Penalizing null recurrent diffusions

We present some limit theorems for the normalized laws (with respect to functionals involving last passage times at a given level up to time t) of a large class of null recurrent diffusions. Our results rely on hypotheses on the L\'evy measure of the diffusion inverse local time at 0. As a special case, we recover some of the penalization results obtained by Najnudel, Roynette and Yor in the (reflected) Brownian setting.


A few notation
We consider a linear regular null recurrent diffusion (X t , t ≥ 0) taking values in R + , with 0 an instantaneously reflecting boundary and +∞ a natural boundary. Let P x and E x denote, respectively, the probability measure and the expectation associated with X when started from x ≥ 0. We assume that X is defined on the canonical space Ω := C(R + → R + ) and we denote by (F t , t ≥ 0) its natural filtration, with F ∞ := t≥0 F t .
We denote by s its scale function, with the normalization s(0) = 0, and by m(dx) its speed measure, which is assumed to have no atoms. It is known that (X t , t ≥ 0) admits a transition density q(t, x, y) with respect to m, which is jointly continuous and symmetric in x and y, that is: q(t, x, y) = q(t, y, x). This allows us to define, for λ > 0, the resolvent kernel of X by: u λ (x, y) = ∞ 0 e −λt q(t, x, y)dt. (1) We also introduce (L a t , t ≥ 0) the local time of X at a, with the normalization: As is well-known, (τ (a) l , l ≥ 0) is a subordinator, and we denote by ν (a) its Lévy measure. To simplify the notation, we shall write in the sequel τ l for τ (0) l and ν for ν (0) . We shall also denote sometimes by µ(t) = µ([t, +∞[) the tail of the measure µ.

Motivations
Our aim in this paper is to establish some penalization results involving null recurrent diffusions. Let us start by giving a definition of penalization: Definition 1. Let (Γ t , t ≥ 0) be a measurable process taking positive values, and such that 0 < E x [Γ t ] < ∞ for any t > 0 and every x ≥ 0. We say that the process (Γ t , t ≥ 0) satisfies the penalization principle if there exists a probability measure Q (Γ) x defined on (Ω, F ∞ ) such that: This problem has been widely studied by Roynette, Vallois and Yor when P x is the Wiener measure or the law of a Bessel process (see [RVY06c] for a synthesis and further references). They showed in particular that Brownian motion may be penalized by a great number of functionals involving local times, supremums, additive functionals, numbers of downcrossings on an interval... Most of these results were then unified by Najnudel, Roynette and Yor (see [NRY09]) in a general penalization theorem, whose proof relies on the construction of a remarkable measure W.
Later on, Salminen and Vallois managed in [SV09] to extend the class of diffusions for which penalization results hold. They proved in particular that under the assumption that the (restriction of the) Lévy measure 1 ν([1,+∞[) ν |[1,+∞[ of the subordinator (τ l , l ≥ 0) is subexponential, the penalization principle holds for the functional (Γ t = h(L 0 t ), t ≥ 0) with h a non-negative and nonincreasing function with compact support. Let us recall that a probability measure µ is said to be subexponential (µ belongs to class S) if, for every t ≥ 0, where µ * 2 denotes the convolution of µ with itself. The main examples of subexponential distributions are given by measures having a regularly varying tail (see Chistyakov [Čis64] or Embrechts, Goldie and Veraverbek [EGV79]): where β ≥ 0 and η is a slowly varying function. When β ∈]0, 1[, we shall say that such a measure belongs to class R. Let us also remark that a subexponential measure always satisfies the following property: The set of such measures shall be denoted by L, hence: Now, following Salminen and Vallois, one may reasonably wonder what kind of penalization results may be obtained for diffusions whose normalized Lévy measure belongs to classes R or L. This is the main purpose of this paper, i.e. we shall prove that the results of Najnudel, Roynette and Yor remain true for diffusions whose normalized Lévy measure belongs to R, and we shall give an "integrated version" when it belongs to L 2 .

