A transience condition for a class of one-dimensional symmetric L\'evy processes

In this paper, we give a sufficient condition for transience for a class of one-dimensional symmetric L\'evy processes. More precisely, we prove that a one-dimensional symmetric L\'evy process with the L\'evy measure $\nu(dy)=f(y)dy$ or $\nu(\{n\})=p_n$, where the density function $f(y)$ is such that $f(y)>0$ a.e. and the sequence $\{p_n\}_{n\geq1}$ is such that $p_n>0$ for all $n\geq1$, is transient if $$\int_1^{\infty}\frac{dy}{y^{3}f(y)}<\infty\quad\textrm{or}\quad \sum_{n=1}^{\infty}\frac{1}{n^{3}p_n}<\infty.$$ Similarly, we derive an analogous transience condition for one-dimensional symmetric random walks with continuous and discrete jumps.


Introduction
Let (Ω, F, P) be a probability space and let {L t } t≥0 be a stochastic process on (Ω, F, P) taking values in R d , d ≥ 1. The process {L t } t≥0 is called a Lévy process if L 0 = 0 P-a.s., if it has stationary and independent increments and if it has càdlàg paths P-a.s. (that is, if its trajectories are rightcontinuous with left limits P-a.s.). Having these properties, every Lévy process can be completely and uniquely characterized through the characteristic function of a single random variable L t , t > 0, that is, by the famous Lévy-Khintchine formula we have where ψ(ξ) = i ξ, b + 1 2 ξ, cξ + R d 1 − exp{i ξ, y } + i ξ, y 1 {|y|≤1} (y) ν(dy).
Here b is a vector in R d , c is a symmetric nonnegative-definite d × d matrix and ν(dy) is a σ-finite Borel measure on R d satisfying ν({0}) = 0 and R d min{1, y 2 }ν(dy) < ∞.
The measure ν(dy), the triplet (b, c, ν) and the function ψ(ξ) are called the Lévy measure, the Lévy triplet and the characteristic exponent of the Lévy process {L t } t≥0 , respectively. Further, recall that the vector b, the matrix c and the Lévy measure ν(dy) correspond to the deterministic part (shift), the continuous (Brownian) part and the jumping part of the Lévy process {L t } t≥0 , respectively.
In this paper, we consider the transience and recurrence property of Lévy processes. A Lévy process {L t } t≥0 is said to be transient if dξ < ∞ for some a > 0 (see [Sat99,Corollary 37.6 and Remark 37.7]). Again, in many situations this criterion is also not applicable. More precisely, for a given Lévy triplet (b, c, ν) it is not always easy to compute the integral part of the characteristic exponent as well as the integral appearing in the Chung-Fuchs criterion. According to this, the aim of this paper is to derive a sufficient condition for transience for Lévy processes in terms of the Lévy triplet. Let us remark that analogous definitions and characterizations of the transience and recurrence property hold also for random walks (see [Dur10,Chapter 4]). Recall that a random walk is a stochastic process {S n } n≥0 defined on a probability space (Ω, F, P) taking values in R d , d ≥ 1, defined by S 0 := 0 and S n : As already mentioned, in this paper we consider the one-dimensional symmetric case only. Note that, according to [Sat99,Theorem 37.8] and [Dur10, Theorem 4.2.13], the limitation to the onedimensional case is not too big restriction since it is well known that every d-dimensional, d ≥ 3, Lévy process and random walk are transient. Further, recall that a stochastic process {X t } t∈T is Theorem 1.1. Let {L t } t≥0 be a one-dimensional symmetric Lévy process with the Lévy measure ν(dy) = f (y)dy or ν({n}) = p n , where the density function f (y) is such that f (y) > 0 a.e. and the sequence {p n } n≥1 is such that p n > 0 for all n ≥ 1.
