On the Existence of Recurrent Extensions of Self-similar Markov Processes

Let $X= (X_t) _{t \geq 0}$ be a self-similar Markov process with values in the non-negative half-line, such that the state $0$ is a trap. We present a necessary and sufficient condition for the existence of a self-similar recurrent extension of $X$ that leaves $0$ continuously. This condition is expressed in terms of the Levy process associated with $X$ by the Lamperti transformation.


Introduction
In his pioneering study [15] of the structure of self-similar Markov processes with state space [0, ∞[, Lamperti posed the problem of determining those self-similar Markov processes that agree with a given self-similar Markov process up to the time the latter process first hits 0. Our goal in this paper is to present a necessary and sufficient condition for the existence of such a "recurrent extension" that, in addition, leaves 0 continuously. Our work was inspired by that of Vuolle-Apiala [22] and Rivero [18]. To state our results precisely, we introduce some notation and recall some of the basic theory of self-similar Markov processes. A Borel right process X = ((X t ) t≥0 , (P x ) x≥0 ) with values in [0, ∞) is self-similar provided there exists H > 0 such that, for each c > 0 and x ≥ 0, the law of the rescaled process (c −H X ct ) t≥0 , is P x/c H when X has law P x . The number H is the order of X, and when there is a need to emphasize H, we shall describe X as being H-self-similar. By the discussion in section 2 of [15], we can (and do) assume that X is a Hunt process; thus in addition to being a right-continuous strong Markov process, the sample paths of X are quasi-left-continuous. One of several zero-one laws developed by Lamperti states that if T 0 := inf{t > 0 : X t = 0}, (1.1) For definiteness, we take X to be realized as the coordinate process X t (ω) = ω(t) on the sample space Ω of all right-continuous left-limited paths from [0, ∞[ to itself. We assume that 0 is a trap for X, so that each of the laws P x governing X is carried by {ω ∈ Ω : ω(t) = 0, ∀t ≥ T 0 (ω)}. The natural filtration on Ω is (F t ), and F ∞ := ∨ t≥0 F t . We write P t f (x) = P t (x, f ) := P x [f (X t )] for the transition semigroup of X, and U q := ∞ 0 e −qt P t dt, q > 0, for the associated resolvent operators. Define, for c > 0, Φ c : Ω → Ω by Φ c ω(t) := c −H ω(ct). The H-self-similarity of X means that identically on Ω.
for each x ≥ 0, and (ii) 0 is not a trap for X.
Typically, X will be realized as the coordinate process on Ω, in which case X t (ω) = ω(t).
What distinguishes X from X is the collection of laws (P x ) x≥0 . For emphasis or clarity we may at times write T 0 instead of T 0 , etc. In view of (1.3), if a recurrent extension X is self-similar, then its order must be the same as that of X. Let X be a recurrent extension of X, with resolvent U q , q > 0. Writing T 0 for the hitting time of 0 by X, we have ψ q (x) := P x [exp(−qT 0 )] = P x [exp(−qT 0 )], and by the strong Markov Excursion theory [13,16,3,10] leads to an expression for U q f (0) in terms of a certain entrance law for X. Let M denote the closure of the zero set {t ≥ 0 : X t = 0}, and let G denote the set of strictly positive left endpoints of the maximal intervals in the complement of M . The excursions of X from 0 are indexed by the elements of G: The excursion e s associated with s ∈ G is the Ω-valued path defined by where θ s is the shift operator on Ω. Let L = (L t ) t≥0 denote X-local time at 0, normalized so that P 0 ∞ 0 e −t dL t = 1. Then there is a σ-finite measure n on (Ω, F ∞ ) such that, for predictable Z ≥ 0 and F ∞ -measurable F ≥ 0, Formula (1.5) determines n uniquely, and under n the coordinate process (X t ) t>0 is a strong Markov process with transition semigroup (P t ) and one-dimensional distributions One can show that there exists ≥ 0 such that t 0 1 {0} (X s ) ds = · L t for all t ≥ 0, P 0 -a.s.; it follows from this and (1.5) (applied to Z s = e −qs and F = T0 0 e −qt f (X t ) dt, first with a general f and then with f = 1) that where n q := which is finite for all q > 0, as follows from (1.5) with F = 1 − exp(−qT 0 ) and Z s = e −qs . The family (n t ) t>0 is an entrance law : n t P s = n t+s for all t, s > 0. The measure n is uniquely determined by (n t ) and (P t ). In this paper we focus on recurrent extensions that leave 0 continuously, referring the interested reader to [22] and [18] for discussions of extensions for which n[X 0 = 0] = 0. We shall produce an H-self-similar recurrent extension of X by constructing a suitable excursion measure n as above. Motivated by the preceding discussion, we make the following definitions. Recall that (P t ) t≥0 is the transition semigroup for X.
