Exponential Ergodicity of Killed Lévy Processes in a Finite Interval

Following Bertoin who considered the ergodicity and exponential decay of Lévy processes in a finite domain [4], we consider general Lévy processes and their ergodicity and exponential decay in a finite interval. More precisely, given Ta = inf{t > 0 : Xt / ∈ (0, a)}, a > 0 and X a Lévy process then we study from spectral-theoretical point of view the killed process P (Xt ∈ ., Ta > t). Under general conditions, e.g. absolute continuity of the transition semigroup of the unkilled Lévy process, we prove that the killed semigroup is a compact operator. Thus, we prove stronger results in view of the exponential ergodicity and estimates of the speed of convergence. Our results are presented in a Lévy processes setting but are well applicable for Markov processes in a finite interval once one can establish Lebesgue irreducibility of the killed semigroup and that the killed process is a doubly Feller process. For example, this scheme is applicable to the work of Pistorius [10].


Introduction and results
In this short note we investigate the ergodic properties of general Lévy processes killed upon exiting a finite interval.Exit from such domains is known as the "doublesided exit problem".We stress that this technique is applicable in the far wider context of Markov processes.So far this problem has been previously considered in some generality by Bertoin in [4] for the case when a Lévy process has only negative jumps.
There Bertoin uses the so called R-theory developed by Tuominen and Tweedie [13] in order to identify the r-positivity of the process and to identify the r-invariant function and measure.Under similar conditions, i.e. the doubly Feller property of the underlying Lévy process, we derive and discuss the exponential ergodicity of the semigroup of the killed Lévy process in the general case, i.e. when our Lévy process can make both positive and negative jumps.Moreover, we connect this topic to the general theory of semigroups and explicitly demonstrate how the main result can be related to general spectral theory.We achieve this by making use of a result by Schilling and Wang [11] on compactness of Markov semigroups and using the classical theory of compact, positive operators.We strongly believe that this approach is perfectly adapted to studying the ergodic properties in the "double-sided exit problem" as it makes use of first and foremost the domain in question namely a compact interval and then of the underlying structure of the one-dimensional Lévy process.
The note is organized as follows: in the first section we introduce the notation and the main results; in the second section we discuss the implications of our results, their conditions and how far they can be extended, also we point out some challenges; in the third section we provide the proof of our results.

Notation and Main Result
We denote by X = (X t ) t≥0 a real-valued Lévy process, i.e. an a.s.right-continuous process with stationary and independent increments.The semigroup of the Lévy process will be denoted by P. We recall that each Lévy process is characterized by its Lévy -Khintchine exponent, i.e.
where σ 2 ≥ 0 is the variance of the Brownian component, γ ∈ R is the linear term and Π is a σ-finite measure which describes the structure of the jumps of X, i.e. their intensity and size.
Fix a > 0. Denote the first hitting time to the closed set R \ (0, a) by Then the Lévy process killed upon exiting (0, a) is a Markov process, see [3, IV, Prop.4, p.46] and its semigroup will be denoted by P .For any q > 0, we will denote its resolvent by Θ q (x, dy) = ∞ 0 e −qt P x (X t ∈ dy) , for x ∈ (0, a) and y ∈ (0, a). (2.3) In the sequel we will call (DF) the assumption that P t is a semigroup of a doubly Feller process, i.e.P t f ∈ C b (R) for any f ∈ L ∞ (R), where C b (R) stands for the continuous bounded functions on R and L ∞ (R) is the set of all bounded measurable functions on R. By Theorem 2.2 in [8], it follows that X is a doubly Feller process when (AC) holds, namely P (X t ∈ dx) << dx, for every t > 0.
Call (F) the assumption that σ > 0 or X is not a negative of subordinator, does not live on a lattice and Π((−a, 0)) > 0 or X is not a subordinator, does not live on a lattice and Π((0, a)) > 0.
We note that condition (F) is very general, whereas (DF) seems slightly more restrictive, see Subsection 3.1 for details.
We are now able to state our main result.We will denote by C 0 ((0, a)) the space of continuous functions on [0, a] which vanish at the boundary.Theorem 2.1.Let (F) holds and a > 0 is fixed.Then P t is a semigroup of a Lebesgue irreducible Markov process with a state space (0, a).If additionally (DF) holds, for each ECP 19 (2014), paper 31.
Remark 2.3.Under the conditions in [4, Theorem 2 ] it is immediately augmented with convergence in total variation and the knowledge of an existing exponential rate of convergence in [4, (v), Th.2].We note that W −ρ in the notation of Bertoin satisfying W −ρ (a) = 0 is an immediate consequence of the fact that P t f (a) = 0, for any f ∈ C ([0, a]) due to the fact that X issued forth from a immediately enters (a, ∞) and Remark 2.4.It seems that the R-theory with all its might in general state space Markov processes is in this particular instance of "double-exit problem" weaker than the application of spectral theory.We believe this is due to the special case of a certain type of smoothing, i.e. the strong Feller property and the compactness of the closure of the domain (0, a).This is due to the fact that those properties imply compactness of the semigroup und thus in particular a gap between the first and the second eigenvalue.
The R-theory does not directly imply this spectral gap property.
3 Discussion and Further Remarks

Condition (F) and (DF)
Condition (DF) is implied by (AC), i.e. the absolute continuity of the transition semigroup of the original Lévy process.Via Fourier inversion it is clear that the transition density p t exists, equals Thus the class when (AC) holds is enormous.It seems that no necessary and sufficient condition for (DF) in terms of the Lévy triplet is known.Condition (F) is explicit in terms of the Lévy triplet and certainly holds when X is of unbounded variation, i.e.
Though in many conceivable examples (DF) implies (F) we have not proved this in generality.