(1. Let us remark that the same transience condition holds also in the case of one-dimensional symmetric random walks. More precisely, let {S n } n≥0 be a one-dimensional symmetric random walk with jumps P(J 1 ∈ dy) = f (y)dy or P(J 1 = n) = p n , where f (y) > 0 a.e. and p n > 0 for all n ≥ 1, then the condition (1.1) implies the transience property of {S n } n≥0 . Also, let us remark that, according to [Sat99,Theorem 38.2] or [She62, Lemma 1.2], the assumptions f (y) > 0 a.e. and p n > 0 for all n ≥ 1 can be relaxed. More precisely, it suffices to demand positivity of f (y) and p n on the complement of a compact set.

Corollary 1.3. A one-dimensional symmetric stable Lévy process or a random walk is transient if
Note that the above corollary implies that the function y −→ y 3 , appearing in the condition (1.1), is optimal in the class of power functions.
The transience and recurrence property of one-dimensional symmetric Lévy processes in terms of the Lévy triplet has already been studied in the literature. Namely, in [Sat99,Theorem 38.3] (see also [She62,Theorem 5]) it has been proved that a one-dimensional symmetric Lévy process {L t } t≥0 with the Lévy measure ν(dy) is recurrent if (1.2) Intuitively, the condition (1.2) measures the speed of divergence of the second moment of ν(dy), and, regarding this speed, it concludes the recurrence property. Clearly, if ν(dy) has finite second moment, then {L t } t≥0 is recurrent. Thus, if the second moment of ν(dy) diverges slow enough, then {L t } t≥0 is recurrent. Similarly, the condition (1.1) measures the speed of divergence of the third moment of ν(dy). If the third moment of ν(dy) diverges fast enough, then {L t } t≥0 is transient.
Recall that a symmetric Borel measure ρ(dy) on R is unimodal if it is finite outside of any neighborhood of the origin and if x −→ ρ(x, ∞) is a convex function on (0, ∞). Equivalently, a symmetric Borel measure ρ(dy) on R is unimodal if it is of the form ρ(dy) = aδ 0 (dy) + f (y)dy, where 0 ≤ a ≤ ∞ and the density function f (y) is symmetric, decreasing on (0, ∞) and it satisfies |y|>ε f (y)dy < ∞ for all ε > 0 (see [Sat99,Chapter 5]). Note that measures with a discrete support are never unimodal. Now, if the Lévy measure ν(dy) is additionally unimodal, the condition (1.2) is also necessary for the recurrence property (see [Sat99,Theorem 38.3] or [She62, Theorem 5]). Also, note that unimodality of the ν(dy) and finiteness of (1.2) imply that f (y) > 0 a.e., and the condition (1.2) is stronger than the condition (1.1) (see Section 4 for the proof). Thus, the condition (1.1) is a generalization of the condition (1.2) in the case when the jumping measure is not unimodal. Finally, we give an example where the condition (1.1) is more suitable than the Chung-Fuchs criterion and the condition (1.2). We consider an example of a Lévy process with "multiple indices of stability". Let {L t } t≥0 be a one-dimensional symmetric Lévy process with the Lévy measure ν({n}) = p n , where p 2n = (2n) −α−1 and p 2n−1 = (2n − 1) −β−1 for n ≥ 1 and α, β ∈ (0, ∞). For a continuous version of such process it suffices to interpolate the points {(i, p i ) : i ∈ Z}. Now, clearly, if α < 1 and β < 1, then the condition (1.1) implies transience of {L t } t≥0 . On the other hand, since ν(dy) is not unimodal, the condition (1.2) is not applicable, and the application of the Chung-Fuchs criterion leads to a non-trivial computation. Also, let us remark that the same example shows that the condition (1.1) is only sufficient for transience. Indeed, assume that α < 1 and β ≥ 1. Then, since β ≥ 1, the condition (1.1) fails to hold. On the other hand, since α < 1, {L t } t≥0 is transient (see Section 4 for the proof). Now, we explain our strategy of proving Theorem 1.1. The proof is divided in three steps. In the first step, by using electrical networks techniques, we prove Theorem 1.1 in the case of a random walk with discrete jumps. In the second step, we prove Theorem 1.1 in the case of a random walk {S n } n≥0 with continuous jumps P(J 1 ∈ dy) = f (y)dy. More precisely, for δ > 0 we define a discretization of {S n } n≥0 as a random walk {S δ n } n≥0 on δZ with the jump distribution P(J δ 1 = δn) := δn+ δ 2 δn− δ 2 f (y)dy, n ∈ Z. Next, by an "approximation approach" we prove that all the random walks {S δ n } n≥0 , δ > 0, are either transient or recurrent at the same time and their transience and recurrence property is equivalent with the transience and recurrence property of {S n } n≥0 . Finally, by using the first step, we prove that the condition (1.1) for {S n } n≥0 implies the transience property of {S 1 n } n≥0 . And this accomplishes the proof of the second step. At the end, in the last step, we consider the case of Lévy processes. By using [Sat99, Theorem 38.2], it suffices to consider the situation of a compound Poisson process. Now, the proof follows from the first and second step. This accomplishes the proof of Theorem 1.1.
