ON THE EXISTENCE OF RECURRENT EXTENSIONS

OF SELF-SIMILAR MARKOV PROCESSES

P.J. Fitzsimmons

University of California San Diego

March 29, 2006

Abstract. Let X = (X_{t})_{t≥0} be a self-similar Markov process with values in [0, ∞[, 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 Lévy process associated with X by the Lamperti transformation.

1. Introduction

In his pioneering study [L72] 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 a recurrent extension that, in addition, leaves 0 continuously. Our work was inspired by that of Vuolle-Apiala [VA94] and Rivero [R05].

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 [L72], 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

(1.1)  T_{0} := inf{t > 0 : X_{t} = 0},

then either P^{x}[T_{0} < ∞] = 0 for all x > 0 or P^{x}[T_{0} < ∞] = 1 for all x > 0. Our interest is in the latter situation, which we assume to be the case throughout the rest of the paper.

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

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 60G18, secondary 60G51, 60J45, 60J55.

Key words and phrases. self-similar, semi-stable, Lamperti transformation, recurrent extension, Cramér condition, excursion.

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

(1.2)  Φ_{c}P^{x}(B) := P^{x}[Φ_{c}^{−1}B] = P^{x/c^{H}} [B],  ∀B ∈ F_{∞}, x ≥ 0, c > 0.

Observe that

(1.3)  T_{0}^{∘}Φ_{c} = c^{−1}T_{0},

identically on Ω.

(1.4) Definition. A Borel right Markov process \overline{X} = (\overline{X}_{t}, \overline{P}^{x}) with state space [0, ∞[ is a recurrent extension of X provided (i) the stopped process ((X_{t∧\overline{T}_{0}})_{t≥0}, \overline{P}^{x}) has the same law as (X_{t}, P^{x}), for each x ≥ 0, and (ii) 0 is not a trap for \overline{X}.

Typically, \overline{X} will be realized as the coordinate process on Ω, in which case \overline{X}_{t}(ω) = ω(t). What distinguishes \overline{X} from X is the collection of laws (\overline{P}^{x})_{x≥0}. For emphasis or clarity we may at times write \overline{T}_{0} instead of T_{0}, etc.

In view of (1.3), if a recurrent extension \overline{X} is self-similar, then its order must be the same as that of X.

Let \overline{X} be a recurrent extension of X, with resolvent \overline{U}^{q}, q > 0. Writing \overline{T}_{0} for the hitting time of 0 by \overline{X}, we have ψ_{q}(x) := \overline{P}^{x}[exp(−q\overline{T}_{0})] = P^{x}[exp(−qT_{0})], and by the strong Markov property of \overline{X} at time \overline{T}_{0},

(1.5)  \overline{U}^{q}f(x) = U^{q}f(x) + ψ^{q}(x)\overline{U}^{q}f(0),  ∀x ≥ 0, q > 0.

Excursion theory [I72, Ma75, B92, FG06] leads to an expression for \overline{U}^{q}f(0) in terms of a certain entrance law for X. Let \overline{M} denote the closure of the zero set \{t ≥ 0 : \overline{X}_{t} = 0\}, and let \overline{G} denote the set of strictly positive left endpoints of the maximal intervals in the complement of \overline{M}. The excursions of \overline{X} from 0 are indexed by the elements of \overline{G}: The excursion e^{s} associated with s ∈ \overline{G} is the Ω-valued path defined by

e_{t}^{s} := \{ \begin{matrix}\overline{X}_{s+t}, & 0 ≤ t < T_{0}^{∘}θ_{s}, \\ 0, & t ≥ T_{0}^{∘}θ_{s}, \end{matrix}

where θ_{s} is the shift operator on Ω. Let \overline{L} = (\overline{L}_{t})_{t≥0} denote \overline{X}-local time at 0, normalized so that \overline{P}^{0} ∫_{0}^{∞} e^{−t} d\overline{L}_{t} = 1. Then there is a σ-finite measure n on (Ω, F_{∞}) such that, for predictable Z ≥ 0 and F_{∞}-measurable F ≥ 0,

(1.6)  \overline{P}^{0} [∑_{s∈\overline{G}} Z_{s} F (e^{s})] = \overline{P}^{0} [∫_{0}^{∞} Z_{t} d\overline{L}_{t}] · n[F].

