, g i| ε for every ε > . Hence g ∈ H , and since C b is an algebra that contains theconstant functions, we conclude by a functional version of the monotone classtheorem that H = L ∞ .(ii) Since strong mixing implies ergodicity, this follows from Birkhoff’s er-godic theorem.We mention that the argument for Theorem 2.1 applies more generally to thelarger class of generalized Ornstein-Uhlenbeck processes considered by Lindnerand Maller [21]. Further, sufficient conditions ensuring exponential ergodicitycan be found in Theorem 4.3 of Lindner and Maller [21], Lee [20], Wang [30],and Kevei [16]. Patie [25] pointed out that the Ornstein-Uhlenbeck type processes U and V are related by a simple time-substitution. We shall see here that the sametransformation, now applied to the stationary process ˜ V , yields a stationaryversion ˜ U of U , and then draw some consequences of this construction.Introduce the additive functional A ( t ) := Z t d s ˜ V ( s ) = ln ˜ I ( t ) − ln ˜ I (0) , t ∈ R ; clearly A : R → R is bijective and we denote the inverse bijection by T . Observethat A ( T ( t )) = t yields the useful identity Z T ( t ) −∞ exp( ˜ ξ s )d s = ˜ I (0)e t for all t ∈ R . (3.1)We also define a measure ν on (0 , ∞ ) by h ν, f i = Z (0 , ∞ ) x f (1 /x ) µ (d x ) , and further introduce an equivalent sigma-finite measure on the underlying prob-ability space (Ω , A , P ) by Q (Λ) = E (cid:18) V (0) Λ (cid:19) , Λ ∈ A . Note that Q ( f (1 / ˜ V (0))) = h ν, f i for every measurable f : (0 , ∞ ) → R + . 5 heorem 3.1. (i) The measure Q (respectively, ν ) is finite if and only if ξ ∈ L ( P ) , and in that case, Q (Ω) = ν ((0 , ∞ )) = E ( ξ ) .(ii) Under Q , ˜ U ( t ) := 1 / ˜ V ( T ( t )) , t ∈ R is a stationary and ergodic Markov process, with one-dimensional marginal ν and the same semigroup as the Ornstein-Uhlenbeck type process U de-fined in (1.1) .(iii) For all functions f, g ∈ L ( ν ) with h ν, g i 6 = 0 , we have lim t →∞ R t f ( ˜ U ( s ))d s R t g ( ˜ U ( s ))d s = h ν, f ih ν, g i Q -a.s. and therefore also P -a.s.Proof. (i) Recall that ˜ V (0) = ˜ I (0) = ˆ I , so Q (Ω) = ν ((0 , ∞ )) = E (1 / ˆ I ) . When ξ ∈ L ( P ) , Equation (3) in [5] gives E (1 / ˆ I ) = E ( ξ ) .Next, suppose that ξ − ∈ L ( P ) and ξ +1 / ∈ L ( P ) , that is the mean E ( ξ ) exists and is infinite. We can construct by truncation of the large jumps of ξ ,an increasing sequence ( ξ ( n ) ) n ∈ N of L´evy processes such that ξ ( n )1 ∈ L ( P ) with E ( ξ ( n )1 ) > and lim n →∞ ξ ( n ) t = ξ t for all t > a.s. In the obvious notation, ˆ I ( n ) decreases to ˆ I as n → ∞ , and lim n →∞ E ( ξ ( n )1 ) = ∞ . We conclude by monotoneconvergence that E (1 / ˆ I ) = ∞ .Finally, suppose that both ξ − / ∈ L ( P ) and ξ +1 / ∈ L ( P ) , so the mean of ξ isundefined. Equivalently, in terms of the L´evy measure, say Π , of ξ , we have Z ( −∞ , − | x | Π(d x ) = Z (1 , ∞ ) x Π(d x ) = ∞ , see Theorem 25.3 in Sato [28]. Using Erickson’s test characterizing L´evy pro-cesses which drift to ∞ when the mean is undefined (see Theorem 15 in Doney[13]), it is easy to decompose ξ into the sum ξ = ξ ′ + η of two independentL´evy processes, such that ξ ′ is a L´evy process with infinite mean and η is acompound Poisson process with undefined mean that drifts to ∞ . The event Λ := { η t > for all t > } has a positive probability (because η is compoundPoisson and drifts to ∞ ). On that event, we have ξ > ξ ′ and thus also, in theobvious notation, ˆ I ˆ I ′ . This yields E (1 / ˆ I, Λ) > E (1 / ˆ I ′ ) P (Λ) , and we have see above that the first term in the product is infinite. We concludethat E (1 / ˆ I ) = ∞ .