General applicability of our results
Our Theorem 2.1 essentially relies on Proposition 4.2 and the irreducibility of the semigroup P t and is independent of the fact that X is a Lévy process.Given that any doubly Feller process killed upon hitting an open set is a doubly Feller, see [5], therefore all we need to know to apply our result is that X is a doubly Feller process, a.s.which implies that the killed process is doubly Feller and that X killed upon exit of (0, a) is a Lebesgue irreducible Markov process.In this vein the results of Pistorius [10] on reflected spectrally one-sided process and its ergodicity can be reduced to the still demanding but yet much shorter task of computation of the resolvent, its properties and the verification of the fact that the reflected killed process satisfies (DF).The R-theory is again superfluous.It is very difficult however to have information on the eigenvalues.Some trivial estimates for ρ 1 exist but any precise analytical way to computing it is elusive.Furthermore, it seems that numerical schemes will be hard to obtain even for Lévy processes due to the difficulty of computing the resolvents.

Applicability to Lévy processes
We believe that our results and methodology is very streamlined in view of the classical spectral and Markov theory it relies on.Some results that need a good guess seem to come naturally thanks to the language and notions we use from analysis.Certainly, not all comes for free and for Lévy processes what needs to be computed to have any information on the first eigenfunction and first eigenvalue, namely W and ρ 1 , is the resolvent Θ q .Once this is done as in [4] (see also [12]) then one can have a grasp on these quantities which by no means ensures that they would be known explicitly.Even in [4], where many quantities are tractable we have no clear way to obtain W −ρ in a closed form and even ρ = ρ 1 .Therefore a new methodology is needed for further progress in this direction.

Proof of Theorem 2.1
We prove Theorem 2.1 in several steps.Taking into accound Theorems 4.4, 4.4 and 4.6 we see that the assertions i), ii) and the first part of the assertions iii) and iv) without the specific form of the co-eigenfunction follow in fact from general theory of compact irreducible semigroups.Thus we will first prove irreducibility and then compactness of the killed semigroup.

Proof of irreducibility
We start with the question of irreducibility as defined in the appendix.
Proof.We need to show, that the resolvent maps a non-trivial and non-negative function to a strictly positve.Fix a generic interval A = (b, c) ⊂ (0, a), x ∈ (0, a).It suffices to show that Θ q 1 A (x) > 0, for each x ∈ (0, a).Clearly this is the case for x ∈ A since the Lévy process is a.s.continuous at any deterministic time.Assume that x / ∈ A and without loss of generality assume that 0 < x ≤ b.Now it is enough to show that with positive probability X enters (b+ , c− ) for some very small > 0 prior to exiting (0, a).
where Y is a Lévy process collecting all jumps of X between ( i 2 , i ) only.Then for all i big enough we obtain that X S ∈ (b + i /2, c − i /2) and therefore Θ q 1 A (x) > 0. So it remains to investigate when P (A i ) > 0, for all i big enough.If X is with infinite variation then by definition Z is as well with infinite variation and from [2, Prop 1.1.]we get that Z has the so-called small deviation property and thus P (A i ) > 0. If X has a bounded variation, i.e. ] we conclude that Z has the small deviation property and thus P (A i ) > 0. However, if b > 0 we add a drift to Y , say Y t = 2bt + Y t which also has as a stopping time ∞ > S > 0 such that Y S ∈ (b + i , c − i ) and clearly Z t = Z t − 2bt is such that b < 0. Applying the same procedure we conclude the statement.If Π(0, ∞) < ∞ then we decompose X t = Y t + Z t with Y being a compound Poisson process collecting all positive jumps of X. From [4, Prop.1] Z killed upon exiting (0, a) is Lebesque irreducible if either σ > 0 or Π(−a, 0) > 0 holds.Therefore, conditioning upon {Y ≡ 0} until Z enters (b, c) we get that Θ q 1 A (x) > 0. However, when the last condition of (F ) is satisfied it may happen that Π(−a, 0) = 0. Then we put Y to be the compound Poisson process collecting the negative jumps only and use the fact that Z is Lebesgue irreducible from [4, Prop.1] in the same fashion as above.