Formula (1.6) 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

(1.7)  n_{t}(B) := n[X_{t} ∈ B, t < T_{0}],  t > 0, B ∈ B_{]0,∞[}.

One can show that there exists ℓ ≥ 0 such that ∫_{0}^{t} 1_{\{0\}}(\overline{X}_{s}) ds = ℓ · \overline{L}_{t} for all t ≥ 0, \overline{P}^{0}-a.s.; it follows from this and (1.6) (applied to Z_{s} = e^{−qs} and F = ∫_{0}^{T_{0}} e^{−qt}f(X_{t}) dt, first with a general f and then with f = 1) that

(1.8)  \overline{U}^{q}f(0) = \frac{ℓf(0) + n^{q}(f)}{qℓ + qn^{q}(1)},  f ∈ bpB_{[0,∞[},

where n^{q} := ∫_{0}^{∞} e^{−qt}n_{t} dt. (Here bpB_{[0,∞[} denotes the class of bounded real-valued Borel functions on [0, ∞[). The denominator in (1.8) is equal to

qℓ + n [∫_{0}^{T_{0}} qe^{−qt} dt] = qℓ + n[1 − e^{−qT_{0}}],

which is finite for all q > 0, as follows from (1.6) 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}).

If the recurrent extension \overline{X} is self-similar, then (i) ℓ = 0 [VA94, p. 551] and (ii) either n[X_{0} = 0] = 0 or n[X_{0} > 0] = 0 [VA94, Thm. 1.2].

(1.9) Definition. A recurrent extension \overline{X} is said to leave 0 continuously provided n[X_{0} > 0] = 0.

It follows easily from (1.6) that \overline{X} leaves 0 continuously if and only if

\overline{P}^{0}[\overline{X}_{s} = 0 for all s ∈ \overline{G}] = 1.

In this paper we focus on recurrent extensions that leave 0 continuously, referring the interested reader to [VA94] and [R05] for discussion 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.

(1.10) Definitions. (a) A measure n on (Ω, F_{∞}) is an excursion measure provided (i) the measure n_{t} defined on B_{]0,∞[} by formula (1.7) is σ-finite for each t > 0, (ii) (X_{t})_{t>0} under n is Markovian with transition operators (P_{t}) and one-dimensional distributions (n_{t}), and (iii) n puts no mass on the zero path [0] : t ↦ 0.

(b) We say that an excursion measure n is self-similar if

(1.11)  Φ_{c}n = c^{−γ}n,  ∀c > 0,

for some γ > 0. Equivalently,

(1.12)  n_{ct}(c^{H}B) = c^{−γ}n_{t}(B),  ∀c > 0, t > 0, B ∈ B_{]0,∞[},

(c) An excursion measure n is admissible provided n[1 − exp(−T_{0})] < ∞.

As noted above, the excursion measure associated with a recurrent extension of X is necessarily admissible.

Itô [I72] showed that the excursion measure n determined by a recurrent extension \overline{X} (as in (1.6)) satisfies a set of six necessary conditions. These conditions are not quite sufficient for the existence of a recurrent extension, but Salisbury [S86a, S86b] 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 [B83]. Vuolle-Apiala [VA94] 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 \overline{X}^{ℓ} (one for each ℓ ≥ 0) such that (1.8) holds. The process \overline{X}^{0} has the additional property that \overline{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 [VA94, (1.5)] (see also [R05, 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

(1.13)  A_{t} := ∫_{0}^{t} X_{s}^{−1/H} ds,  t ≥ 0,

and its right continuous inverse τ defined by

(1.14)  τ(u) := inf{t > 0 : A_{t} > u},  u ≥ 0,

in which we follow the usual convention that inf ∅ = +∞. According to [L72, Thm. 4.1], the [−∞, ∞[-valued process

(1.15)  Z_{u} := log X_{τ(u)},  u ≥ 0,

is a Lévy process (i.e., a process with stationary independent increments). Moreover, by [L72, Lem. 3.2]. we have either

(1.16)  P^{x}[X_{T_{0}−} > 0, T_{0} < ∞] = 1,  ∀x > 0,

or

(1.17)  P^{x}[X_{T_{0}−} = 0, T_{0} < ∞] = 1,  ∀x > 0.