(ii) It is convenient to view now Ω as the space of c`adl`ag paths ω : R → (0 , ∞ ) endowed with the usual shift automorphisms ( θ t ) t ∈ R , i.e. θ t ( ω ) = ω ( t + · ) , and P as the law of ˜ V . We have seen in Theorem 2.1(i) that P is ( θ t ) -ergodic.General results due to Maruyama and Totoki on time changes of flows basedon additive functionals show that the measure Q is invariant for the time-changed flow of automorphisms ( θ ′ t ) t ∈ R , where θ ′ t ( ω ) := ω ( T ( t ) + · )) . See The-orems 4.1(iii) and 4.2 in [29]. Further, ergodicity is always preserved by suchtime substitutions, see Theorem 5.1 in [29]. This shows that ( ˜ V ( T ( t ))) t > is astationary ergodic process under Q . 6n the other hand, time substitution based on an additive functional alsopreserves the strong Markov property, so ( ˜ V ( T ( t ))) t > is a Markov process under Q . By stationarity, ( ˜ V ( T ( t ))) t ∈ R is Markov too. Composing with the inversion x /x , we conclude that ˜ U is a stationary and ergodic Markov process under Q . It remains to determine the semigroup of ˜ U , and for this, we simply recallfrom Theorem 1.4 of Patie [25] that the processes U and V can be related bythe same time-substitution as that relating ˜ U and ˜ V . As a consequence, ˜ U and U have the same semigroup.(iii) Under Q , this is a consequence of (ii) and Hopf’s ratio ergodic theo-rem. See also Lemma 5.1 in [29]. The measures P and Q being equivalent, thestatement of convergence also holds P -a.s. We mention that, alternatively, thiscan also be deduced from Birkhoff’s ergodic theorem for ˜ V (Theorem 2.1(ii)) bychange of variables. Remark 1. (i) In the case ξ ∈ L ( P ) , Theorem 3.1(i-ii) agrees with theresults in [5]; the arguments in the present work are however much simpler.We stress that one should not conclude from Theorem 3.1(i-ii) that U ( t ) then converges in distribution to the normalized version of ν . Actually thisfails when the L´evy process is lattice-valued (i.e. ξ t ∈ r Z a.s. for some r > , think for instance of the case when ξ is a Poisson process), becausethen the Ornstein-Uhlenbeck type process U is periodic.(ii) Inverting the transformation `a la Ornstein-Uhlenbeck incites us to set ˜ X ( t ) := t ˜ U (ln t ) = t/ ˜ V ( T (ln t )) , t > , and the calculation in the proof of Theorem 3.1(ii) yields the expression `ala Lamperti ˜ X ( t ) := exp( ˜ ξ ˜ τ ( t ) ) , with ˜ τ : (0 , ∞ ) → R the inverse of the exponential functional ˜ I . Theorem3.1(ii) entails that under Q , ˜ X is a self-similar version X . We refer to[4] for an alternative similar construction which does not require workingunder an equivalent measure.(iii) If we write G for the infinitesimal generator of the Feller process V , thenthe stationary of the law µ is is characterized by the identity h µ, G f i = 0 for every f in the domain of G . Informally , according to a formula ofVolkonskii (see (III.21.6) in [27]), the infinitesimal generator G ′ of thetime-changed process V ◦ T is given by G ′ f ( x ) = x G f ( x ) , so the measure µ ′ (d x ) := x − µ (d x ) fulfills h µ ′ , G ′ f i = 0 for every f in the domain of G ,and thus should be invariant for the time-changed process V ◦ T . We thenrecover the assertion that ν is invariant for ˜ U = 1 / ( ˜ V ◦ T ) . We conclude this section by discussing recurrence. Recall first that thesupport of the stationary law µ of the generalized Ornstein-Uhlenbeck process V is always an interval, say I ; see Haas and Rivero [15] or Lemma 2.1 in[3]. More precisely, excluding implicitly the degenerate case when ξ is a puredrift, I = [0 , /b ] if ξ is a non-deterministic subordinator with drift b > , The application of Volkonskii’s formula is not legitimate, since the function x /x isnot bounded away from . = [1 /b, ∞ ) if ξ is non-deterministic and of finite variation L´evy process withno positive jumps and drift b > , and I = [0 , ∞ ) in the remaining cases.Writing I o for the interior of I , it is further readily checked that V ( t ) ∈ I o forall t > a.s. whenever V (0) ∈ I o . Corollary 3.1.