As noted above, the excursion measure associated with a recurrent extension of X is necessarily admissible.
Itô [13] showed that the excursion measure n determined by a recurrent extension X (as in (1.5)) satisfies a set of six necessary conditions. These conditions are not quite sufficient for the existence of a recurrent extension, but Salisbury [20,21] discovered that if two of the conditions are strengthened, then the resulting set of conditions is sufficient (and still necessary). These results hold for processes with very general state spaces. The special case of [0, ∞[-valued Feller processes was treated by Blumenthal in [2]. Vuolle-Apiala [22] verified that the conditions imposed by Blumenthal are satisfied in the setting of self-similar Markov processes on [0, ∞[. Thus, if n is an admissible excursion measure, then X admits recurrent extensions X (one for each ≥ 0) such that (1.7) holds. The process X 0 has the additional property that U q (x, 1 {0} ) = 0 for all x ≥ 0, a condition that is necessary for self-similar recurrent extensions, as was noted above. By [22, (1.5)][ (see also [18,Lem. 2]), this extension is self-similar if and only if n is self-similar. Our hypotheses will be stated in terms of the Lévy process associated with X by the Lamperti transformation. For this consider the continuous additive functional A defined by and its right continuous inverse τ defined by in which we follow the usual convention that inf ∅ = +∞. According to [15,Thm. 4 is a Lévy process (i.e., a process with stationary independent increments). Moreover, by [ (1.14) If (1.13) holds then the random variable ζ := A T0− is exponentially distributed (with rate δ > 0, say) and there is a real-valued Lévy process Z independent of ζ such that Z is Z killed at time ζ: If (1.14) holds then ζ = A T0− = +∞, P x -a.s. for all x > 0, and Z is a real-valued Lévy process that drifts to −∞. Let Q z denote the law of Z under the initial condition Z 0 = z. The process Z = (Z t , Q z ) is referred to as the Lévy process underlying X. We shall write (Q t ) for the transition semigroup of Z. is finite for all u ≥ 0; see [1,Thm. 1]. Define the inverse of τ : For each x > 0, the process defined by holds.
(b) There is at most one self-similar recurrent extension that leaves 0 continuously.

Remark 1. (a)
To see why the condition 0 < κ < 1/H is natural, observe that if there is to be a recurrent extension X of X, then the inverse local time of X at 0 is a stable subordinator of index γ := κH. (c) The meaning of the parameter k in (VA) was clarified by Rivero [18], who introduced the following condition (R), expressed in terms of the underlying Lévy process Z: The law of Z 1 is not supported by a lattice rZ; for all real t 1 < t 2 < · · · < t n and x 1 , x 2 , . . . , x n . Thus Y = (Y t ) under Q is a stationary Markov process with random times of birth and death (namely, α and β), with one-dimensional distributions (while alive) all equal to ξ, and with transition probabilities those of the Lévy process Z. For the construction and various properties of such processes the reader is referred to [17,6,11].
We now use a time-reversal argument to show that ξ is in fact invariant for Z. To this end define a semigroup ( Q κ t ) by the formula (2.5) and observe that ( Q κ t ) is the semigroup dual to (Q t ) with respect to ξ. That is, cf. the computation in (2.15) below. Thus The moment generating function Q 0 [exp(λZ t )], λ ∈ R, is necessarily of the form where ψ : R →] − ∞, +∞]. By Hölder's inequality, log ψ is convex (strictly convex on the interior of the interval where it is finite). Now either Z drifts to −∞ (in which case ψ(0) = 0 But by (2.2) and Tonelli's theorem we also have m = ∞ 0 n t dt, so by the uniqueness of such a representation [11, (5.25)], η t = n t for all t > 0. Let Ω + be the space of right-continuous left-limited paths from ]0, ∞[ to [0, ∞[, and let n + be the image of n under the mapping Ω ω → ω| ]0,∞[ ∈ Ω + . Let F + ∞ be the σ-field on Ω + generated by the coordinate maps X + t , t > 0. Theorem (2.1) of [14] now tells us that if Π : W → Ω + denotes the (inverse Lamperti) transformation w → (t → exp(Y K(t,w) (w)), t > 0) then In particular for all t > 0, from which it follows that lim u↓α Y u = −∞, Q-a.s. By time reversal (as in (2.9)), this means that the dual process Z κ does not jump to −∞ ( Q z -a.s. for ξ-a.e. z ∈ R). That is, ψ(κ) = 0; equivalently, Conversely, suppose that (1.19) holds for some κ > 0. Then ξ(dz) = e −κz dz is an invariant measure for Z: (2.13) Let (Y, Q) be the Kuznetsov process for Z and ξ as before. Because ξ is invariant, Q[α > −∞] = 0; see [11, (6.7)]. Making use of (2.5) and the discussion following (2.6) we see that shows that m is purely excessive for X, so there is a uniquely determined entrance law (η t ) t>0 such that m = ∞ 0 η t dt. As before let Ω + be the space of right-continuous left-limited paths from ]0, ∞[ to [0, ∞[. Let n + be the measure on Ω + under which the coordinate process (X + t ) t>0 is Markovian with one-dimensional distributions (η t ) t>0 and transition semigroup (P t ); see [12] for the construction of such measures. Then by [14,Thm. 2.1], writing Π : W → Ω + for the map w → (t → exp(Y K(t,w) (w)), t > 0), we have (2.14) In particular, the choice B = B 0 := {ω ∈ Ω + : lim t↓0 ω(t) = 0} in (2.14) shows that that n + is carried by B 0 because Q is carried by {w ∈ W : lim t↓−∞ w(t) = −∞, α(w) = −∞}.