If (1.16) holds then the random variable ζ := A_{T_{0}−} is exponentially distributed (with rate δ > 0, say) and there is a real-valued Lévy process \overline{Z} independent of ζ such that Z is \overline{Z} killed at time ζ:

(1.18)  Z_{u} = \{ \begin{matrix}\overline{Z}_{u}, & 0 ≤ u < ζ; \\ −∞, & u ≥ ζ. \end{matrix}

If (1.17) holds then ζ = A_{T_{0}−} = +∞, 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.

Because A and τ are strictly increasing on their respective domains of finiteness, the Lamperti transformation can be inverted as follows. Let (Z_{t}, Q^{z}) be a [−∞, ∞[-valued Lévy process that drifts to −∞ in case ζ := inf{t : Z_{t} = −∞} = +∞, a.s. (The state −∞ is a trap, and serves as the cemetery state for Z.) Then

(1.19)  τ(u) := ∫_{0}^{u} exp(Z_{s}/H) ds

is finite for all u ≥ 0; see [BY05, Thm.1]. Define the inverse of τ:

(1.20)  A(t) = A_{t} := inf{u : τ(u) > t},  t ≥ 0.

For each x > 0, the process defined by

(1.21)  exp(Z_{A(t)}),  t ≥ 0,

has, under Q^{log x}, the same law as X under P^{x}.

Here is the main result of the paper. Define I := τ(∞) = ∫_{0}^{∞} exp(Z_{v}/H) dv.

(1.22) Theorem. (a) There is an excursion measure n satisfying (1.11) such that n[X_{0} > 0] = 0 if and only if κ := γ/H > 0 satisfies Cramér’s condition:

(1.23)  Q^{0}[exp(κZ_{t})] = 1,  for some (or all) t > 0.

(b) The H-self-similar Markov process X admits a self-similar recurrent extension that leaves 0 continuously if and only if (1.23) holds for some κ ∈]0, 1/H[ and

(1.24)  Q^{0} [I^{κH−1}] < ∞.

There is at most one self-similar recurrent extension that leaves 0 continuously.

(1.25) Remarks. (a) If the excursion measure promised by part (a) of Theorem (1.22) is to correspond to a self-similar recurrent extension \overline{X}, then κH must lie in ]0, 1[, because the inverse local time of \overline{X} at 0 is a stable subordinator of index κH.

(b) Let C_{0}]0, ∞[ denote the class of continuous real-valued functions f on ]0, ∞[ such that lim_{x→0} f(x) = lim_{x→+∞} f(x) = 0. Vuolle-Apiala [VA94] introduced the following condition (VA): There exists k > 0 such that the limit

(VA.1)  lim_{x→0+} \frac{P^{x}[1 − exp(−T_{0})]}{x^{k}}

exists in ]0, ∞[, and the limit

(VA.2)  lim_{x→0+} \frac{U^{q}f(x)}{x^{k}}

exists for all q > 0 and all f ∈ C_{0}]0, ∞[, and is strictly positive for at least one non-negative f ∈ C_{0}]0, ∞[. Vuolle-Apiala showed that (VA) implies the existence (and uniqueness) of a selfsimilar recurrent extension of X that leaves 0 continuously.

(c) The meaning of the parameter k in (VA) was clarified by Rivero [R05], who introduced the following condition (R), expressed in terms of the underlying Lévy process Z:

(R.1)  The law of Z_{1} is not supported by a lattice rZ;

there exists θ > 0 such that

(R.2)  Q^{0}[exp(θZ_{t})] = 1, for some (or all) t > 0 and
(R.3)  Q^{0}[Z_{1}^{+} exp(θZ_{1})] < ∞.