The Ornstein-Uhlenbeck type process U is topologically recur-rent, in the sense that for every x > with /x ∈ I o , U visits every neighborhoodof x a.s., no matter its initial value U (0) .Proof. It follows from (2.1) and the Portmanteau theorem that every point x ∈ I o is topologically recurrent for the generalized Ornstein-Uhlenbeck process V . Plainly, this property is preserved by time-substitution. We assume throughout this section that ξ L ( P ) , so ν (and also Q ) is aninfinite measure. Aaronson’s ergodic theorem (see, e.g. Theorem 2.4.2 in [1])states that for every f ∈ L ( ν ) , f > , and every potential normalizing function a : R + → (0 , ∞ ) , one always have either lim sup t →∞ a ( t ) Z t f ( ˜ U ( s ))d s = ∞ a.s.or lim inf t →∞ a ( t ) Z t f ( ˜ U ( s ))d s = 0 a.s.Without further mention, we shall henceforth implicitly work under the prob-ability measure P , and say that a family ( Y ( t )) t> of random variables has anon-degenerate limit in distribution as t → ∞ if Y ( t ) converges in law towardssome not a.s. constant random variable.Motivated by the famous Darling-Kac’s theorem, the purpose of this sectionis to provide an explicit necessary and sufficient condition in terms of the L´evyprocess ξ for the existence of a normalizing function a : R + → (0 , ∞ ) such thatthe normalized occupation measure of U converges in distribution as t → ∞ toa non-degenerate limit.We start with the following simple observation. Lemma 4.1.
The following assertions are equivalent(i) For every f ∈ L ( ν ) with h ν, f i 6 = 0 , a ( t ) Z t f ( ˜ U ( s ))d s, t > has a non-degenerate limit in distribution as t → ∞ .(ii) (cid:16) T ( t ) a ( t ) (cid:17) t> has a non-degenerate limit in distribution as t → ∞ .Proof. Note that the identity function g ( x ) ≡ /x always belongs to L ( ν ) , actu-ally with h ν, g i = 1 , and R t d s/ ˜ U ( s ) = T ( t ) . The claim thus follows from Hopf’sratio ergodic theorem (Theorem 3.1(iii)) combined with Slutsky’s theorem.8or the sake of simplicity, we shall focus on the case when the sought nor-malizing function a : (0 , ∞ ) → (0 , ∞ ) is regularly varying at ∞ with index α ∈ (0 , . Recall from the Darling-Kac theorem (Theorem 8.11.3 in [7]) thatthis is essentially the only situation in which interesting asymptotic behaviorscan occur. Recall also from Theorem 1.5.12 in [7] that a then possesses anasymptotic inverse b : (0 , ∞ ) → (0 , ∞ ) , in the sense that a ( b ( t )) ∼ b ( a ( t )) ∼ t as t → ∞ , such that b is regularly varying at ∞ with index /α .We may now state the main result of this work, which specifies the Darling-Kac theorem for Ornstein-Uhlenbeck type processes. Theorem 4.1.
The following assertions are equivalent:(i) b ( t ) − ξ t has a non-degenerate limit in distribution as t → ∞ .(ii) Let f ∈ L ( ν ) with h ν, f i 6 = 0 . Then a ( t ) Z t f ( ˜ U ( s ))d s, t > has a non-degenerate limit in distribution as t → ∞ .In that case, the limit in (i) is a positive α -stable variable, say σ , with E (exp( − λσ )) = exp( − cλ α ) for some c > , and the limit in (ii) has the law of h ν, f i σ − α (and is thusproportional to a Mittag-Leffer variable with parameter α ). Remark 2.