Identifying Ω with B 0 and F ∞ with F + ∞ ∩ B 0 , we see that n + induces an excursion measure n on (Ω, F ∞ ) as in Definition 3(a). Now write Ψ x : w → (w(t) + x) t∈R for the spatial translation operator on W ; a check of finite dimensional distributions shows that This, in combination with (2.14), implies that n is self-similar, with similarity index γ = κH. From the proof of Lemma 3 in [18], we have (2.15) where I := ∞ 0 exp(Z v /H) dv. Lemma 2 of [19] tells us that Q 0 [I κH−1 ] < ∞ because κH ∈ ]0, 1[, thereby guaranteeing the admissibility of n.
(b) Turning to the uniqueness assertion, let X 1 and X 2 be self-similar recurrent extensions of X that leave 0 continuously. In view of (1.4) and (1.7), the resolvent (and so the distribution) of X j is uniquely determined by the associated entrance law (n j t ), j = 1, 2. Because of a uniqueness theorem [11, (5.25)] cited earlier, each entrance law is in turn uniquely determined by its integral m j = ∞ 0 n j t dt. But as noted at the beginning of this section, m j has the form m j (dx) = C j x −1+(1−γ)/H dx, where γ = κH. (Both X 1 and X 2 satisfy (1.19), by the part of Theorem 1 already proved.) It follows that n 1 t = (C 1 /C 2 )n 2 t for all t > 0, so X 1 and X 2 have the same resolvent.
Remark 2. The argument just given for the "if" portion of part (a) of Theorem 1 proves a bit more than is asserted in the statement of the theorem. Namely, if κ > 0 is such that (1.19) holds, then there is a self-similar excursion measure n (with index γ := κH) such that n[X 0 > 0] = 0. Notice that (1.3) implies the existence of a constant C ∈]0, ∞] such that If, as in the proof above, we normalize n so that n

An Application
Suppose that our self-similar Markov process X satisfies (1.13). In this section we shall apply Theorem 1 to obtain the following improvement of Theorem 6(i) of Chaumont and Rivero [4]. Then h : x → x −κ is a purely excessive function for X, and the h-transform process X h with laws is "X conditioned to converge to 0": Proof. Let us start with the Lévy process Z := −Z, the dual of Z with respect to Lebesgue measure on R, and apply the inverse Lamperti transformation to obtain a self-similar Markov process X = (( X t ) t≥0 , ( P x ) x≥0 ) on [0, ∞[ with 0 as a trap. The process X is in weak duality with X with respect to the measure η(dx) := x −1+1/H dx; see [23]. By hypothesis, X satisfies the condition (1.19) of part (a) of Theorem 1. In particular, P x [ T 0 < ∞] = 1 for all x > 0.
(Otherwise, the Lévy process Z has infinite lifetime and lim sup t→+∞ Z t = +∞, a.s., hence the moment generating function ψ of Z (defined by analogy with (2.7)) vanishes at 0 as well as at κ. Since ψ in convex on [0, κ], this yields ψ (0+) ∈ [−∞, 0[, implying that Z drifts to −∞, a contradiction.) By the proof of Theorem 1 (see Remark 2), applied to X, there is an X-excursion measure n under which the coordinate process on (Ω, F ∞ ) is a strong Markov process with transition semigroup ( P t ) (that of X) and entrance law ( n t ) t>0 , say. Moreover, By Nagasawa's theorem, as found for example in the section 4 of [7] (see also [5] and the appendix of [8]), the process X defined by X t := X (T0−t)− , if 0 ≤ t < T 0 , 0, otherwise, (3.5) is Markovian under n, with transition semigroup i.e., the h-transform of (P t ) corresponding to h(x) = x −κ . In other words, the image n of n under the mapping ω → X(ω) is an X h -excursion measure. Because n[T 0 = ∞] = 0, we have, for > 0, It follows that P x h lim sup t↑T0 X t > 0 = 0, first for m-a.e. x > 0, and then for all x > 0 by the reasoning used above.