Suppose that (R) holds. Rivero showed that if θH ≥ 1, then condition (VA) must fail, while if 0 < θH < 1 then (VA) holds for k = θ. Conversely, Rivero showed that if (VA) holds then 0 < kH < 1 and (R.2) and (R.3) hold with θ = k. Thus, modulo the technical condition (R.1), (VA) and (R) are equivalent, for k = θ ∈]0, 1/H[. Moreover, under (VA), k = θ = κ (as in (1.23)).

(d) The essentially equivalent conditions (VA) and (R) imply our conditions (1.23) (with 0 < κH < 1) and (1.24). It is not clear how broad the gap is between these conditions.

2. Proof of Theorem (1.22).

(a) Let us first suppose that X admits an excursion measure n such that n[X_{0} > 0] = 0 and such that the scaling property

(2.1)  Φ_{c}n = c^{−γ}n,  ∀c > 0,

holds for some γ > 0. The “mean occupation measure” m defined on the Borel subsets of ]0, ∞[ by

(2.2)  m(B) := n ∫_{0}^{T_{0}} 1_{B}(X_{s}) ds,  B ∈ B_{]0,∞[},

is an X-excessive measure; see [FG06, (4.5)]. As a consequence of (2.1), m takes the form

(2.3)  m(dx) = Cx^{−1+(1−γ)/H} dx,

for some constant 0 < C < ∞; see [R05, Lem. 3]. [Briefly, (2.1) implies that m scales as follows: m(b^{H}B) = b^{1−γ}m(B); this in turn implies (2.3).] By the theory of time changes for Markov processes, as found for example in [FG88], the measure ν := dx ↦ x^{−1−γ/H} dx is therefore excessive for the time-changed process t ↦ X_{τ(t)}. Define κ := γ/H. Then ξ : dz ↦ e^{−κz} dz (the image of ν under the mapping x ↦ log x) is excessive for the Lévy process Z.

Let (Y_{t}, ℚ) denote the Kuznetsov measure associated with Z and ξ. In more detail, let W be the space of paths w : R → [−∞, +∞[ that are R-valued and right-continuous with left limits on an open interval ]α(w), β(w)[ and that hold the value −∞ outside that interval. (Thus −∞ serves as cemetery state for Y.) The coordinate maps Y_{t}(w) := w(t), t ∈ R, generate G := σ{Y_{t} : t ∈ R}, and ℚ is the unique measure on (W, G) not charging the dead path [−∞] : t ↦ −∞, and such that

(2.4)  \begin{aligned}ℚ[Y_{t_{1}} ∈ dx_{1}, Y_{t_{2}} & ∈ dx_{2}, …, Y_{t_{n}} ∈ dx_{n}] \\ & = ξ(dx_{1})Q_{t_{2}−t_{1}}(x_{1}, dx_{2}) ⋯ Q_{t_{n}−t_{n−1}}(x_{n−1}, dx_{n}), \end{aligned}

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 [Mi79, F87b, G90].

We now use a time-reversal argument to show that ξ is in fact invariant for Z. To this end define a semigroup (Q̂_{t}^{κ}) by the formula

(2.5)  Q̂_{t}^{κ}f(x) := Q^{0}[f(x − Z_{t})e^{κZ_{t}}],

and observe that (Q̂_{t}^{κ}) is the semigroup dual to (Q_{t}) with respect to ξ. That is,

∫_{R} f · Q_{t}g dξ = ∫_{R} g · Q_{t}^{κ}f dξ, t > 0, f, g ∈ pB_{R};

cf. the computation in (2.15) below. Thus

(2.6)  \begin{aligned}ℚ[Y_{t_{1}} ∈ dx_{1}, Y_{t_{2}} & ∈ dx_{2}, …, Y_{t_{n−1}} ∈ dx_{n−1}, Y_{t_{n}} ∈ dx_{n}] \\ & = ξ(dx_{n})Q̂_{t_{n}−t_{n−1}}^{κ}(x_{n}, dx_{n−1}) ⋯ Q̂_{t_{2}−t_{1}}^{κ}(x_{2}, dx_{1}), \end{aligned}