The paper is organized as follows. In Section 2, we give a proof of Theorem 1.1 for the case of discrete jumps. In Section 3, by using the results from Section 2, we proof Theorem 1.1 for the case of continuous jumps. Finally, in Section 4, we discuss some properties of the condition (1.1).

Discrete case
In this section, we prove the main step of the proof of Theorem 1.1. More precisely, we derive a sufficient condition for transience for one-dimensional symmetric random walks on Z.
Theorem 2.1. Let {S n } n≥0 be a one-dimensional symmetric random walk on Z with jumps P(J 1 = n) = p n , where the sequence {p n } n≥1 is such that p n > 0 for all n ≥ 1. Then the random walk Note that the same transience condition also holds in the case of a one-dimensional symmetric Lévy process {L t } t≥0 with a discrete supported Lévy measure ν({n}) = p n , where p n > 0 for all n ≥ 1. Indeed, first note that where {S n } n≥0 is a random walk with jumps P(J 1 = n) := 1 ν(Z) p n and {P t } t≥0 is the Poisson process with parameter ν(Z) independent of {S n } n≥0 . Now, the desired result follows from the definition of transience in terms of sojourn times.
The proof of Theorem 2.1 is based on techniques and results from electrical networks. Let us introduce some notation we need. A graph is a pair Note that the Markov chain {X n } n≥0 is irreducible (that is, ∞ n=1 P(X n = v|X 0 = u) > 0 for all u, v ∈ V (G)) and it is reversible (that is, there exists a nontrivial measure π(dy) on V (G), such that π(u)q uv = π(v)q vu for all u, v ∈ V (G)). Indeed, irreducibility easily follows from connectedness of the graph G and for the reversibility measure we can take π := c. Also, let us remark that to every irreducible and reversible time-homogeneous Markov chain on a discrete state space S given by the transition kernel q uv , u, v ∈ S, and a reversibility measure π(dy) we can join a network N = (G, c). Indeed, put V (G) = S, the vertices u and v are adjacent if q uv > 0, the graph G is connected because of irreducibility of the corresponding Markov chain and the conductance is defined by c(e uv ) = π(u)q uv .
Further, let u 0 ∈ V (G) be an arbitrary vertex of the network N . A flow from u 0 to ∞ is a function θ : The energy of the flow is defined by Next, recall that a state u of a time-homogeneous Markov chain {X n } n≥0 on a discrete state space S is called transient if ∞ n=1 P(X n = u|X 0 = u) < ∞ and it is called recurrent if ∞ n=1 P(X n = u|X 0 = u) = ∞. If every state is transient (resp. recurrent) the chain itself is called transient (resp. recurrent). It is well known that every irreducible Markov chain is either recurrent or transient (see [MT93,Theorem 8 .1.2]). Finally, the main tool for proving Theorem 2.1 is given in the following theorem. i = j or |i − j| ≥ 2 2 −2|i| , 0 < i < j = i + 1 or j < j + 1 = i < 0.