Statement of the main results
Let a ≥ 0, g (t) a := sup{u ≤ t; X u = a} and (F t , t ≥ 0) be a positive and predictable process such that Theorem 2.
1. If ν belongs to class L, then and 2. If ν belongs to class R: and if F is decreasing: Remark 3. Point 2. does not hold for every ν ∈ L. Indeed, otherwise, taking a = 0 and F t = 1 {L 0 t ≤ℓ} with ℓ > 0, one would obtain: a relation which is known to hold if and only if ν ∈ S, see [EGV79] or [Sat99,p.164].
Remark 4. If (X t , t ≥ 0) is a positively recurrent diffusion, then +∞ 0 ν([s, +∞[)ds = m(R + ) and the limit in Point 1. equals: In the following penalization result, we shall choose the weighting functional Γ according to ν: Theorem 5. Assume that: a) either ν belongs to class L, and Γ t = Then, the penalization principle is satisfied by the functional (Γ t , t ≥ 0), i.e. there exists a probability measure Q (F ) x on (Ω, F ∞ ), which is the same in both cases, such that, Furthermore: is weakly absolutely continuous with respect to P x : 2. Define g a := sup{s ≥ 0, X s = a}. Then, under Q (F ) x : i) g a is finite a.s., ii) conditionally to g a , the processes (X t , t ≤ g a ) and (X ga+t , t ≥ 0) are independent, iii) the process (X ga+u , u ≥ 0) is transient, goes towards +∞ and its law does not depend on the functional F .
We shall give in Theorem 21 a precise description of Q (F ) x through an integral representation.
Remark 6. The main example of diffusion satisfying Theorems 2 and 5 is of course the Bessel process with dimension δ ∈]0, 2[ reflected at 0. Indeed, setting β = 1 − δ 2 ∈]0, 1[, the tail of its Lévy measure at 0 equals: Remark 7. Let us also mention that this kind of results no longer holds for positively recurrent diffusions. Indeed, it is shown in [Pro10] that if (X t , t ≥ 0) is a recurrent diffusion reflected on an interval, then, under mild assumptions, the penalization principle is satisfied by the functional (Γ t = e −αL 0 t , t ≥ 0) with α ∈ R, but unlike in Theorem 5, the penalized process so obtained remains a positively recurrent diffusion.
Example 8. Assume that ν ∈ R and let h be a positive and decreasing function with compact support on R + .
h(u)q(u, 0, a)du < ∞ and therefore: and the martingale (M t (g a ), t ≥ 0) is given by: • One may also take for instance These were the first kind of weights studied by Roynette, Vallois and Yor, see [RVY06a] and [RVY06b].

Organization
The remainder of the paper is organized as follows: • In Section 2, we introduce some notation and recall a few known results that we shall use in the sequel. They are mainly taken from [Sal97] and [SVY07].
• Section 3 is devoted to the proof of Theorem 2. The two Points 1. and 2. are dealt with separately: when ν ∈ R, the asymptotic is obtained via a Laplace transform and a Tauberien theorem, while in the case ν ∈ L, we shall use a basic result on integrated convolution products.
• Section 4 gives the proof of Point 1. of Theorem 5, which essentially relies on a meta-theorem, see [RVY06c].
• In Section 5, we derive a integral representation for the penalized measure Q (F ) x which implies Point 2. of Theorem 5.
• Finally, Section 6 is devoted to prove that, with our normalizations, the process (N (a) t

Preliminaries
In this section, we essentially recall some known results that we shall need in the sequel.
• Let T a := inf{u ≥ 0; X u = a} be the first passage time of X to level a. Its Laplace transform is given by Since (X t , t ≥ 0) is assumed to be null recurrent, we have for where ∂ is a cemetary point. We denote by q(t, x, y) its transition density with respect to m: • We also introduce (X ↑a t , t ≥ 0) the diffusion ( X t , t ≥ 0) conditionned not to touch a, following the construction in [SVY07]. For x > a and F t a positive, bounded and F t -measurable r.v.: By taking F t = f (X t ), we deduce in particular that, for x, y > a: and m ↑a (dy) = (s(y) − s(a)) 2 m(dy).
Letting x tend towards a, we obtain: where P y (T a ∈ dt) =: n y,a (t)dt.
• We finally define (X x,t,y u , u ≤ t) the bridge of X of length t going from x to y. Its law may be obtained as a h-transform, for u < t: (3) With these notation, we may state the two following Propositions which are essentially due to Salminen.

The law of g (t)
a := sup{u ≤ t; X u = a} is given by: a , X t ) reads : We now study the pre-and post-g a -process: Proof. i) Point (i) follows from Proposition 5.5 of [Mil77] applied to the diffusion ii) Point (ii) is taken from [Sal97]. iii) As for Point (iii), still from [Sal97], conditionnally to g (t) a = u and X t = y > a, we have: But the bridges of X et X ↑ have the same law. Indeed, for y, x > a: and the result follows by letting x tend toward a.

Study of asymptotics
The aim of this section is to prove Theorem 2. We start with the case ν ∈ R.
3.1 Proof of Theorem 2 when ν ∈ R Let (F t , t ≥ 0) be a decreasing, positive and predictable process such that Our approach in this section is based on the study of the Laplace a . Indeed, from Propositions 9 and 10, we may write, applying Fubini's Theorem: We shall now study the asymptotic (when λ → 0) of each term separately. To this end, we state and prove two Lemmas.

The Laplace transform of t → ν (a) ([t, +∞[)
Lemma 11. The following formula holds: Proof. Since τ is a subordinator and m has no atoms, from the Lévy-Khintchine formula: Then, from the classic relation: Since both terms are positive, we may let ε → 0 to obtain: Remark 12. Since we assume that (X t , t ≥ 0) is a null recurrent diffusion, we have m(R + ) = +∞ and from Salminen [Sal93]: Thus, from the monotone convergence theorem, the function t → ν (a) ([t, +∞[) is not integrable at +∞. On the other hand, if (X t , t ≥ 0) is positively recurrent, we obtain: We now study the asymptotic of the first hitting time of X to level a.