The moment generating function Q^{0}[exp(λZ_{t})], λ ∈ R, is necessarily of the form

(2.7)  Q^{0}[exp(λZ_{t})] = exp(tψ(λ)),

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 and Q^{0}[Z_{1}] = ψ′(0+) ∈ [−∞, 0[) or jumps to −∞ (in which case ψ(0) < 0). Clearly 1 ≥ Q̂_{t}^{κ}1(x) = exp(ψ(κ)t), so ψ(κ) ≤ 0. If ψ(κ) < 0 then Q_{κ}^{0}[Ẑ_{t}^{κ} = −∞] = 1 − exp(ψ(κ)t) > 0. If ψ(κ) = 0 then Q̂_{κ}^{0}[Ẑ_{t}^{κ}] = −Q^{0}[Z_{t}e^{κZ_{t}}] = −ψ′(κ−) ∈ [−∞, 0[ by convexity. Therefore Ẑ^{κ} either jumps to −∞ at a finite time (if ψ(κ) < 0) or drifts to −∞ (if ψ(κ) = 0). In either case, the exponential integral

Î:= ∫_{0}^{∞} exp(Ẑ_{s}^{κ}/H) ds

is finite, Q̂_{κ}^{z}-a.s., for all z ∈ R. Thus if we define

(2.8)  ρ(u) := ∫_{−∞}^{u} exp(Y_{s}/H) ds,  u ∈ R,

then

(2.9)  ℚ[ρ(u) = ∞, α < u < β] = ∫_{R} Q̂_{κ}^{z}[Î = ∞] ξ(dz) = 0,

by [Mi79, (4.7)]. In particular,

lim_{u↓α} ρ(u) = 0, ℚ-a.s.

It follows that the inverse process C defined by

C(t) := inf{u : ρ(u) > t}, t ≥ 0,

is strictly increasing and continuous on [0, ρ(∞)[. Recalling from section 1 the discussion of the (inverse) Lamperti transformation, we see that the measure m is proportional to the image under the mapping z ↦ exp(z) of the Revuz measure of the CAF τ relative to the Z-excessive measure ξ. By a result of Kaspi [K88, (2.3), (2.8)], the formula

(2.10)  η_{t}(f) := ℚ[f(exp(Z_{C(t)})), 0 < C(t) ≤ 1],  f ∈ pB_{]0,∞[}, t > 0,

defines an entrance law for (P_{t}) and

(2.11)  m = ∫_{0}^{∞} η_{t} dt.

But by (2.2) and Tonelli’s theorem we also have m = ∫_{0}^{∞} n_{t} dt, so by the uniqueness of such a representation [G90, (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 [K88] now tells us that if Π : W → Ω^{+} denotes the (inverse Lamperti) transformation w ↦ (t ↦ exp(Y_{C(t,w)}(w)), t > 0) then

(2.12)  n^{+}[A, t < T_{0}] = ℚ[Π^{−1}(A), 0 < C(t) ≤ 1],  A ∈ F_{∞}^{+}.

In particular

ℚ [lim sup_{u↓α} Y_{u} > −∞, 0 < C(t) ≤ 1] = n^{+} [lim sup_{s↓0} X_{s}^{+} > 0, t < T_{0}^{+}]
= n[X_{0} > 0, t < T_{0}] = 0,

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 Ẑ^{κ} does not jump to −∞ (Q̂^{z}-a.s. for ξ-a.e. z ∈ R). That is, ψ(κ) = 0; equivalently,

Q^{0}[e^{κZ_{t}}] = 1, t > 0,

which is (1.23).

Conversely, suppose that (1.23) holds for some κ > 0. Then ξ(dz) = e^{−κz} dz is an invariant measure for Z:

(2.13)

ξQ_{t}(f) = ∫_{R} e^{−κz}Q^{z}[f(Z_{t})] dz = ∫_{R} e^{−κz}Q^{0}[f(z + Z_{t})] dz
= Q^{0} [∫_{R} e^{−κz}f(z + Z_{t}) dz] = Q^{0} [∫_{R} e^{κ(Z_{t}−y)}f(y) dz]
= ∫_{R} e^{−κy}f(y)Q^{0}[e^{κZ_{t}}] dy = ∫_{R} e^{−κy}f(y), dy = ξ(f).