In the case when ξ is a subordinator, Caballero and Rivero provedthat the assertion (i) in Theorem 4.1 is equivalent to the assertion (i) of Lemma4.1 with the weak limit there given by a Mittag-Leffler distribution; see Propo-sition 2 in [9]. Thus in that special case, Theorem 4.1 follows directly fromProposition 2 in [9] and the present Lemma 4.1.Proof. Assume (i); it is well-known that the non-degenerate weak limit σ of b ( t ) − ξ t is an α -stable variable, which is necessarily positive a.s. since ξ driftsto ∞ . Recall that A ( t ) = ln ˜ I ( t ) − ln ˜ I (0) and write ξ t = ln ˜ I ( t ) − ln ˜ V ( t ) . We deduce from the stationarity of ˜ V and Slutsky’s theorem that there is theweak converge b ( t ) − A ( t ) = ⇒ σ as t → ∞ . Using the assumption that b is an asymptotic inverse of a and recalling that b is regularly varying with index /α , this entails by a standard argument that a ( t ) − T ( t ) = ⇒ σ − α as t → ∞ , and we conclude from Lemma 4.1 (it is well-known that σ − α is proportionalto a Mittag-Leffler variable with parameter α ; see for instance Exercise 4.19 inChaumont and Yor [12]).Conversely, if (ii) holds for some f ∈ L ( ν ) with h ν, f i 6 = 0 , then by Hopf’sergodic theorem and Lemma 4.1, a ( t ) − T ( t ) = ⇒ G as t → ∞ , G . The same argument as above yields b ( t ) − ξ t = ⇒ G − /α as t → ∞ , and G − /α has to be a positive α -stable variable.More precisely, the argument of the proof shows that when (i) is satisfied, theweak convergence in (ii) holds independently of the initial value ˜ U (0) . That is,equivalently, one may replace ˜ U by U , the starting point U (0) being arbitrary. Acknowledgment:
I would like to thank V´ıctor Rivero for pointing atimportant references which I missed in the first draft of this work.
References [1] Jon Aaronson.
An introduction to infinite ergodic theory , volume 50 of
Mathematical Surveys and Monographs . American Mathematical Society,Providence, RI, 1997.[2] Anita Behme and Alexander Lindner. On exponential functionals of L´evyprocesses.
J. Theoret. Probab. , 28(2):681–720, 2015.[3] Anita Behme, Alexander Lindner, and Makoto Maejima. On the rangeof exponential functionals of L´evy processes. In
S´eminaire de Probabilit´esXLVIII , volume 2168 of
Lecture Notes in Math. , pages 267–303. Springer,Cham, 2016.[4] Jean Bertoin and Mladen Savov. Some applications of duality for L´evyprocesses in a half-line.
Bull. Lond. Math. Soc. , 43(1):97–110, 2011.[5] Jean Bertoin and Marc Yor. The entrance laws of self-similar Markovprocesses and exponential functionals of L´evy processes.
Potential Anal. ,17(4):389–400, 2002.[6] Jean Bertoin and Marc Yor. Exponential functionals of L´evy processes.
Probab. Surv. , 2:191–212, 2005.[7] N. H. Bingham, C. M. Goldie, and J. L. Teugels.
Regular variation , vol-ume 27 of
Encyclopedia of Mathematics and its Applications . CambridgeUniversity Press, Cambridge, 1989.[8] M. E. Caballero and L. Chaumont. Weak convergence of positive self-similar Markov processes and overshoots of L´evy processes.
Ann. Probab. ,34(3):1012–1034, 2006.[9] Mar´ıa Emilia Caballero and V´ıctor Manuel Rivero. On the asymptoticbehaviour of increasing self-similar Markov processes.
Electron. J. Probab. ,14:865–894, 2009.[10] Philippe Carmona, Fr´ed´erique Petit, and Marc Yor. On the distributionand asymptotic results for exponential functionals of L´evy processes. In
Exponential functionals and principal values related to Brownian motion ,Bibl. Rev. Mat. Iberoamericana, pages 73–130. Rev. Mat. Iberoamericana,Madrid, 1997. 1011] Lo¨ıc Chaumont, Andreas Kyprianou, Juan Carlos Pardo, and V´ıctorRivero. Fluctuation theory and exit systems for positive self-similar Markovprocesses.