Proof of Point 2. of Theorem 2
We now let λ tend toward 0 in (6). Observe first that, from our hypothesis on (F u , u ≥ 0): Then, from Lemmas 11 and 13, we obtain Therefore, for every x ≥ 0: and Point 2. follows from the Tauberian theorem (9) since t −→ E x F g (t) a is decreasing.

Proof of Theorem 2 when ν ∈ L
Let (F t , t ≥ 0) be a positive and predictable process such that From Propositions 9 and 10 we have the decomposition: But, inverting the Laplace transform (11), we deduce that: hence, we may rewrite: As in the previous section, the study of the asymptotic (when t → +∞) will rely on a few Lemmas.
Observe now that, since ν * q(t) = t 0 ν([u, +∞[)q(t − u, 0, 0)du = 1, we have from Fubini-Tonelli: Let ε > 0. There exists A > 0 such that, for every s ≥ A: Integrating this relation, we deduce that, for t > A: Therefore: and it only remains to divide both terms by t 0 ν(s)ds and let t tend toward +∞ to conclude, thanks to Remark 15, that:

Proof of Point 1. of Theorem 2
Going back to (12), we have, with f (u) = P x,u,a (F u )q(u, x, a) and ν (a) (u) = ν (a) ([u, +∞[): From Lemmas 14 and 16, we deduce that: Then, i) for every s ≥ 0 and Λ s ∈ F s : ii) there exists a probability measure Q on (Ω, F ∞ ) such that for every s ≥ 0: In the following, we shall use Biane-Yor's notations [BY87]. We denote by Ω loc the set of continuous functions ω taking values in R + and defined on an interval [0, ξ(ω)] ⊂ [0, +∞]. Let P and Q be two probability measures, such that P(ξ = +∞) = 0. We denote by P • Q the image measure P ⊗ Q by the concatenation application : To simplify the notations, we define the following measure, which was first introduced by Najnudel, Roynette and Yor [NRY09]: Definition 19. Let W x be the measure defined by: du q(u, x, a)P x,u,a • P ↑a a + (s(x) − s(a)) + P ↑a x W x is a sigma-finite measure with infinite mass.
This measure enjoys many remarkable properties, and was the main ingredient in the proof of the penalization results they obtained for Brownian motion. A similar construction was made by Yano, Yano and Yor for symmetric stable Lévy processes, see [YYY09]. With this new notation, we shall now write:

Proof of Point i) of Theorem 5
Let 0 ≤ u ≤ t. Using Biane-Yor's notation, we write: hence, from the Markov property, denoting F g (t) a = F (X s , s ≤ t): Let us assume first that ν ∈ R and that (F t , t ≥ 0) is decreasing. Then, from Theorem 2 with On the other hand, if ν ∈ L and Γ t = t 0 F g (s) a ds, a similar computation gives: Therefore, to apply Theorem 18, it remains to prove that: We shall make a direct computation, applying Proposition 9: • if x ≤ a, then, for y > a, P x (T a > t, X t ∈ dy) = 0 since X has continuous paths, and the same computation leads to: = 1, and the proof is completed.
Remark 20. Consider the martingale (N (a) t = (s(X t ) − s(a)) + − L a t , t ≥ 0). We apply the balayage formula to the semimartingale ((s(X t ) − s(a)) + , t ≥ 0): Therefore, the martingale (M t (F ga ), t ≥ 0) may be rewritten: Finally, Point 2. of Theorem 5 is a direct consequence of the following result: admits the following integral representation: Proof. Let G, H and ϕ be three Borel bounded functionals. We write: On the one hand, I 2 equals On the other hand, from Propositions 9 and 10: We now separate the two cases g (t) a > 0 as in relation (5).
• First, when g (t) a = 0 and x ≤ a, this term is null. Indeed, for x ≤ a < y, P x (T a > t, X t ∈ dy) = 0 since X has continuous paths. Next, for x > a: x [H(X s , s ≥ 0)] .
• Second, when g Our aim now is to prove (14). To this end, we shall use the following representation of the resolvent kernel u λ (x, y) (see [BS02,p.19]): where ψ λ and ϕ λ are the fundamental solutions of the generalized differential equation such that ψ λ is increasing (resp. ϕ λ is decreasing) and the Wronskian ω λ is given, for all z ≥ 0 by: Note that since m has no atoms, the meaning of (15) is as follows: ∀y ≥ x, λ • Assume first that x ≤ a. • Now, let us suppose that x > a. We have, with the same computation: On the one hand: On the other hand: Finally, gathering both terms, we obtain for x > a: which is the desired result (14) from the definition of the Wronskian.