Let (Y, ℚ) be the Kuznetsov process for Z and ξ as before. Because ξ is invariant, ℚ[α > −∞] = 0; see [G90, (6.7)]. Making use of (2.5) and the discussion following (2.6) we see that

Q̂_{κ}^{0}[Ẑ_{t}^{κ}] = −tψ′(κ−) ∈ [−∞, 0[,

so that Ẑ^{κ} drifts to −∞. As before, (2.6) holds; consequently the time-reversal argument used before to show the finiteness of the integral in (2.8) now permits us to deduce that ℚ is carried by {α = −∞, lim_{t↓α} Y_{t} = −∞}.

Define m(dx) = x^{−1−κ+1/H} dx. The Revuz measure of the continuous additive functional τ(u) := ∫_{0}^{u} exp(Z_{v}/H) dv, relative to ξ, is the measure μ(dz) := e^{z/H} ξ(dz) on R. The image of μ under the mapping z ↦ exp(z) is the measure m; another application of Kaspi’s theorem 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 coordinate process (X_{t}^{+})_{t>0} is Markovian with one-dimensional distributions (η_{t})_{t>0} and transition semigroup (P_{t}); see [GG87] for the construction of such measures. Then by [K88, Thm. 2.1], writing Π : W → Ω^{+} for the map w ↦ (t ↦ exp(Y_{C(t,w)}(w)), t > 0), we have

(2.14)  n^{+}[B, t < T_{0}] = ℚ[Π^{−1}(B), 0 < C(t) ≤ 1],  B ∈ F_{∞}^{+}.

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 (1.10). 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

Ψ_{x}ℚ = e^{−κx}ℚ.

This, in combination with (2.14), implies that n is self-similar, with similarity index γ = κH. This concludes the proof of part (a) of Theorem (1.22).

(b) For any self-similar excursion measure n (with γ = κH) we have, from the proof of Lemma 3 in [R05], that

(2.15)  n[1 − exp(−T_{0})] = CH · Γ(1 − κH) · Q^{0}[I^{κH−1}],

where the constant C is as in (2.3). Thus, a self-similar excursion measure n is admissible if and only if (1.24) holds.

Finally, turning to the uniqueness assertion, let \overline{X}^{1} and \overline{X}^{2} be self-similar recurrent extensions of X that leave 0 continuously. In view of (1.5) and (1.8), the resolvent (and so the distribution) of \overline{X}^{j} is uniquely determined by the associated entrance law (n_{t}^{j}), j = 1, 2. Because of a uniqueness theorem [G90, (5.25)] cited earlier, each entrance law is in turn uniquely determined by its integral m^{j} = ∫_{0}^{∞} n_{t}^{j} 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 \overline{X}^{1} and \overline{X}^{2} satisfy (1.23), by the part of Theorem (1.22) already proved.) It follows that n_{t}^{1} = Cn_{t}^{2} for all t > 0 (C = C_{1}/C_{2}), so \overline{X}^{1} and \overline{X}^{2} have the same resolvent. □

(2.16) Remark. If n is a self-similar excursion measure (with similarity index γ) then (1.3) implies that there is a constant C̃ ∈]0, ∞] such that

(2.17)  n[T_{0} > t] = C̃t^{−γ},  t > 0.

In particular, if n is the excursion measure guaranteed by part (a) of Theorem (1.22), then (2.15) and (2.17) combine to show that

C̃ = CHQ^{0}[I^{γ−1}],

and so condition (1.24) holds if and only if n[T_{0} > 1] < ∞.

3. An Application.

Suppose that our self-similar Markov process X satisfies (1.16). In this section we shall apply part (a) of Theorem (1.22) to obtain the following improvement of Theorem 6(i) of Chaumont and Rivero [CR06].