Ann. Probab. , 40(1):245–279, 2012.[12] Lo¨ıc Chaumont and Marc Yor.
Exercises in probability. A guided tour frommeasure theory to random processes via conditioning. 2nd ed.
Cambridge:Cambridge University Press, 2nd ed. edition, 2012.[13] Ronald A. Doney.
Fluctuation theory for L´evy processes , volume 1897 of
Lecture Notes in Mathematics . Springer, Berlin, 2007. Lectures from the35th Summer School on Probability Theory held in Saint-Flour, July 6–23,2005, Edited and with a foreword by Jean Picard.[14] K. Bruce Erickson and Ross A. Maller. Generalised Ornstein-Uhlenbeckprocesses and the convergence of L´evy integrals. In
S´eminaire de Prob-abilit´es XXXVIII , volume 1857 of
Lecture Notes in Math. , pages 70–94.Springer, Berlin, 2005.[15] B´en´edicte Haas and V´ıctor Rivero. Quasi-stationary distributions and Ya-glom limits of self-similar Markov processes.
Stochastic Process. Appl. ,122(12):4054–4095, 2012.[16] P´eter Kevei. Ergodic properties of generalized Ornstein-Uhlenbeck pro-cesses.
Stochastic Processes and their Applications , pages –, 2017.[17] A. Kuznetsov, J. C. Pardo, and M. Savov. Distributional properties ofexponential functionals of L´evy processes.
Electron. J. Probab. , 17:no. 8,35, 2012.[18] John Lamperti. Semi-stable stochastic processes.
Trans. Amer. Math. Soc. ,104:62–78, 1962.[19] John Lamperti. Semi-stable Markov processes. I.
Z. Wahrscheinlichkeits-theorie und Verw. Gebiete , 22:205–225, 1972.[20] Oesook Lee. Exponential ergodicity and β -mixing property for generalizedOrnstein-Uhlenbeck processes. Theoretical Economics Letters , 2(1):21–25,2012.[21] Alexander Lindner and Ross Maller. L´evy integrals and the stationar-ity of generalised Ornstein-Uhlenbeck processes.
Stochastic Process. Appl. ,115(10):1701–1722, 2005.[22] Krishanu Maulik and Bert Zwart. Tail asymptotics for exponential func-tionals of L´evy processes.
Stochastic Process. Appl. , 116(2):156–177, 2006.[23] S. P. Meyn and R. L. Tweedie.
Markov chains and stochastic sta-bility . Communications and Control Engineering Series. Springer-Verlag London Ltd., London, 1993. Also available for download from http://probability.ca/MT/ free of charge.[24] Juan Carlos Pardo and V´ıctor Rivero. Self-similar Markov processes.
Bol.Soc. Mat. Mexicana (3) , 19(2):201–235, 2013.1125] Pierre Patie. q -invariant functions for some generalizations of the Ornstein-Uhlenbeck semigroup. ALEA, Lat. Am. J. Probab. Math. Stat. , 4:31–43,2008.[26] V´ıctor Rivero. Recurrent extensions of self-similar Markov processes andCram´er’s condition.
Bernoulli , 11(3):471–509, 2005.[27] L. C. G. Rogers and David Williams.
Diffusions, Markov processes, andmartingales. Vol. 1 . Wiley Series in Probability and Mathematical Statis-tics: Probability and Mathematical Statistics. John Wiley & Sons, Ltd.,Chichester, second edition, 1994. Foundations.[28] Ken-iti Sato.
L´evy processes and infinitely divisible distributions , volume 68of
Cambridge Studies in Advanced Mathematics . Cambridge UniversityPress, Cambridge, 2013. Translated from the 1990 Japanese original, Re-vised edition of the 1999 English translation.[29] Haruo Totoki. Time changes of flows.
Mem. Fac. Sci. Kyushu Univ. Ser.A , 20:27–55, 1966.[30] Jian Wang. On the exponential ergodicity of L´evy driven Ornstein-Uhlenbeck processes.