Compactness of P t
In the following theorem we demonstrate the compactness of the semigroup by following the ideas of [11].Similar ideas can be found in the proof of Theorem BIV 2.5 in [1].In order to make this work self-contained and to make these ideas more widely konwn to the the probabilistic community we provide a complete proof of this very useful result.Proposition 4.2.Assume that (P t ) t≥0 is a semigroup of a doubly Feller process, then for every t > 0 the operator P a] satisfies that µ t (N ) = 0 then P t (x, N ) = 0 for Lebesgue all x.Using the strong Feller property we conclude that x → P t 1 N (x) = P t (x, N ) is continuous and we thus conclude that P t (x, N ) = 0 for all x ∈ [0, a].Therefore P t (x, •) is absolutely continuous with respect to µ t and has a Radon-Nikodym density p t (x, y).Now, for any u ∈ L ∞ (µ t ), define the measurable set and note that ũ is bounded and Borel measurable.We define due to the strong Feller property.We now need to show that the image of ).First observe that by the Banach-Alaoglu theorem U is weak*-compact and therefore every sequence (u j ) j∈N ⊂ U contains a weak*-subsequence (u j k ) k∈N and for suitable u ∈ L ∞ (µ t ) the limit Moreover, for every k, l, m ∈ N with k, l ≥ m |P 2t u j k − P 2t u j l | ≤ P t P t u j k − P t u j l ≤ P t sup k,l≥m |P t u j k − P t u j l | .
Note that h m := sup k,l≥m |P t u j k − P t u j l | decreases to 0 as m → ∞ and so does the sequence (P t h m ) m∈N as a consequence of the dominated convergence theorem.According to the strong Feller property the functions P t h m are continuous and thus by Dini's theorem we get uniform convergence, which means that (P 2t u j k ) k∈N is Cauchy in C([0, a]) and the proof is complete as C([0, a]) ⊂ L ∞ (µ t ).
We are now ready to prove compactness for our semigroups.First we show that started from any x ∈ (0, a) If X is with infinite variation then T a a.s = Ta is immediate as the two-half planes are regular for X, i.e.X enters immediately in R + and R − provided it starts from zero.Let X be of bounded variation and X satisfies (F) and (DF).We next prove that X enters immediately in (a, ∞) conditional on {X Ta = a} with the other case, i.e. {X Ta = 0} studied in the same way.If P (X Ta = a) > 0 then it follows that X creeps up which implies that the ascending ladder height process H + t = δ + t + jumps with δ + > 0. This implies that R + is regular for X, see [7,Th.22,p.61]and therefore conditioned on {X Ta = a}, X enters immediately after T a the set (a, ∞).Thus T a a.s = Ta .However, since X satisfies (DF) and any doubly Feller process remains doubly Feller upon hitting an open set (recall T a a.s = Ta ), see [5], we can use Proposition 4.2 above to conclude compactness.
ECP 19 (2014), paper 31.BIII 3.1 in [1] irreducibility of the semigroup (T t ) t≥0 is defined by the requirement that for every given 0 < f ∈ E and φ ∈ E there is some t 0 > 0 such that T t0 f, φ > 0 or equivalently by the property that there is some λ > 0 such that for every 0 < f ∈ E  ).Suppose that A is the generator of an irreducible positive semigroup (T t ) t≥0 on the Banach space E of continuous functions on some locally compact space which vanish at infinity.Then the following assertions are true: 1) The spectrum σ(A) of A is not empty.
2) every positive eigenfunction of A is strictly positive 3) if ker(s(A) − A) is contains a positive element then dim ker(s(A) − A) ≤ 1. 4) if s(A) is a pole of the resolvent then it is algebraically simple.The residue has the form P = φ ⊗ u where φ ∈ E and u ∈ E are strictly positive eigenelements of A and A, respectively, satisfying (φ, u) = 1.
The influence of the generator A upon the spectral properties of the semigroups is content of the following result, where we denote by σ(B) the spectrum of an operator B.
The previous results in combination with compactness of the semigroup the following asymptotic result is true: Theorem 4.6 (compare Theorem BIV 2.1 and Corollary BIV 2.1 in [1]).Let (T t ) t≥0 be a compact irreducible Sub-Markov-semigroup on the Banach space E = C 0 (X) for some locally compact space X with generator A. Then the spectrum A is discrete σ(A) = {−ρ 1 , −ρ 2 , . . .} with ρ n+1 ≥ ρ n for n > 1 and R ∈ ρ 1 < ρ 2 and there exists a strictly positive continuous function h and a strictly positive bounded measure ν on K such that for every δ ∈ (0, ρ 1 − ρ 2 ) and some M δ ≥ 1 and all t ≥ 0 e ω(T )t T t − ν ⊗ h ≤ M δ e −δt , where ω(T ) := inf{w | ∃M w ∀t ≥ 0 : T t ≤ M w e −wt } is the growth bound of the semigroup and under the above conditions the growth bound coincides with the spectral radius of the semigroup and the spectral radius of the generator.

e
−λt T t f dt is strictly positive.Irreducible positive semigroups have some fundamental spectral properties, which are usually referred to results of Perron-Frobenius or Krein-Rutman type.We denote by σ(B) the spectrum of B, by r(B) the spectral radius and by s(B) the spectral bound of an operator B, i.e. s(B) := sup{ λ | λ ∈ σ(B)}.