(3.1) Theorem. Suppose there exists κ > 0 such that

(3.2)  Q^{0}[exp(−κZ_{t})] = 1,  for some (or all) t > 0.

Then h : x ↦ x^{−κ} is a purely excessive function for X, and the h-transform process X^{h} with laws

(3.3)  P_{h}^{x}[F, t < T_{0}] = x^{κ} · P^{x}[F X_{t}^{−κ}],  t > 0, F ∈ bF_{t}, x > 0,

is “X conditioned to converge to 0”:

(3.4)  P_{h}^{x}[X_{T_{0}−} = 0, T_{0} < ∞] = 1,  ∀x > 0.

Proof. Let us start with the Lévy process Ẑ := −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 X̂ = ((X̂_{t})_{t≥0}, (P̂^{x})_{x≥0}) on [0, ∞[ with 0 as a trap. The process X̂ is in weak duality with X with respect to the measure η(dx) := x^{−1+1/H} dx; see [VG86]. By hypothesis, X̂ satisfies the condition (1.23) of part (a) of Theorem (1.22). In particular, P̂^{x}[T̂_{0} < ∞] = 1 for all x > 0. (Otherwise, the Lévy process Ẑ has infinite lifetime and lim sup_{t→+∞} Ẑ_{t} = +∞, a.s., hence the moment generating function ψ̂ of Ẑ (defined by analogy with (2.7)) vanishes at 0 as well as at κ. Since ψ̂ in convex on [0, κ], this yields ψ̂′(0+) ∈ [−∞, 0[, implying that Ẑ drifts to −∞, a contradiction.) By Theorem (1.22)(a), applied to X̂, there is an X̂-excursion measure n̂ under which the coordinate process on (Ω, F_{∞}) is a strong Markov process with transition semigroup (P̂_{t}) (that of X̂) and entrance law (n̂_{t})_{t>0}, say. Moreover,

(3.5)  m(B) := ∫_{0}^{∞} n̂_{t}(B) dt = C ∫_{B} x^{−1−κ+1/H} dx,  ∀B ∈ B_{]0,∞[},

for some constant 0 < C < ∞. By Nagasawa’s theorem, as found for example in the section 4 of [F87a] (see also [D85] and the appendix of [F99]), the process X̃ defined by

(3.6)  X̃_{t} := \{ \begin{matrix}X_{(T_{0}−t)−}, & if 0 ≤ t < T_{0}, \\ 0, & otherwise, \end{matrix}

is Markovian under n̂, with transition semigroup

(3.7)  P_{t}^{h}f(x) := x^{κ}P^{x}[f(X_{t})X_{t}^{−κ}; t < T_{0}]

i.e., the h-transform of (P_{t}) corresponding to h(x) = x^{−κ}. In other words, the image ñ of n̂ under the mapping ω ↦ X̃(ω) is an X^{h}-excursion measure. Because ñ[T_{0} = ∞] = 0, we have, for ϵ > 0,

0 = ñ[ϵ < T_{0} = ∞] = ñ [P_{h}^{X_{ϵ}}[T_{0} = ∞]; T_{0} > ϵ] ,

so P_{h}^{x}[T_{0} = ∞] = 0, first for Lebesgue a.e. x > 0, then for all x > 0 because this probability does not depend on x—apply (1.2) and (1.3) to X^{h}. Similarly,

ñ[X_{0} ≠ 0] = n̂ [lim sup_{t↑T̃_{0}} X̃_{t} > 0] = n̂[X_{0} ≠ 0] = 0,

so if ϵ > 0,

0 = ñ [{T_{0} > ϵ} ∩ θ_{ϵ}^{−1}{lim sup_{t↑T_{0}} X_{t} > 0}] = ñ [P_{h}^{X_{ϵ}} [lim sup_{t↑T_{0}} X_{t} > 0]; T_{0} > ϵ] .

It follows that

P_{h}^{x} [lim sup_{t↑T_{0}} X_{t} > 0] = 0,

first for m-a.e. x > 0, and then for all x > 0 by the reasoning used above.

□

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