Recall that flow has to be antisymmetric, hence we define θ(v, u) := −θ(u, v). Next, note that According to this, θ is a flow from 0 to ∞. Finally, since v∈Z θ(0, v) = θ(0, 1) + θ(0, −1) = 1, θ is a unit flow from 0 to ∞. Now, let us prove that the energy of the flow θ is bounded from above by (2.1). We have Note that, from the symmetry of the distribution {p n } n∈Z and the definition of the flow θ, the second and the third therm equal 1 8p 1 . Next, again from the symmetry of the distribution {p n } n∈Z and the symmetry of the function θ 2 , we have Note that θ(u, u + w) = 0 when u + w ≥ 4u, except for u = 0 and w = 1. Thus where ⌈x⌉ denotes the smallest integer not less than x. Now, since for u ∈ B i , i ≥ 2, (that is, for This yields This accomplishes the proof of Theorem 2.1.

Continuous case
In this section, we prove Theorem 1.1 in the case of continuous jumps. As in the case of discrete jumps, the main step is to consider the random walk case.
Theorem 3.1. Let {S n } n≥0 be one-dimensional symmetric random walk with jumps P(J 1 ∈ dy) = f (y)dy, where the probability density function f (y) is such that f (y) > 0 a.e. Then the random Again, similarly as in the case of discrete jumps, the transience condition for the Lévy process case can be easily derived from the random walk case. Indeed, let {L t } t≥0 be a one-dimensional symmetric Lévy process with the Lévy measure ν(dy) = f (y)dy, where the density f (y) is such that f (y) > 0 a.e. Then, first note that, according to [Sat99,Theorem 38.2], without loss of generality we can assume that ν(R) < ∞. Thus, where {S n } n≥0 is a random walk with continuous jumps P(J 1 ∈ dy) := 1 ν(R) f (y)dy and {P t } t≥0 is the Poisson process with parameter ν(R) independent of {S n } n≥0 . Now, the desired result follows from the definition of transience in terms of sojourn times.
Before In the following proposition, we characterize the transience and recurrence property of Lévy process in terms of càdlàg paths. Proof. The proof follows directly from the definition of the transience and recurrence properties. Now, let us recall the notion of characteristics of a semimartingale (see [JS03]). Let (Ω, F, {F t } t≥0 , P, {S t } t≥0 ), {S t } t≥0 in the sequel, be a one-dimensional semimartingale and let h : R −→ R be a truncation function (that is, a continuous bounded function such that h(x) = x in a neighborhood of the origin). We define two processeš where the process {∆S t } t≥0 is defined by ∆S t := S t − S t− and ∆S 0 := S 0 . The process {S(h) t } t≥0 is a special semimartingale. Hence, it admits the unique decomposition where {S(h) t } t≥0 is a local martingale and {S(h) t } t≥0 is a predictable process of bounded variation. δ (s,∆Ss(ω)) (ds, dy) of the process {S t } t≥0 and let {C t } t≥0 be the quadratic co-variation process for martingale appearing in (3.2), then (B,C, N ) is called the modified characteristics of the semimartingale {S t } t≥0 (relative to h(x)).
Proposition 3.4. Let S = {S n } n≥0 be a one-dimensional random walk with continuous jumps P(J 1 ∈ dy) = f (y)dy. For δ > 0, let S δ = {S δ n } n≥0 be a random walk on δZ with discrete jumps Further, let {P t } t≥0 be the Poisson process with parameter 1 independent of S and S δ , δ > 0, and for all a > 0. Thus, by Proposition 3.2, the transience and recurrence property of the random walks S δ , δ > 0, is equivalent with the transience and recurrence property of the random walk S.
At the end, we prove Theorem 3.1.
Therefore, we have proved the desired result.