The type problem for a class of inhomogeneous random walks : a series criterion by a probabilistic argument Basile de Loynes

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Distributed under a Creative Commons Attribution - NonCommercial - NoDerivatives| 4.0 International License The type problem for a class of inhomogeneous random walks : a series criterion by a probabilistic argument

Basile de Loynes1

1Ensai - Campus de Ker-Lann, Rue Blaise Pascal , BP 37 203 , 35 712 Bruz Cedex, France , [email protected]

Abstract In the classical framework, a random walk on a group is a Markov chain with independent and identically distributed increments. In some sense, random walks are time and space homogeneous. In this paper, a criterion for the recurrence or transience of Markov additive processes is given. This criterion is deduced using Fourier analysis and a perturbation argument of a Markov operator. The latter extends the results of the literature since it does not involve a quasi-compacity condition on the operator. Finally, this criterion is applied to a new family of model of strongly drifted random walks on the lattice.

Key words Weakly inhomogeneous random walks . Recurrence . Transience . Random Walks with Random Transition Probabilities . Markov additive processes . Fourier analysis . Operator perturbations

Mathematics Subject Classification (2000) 60F15 . 60J05 . 60J10 . 60J15 . 60J45 . 60J55 . 60K15

Contents

1 The type problem for Markov additive processes3 1.1 Markov additive processes...... 4 1.2 Basic assumptions...... 5 1.3 The type problem : a series criterion...... 8 1.4 Few comments on the main theorem...... 9

2 Applications : Markov additive processes with strong local drift9

3 Proof of the main theorem 14 3.1 Estimates far from the origin...... 14 3.2 Estimates in the neighborhood of the origin...... 14

Introduction

In the classical framework, a random walk on a group G is a discrete time stochastic process (Zn)n≥0 defined as the product of independent and identically distributed (i.i.d.) random

1 variables (ξn)n≥ . More precisely, for any g G, we set Z = g and 1 ∈ 0 Zn = gξ ξn. 1 ··· Random walks on groups are obviously Markov chains that are adapted to the group structure in the sense that the underlying Markov operator is invariant under the group action of G on itself. Thus, this homogeneity naturally gives rise to deep connections between stochastic properties of the random walk and algebraic properties of the group. Starting with the seminal paper of P`olya, [37], a large part of the literature is devoted to the study of these connections in this homogeneous case (for instance [29, 31, 18, 21, 43, 40, 23, 22,2] and references therein). In this paper, we aim at investigating “weakly” inhomogeneous random walks. It turns out that there are at least two ways to introduce inhomogeneity. First, we can consider “spatial” inhomogeneity by weakening the group structure, replacing it, for instance, by a directed graph or semigroupoid structure (see [4, 16, 17, 35, 14]). Secondly, we can study “temporal” inhomo- geneous random walks by introducing a notion of memory as in the model of reinforced [34, 42], excited [38,3], self-interacting [11, 36] or also persistent random walks [8,7,9]. All these models belong to the larger class of stochastic processes with long range dependency. The strategy in the sequel consists of making use of the connection between such stochastic processes and Random Walk with Random Transition Probabilities — or for short, RWRTP — the terminology comes from [20]. Roughly speaking, a RWRTP consists of a dynamical system (Ω,T ), endowed with a quasi-invariant (preimages by T of null measure sets have null ω measure) probability measure λ, together with a family µ ω∈Ω on a group G. The dynamics ω { } 1 dictate the way the measures of µ ω∈Ω can be convoluted . More precisely, choosing at random (with respect to the probability{ } measure λ) a trajectory ω Ω, we are interested in the asymptotic properties of the sequence of convoluted measures µω ∈µT ω µT 2ω µT nω. As a matter of facts, along a trajectory of the dynamical system, the resulting∗ ∗ stochastic∗ · · · ∗ process has independent but, in general, not stationary increments. If in addition, the dynamical system admits a stationary probability measure, in mean, the increments are no longer independent, however, they are in some sense stationary. As it is noticed in [20], this model is actually a generalization of other well known models such as Random Walk With Internal Degree of Freedom — see [27] — or also in the modern formulation Markov additive processes (or semi-Markov processes, Markov random walks), for short MAP — see [26, 19, 32, 30,1, 41, 24]. In this context, many results have been proved. Though, the basic assumptions are generally too much restrictive to encompass the class of very inhomogeneous random walks. Generally speaking, in the context of Markov additive processes, we may introduce the d Fourier transform operator, denoted by t, t R , which is a continuous perturbation of the Markov operator P˚ defining the underlyingF dynamics.∈ The rate of the return probability is then estimated, under mild conditions, by inverse Fourier transform. In the literature, the Markov operator P˚ is usually supposed to admit an ergodic invariant probability measure and to be quasi-compact. In addition, by a perturbation argument the Fourier transform operator remains quasi-compact for all t in a neighborhood of the origin. It allows, under suitable moment conditions on the system of probability measures, to derive a Taylor expansion at the second order of the perturbated dominating eigenvalue γ(t) (whose the coefficients are given roughly by the mean and the variance operators). Finally, under an assumption on the spectrum of the Fourier transform operator outside a neighborhood of the origin, it can be concluded that all the needed stochastic information is actually contained in the nature of the singularity at zero of (1 γ(t))−1 (note γ(0) = 1). This has to be compared to the classical context of random − 1In this sense, it is comparable to Random Dynamical Systems: instead of randomly composing transforma- tions, we randomly compose random (continuous) group homomorphisms.

2 walks for which there is an integral test criterion involving a singularity of this kind — see for instance [10] and [39]. In this paper, the quasi-compacity condition of P˚ is dropped. It is only assumed that P˚ is irreducible, recurrent and aperiodic. The condition on the spectrum of t for large perturbations remains similar but the nature of the singularity at the origin is analyzedF via probabilistic −1 methods (there is also a kind of Chung and Fuchs integral test involving (I t) but, without a strong knowledge on the spectrum of P˚, it seems impracticable, see−F Section 1.4). These estimates give rise to a criterion for the type problem in terms of the summability of a series. More precisely, introduce the following quantities for n 1 and ω Ω ≥ ∈ n−1 1 X h ∗ T kω T kω ∗ T kω i Σn := Σn(ω) = µ[(0) ( µ[(0)) ( µ[(0)) , −n ∇ ∇ − ∇ ∇ k=0 n−1 1 X T kω ∆n := ∆n(ω) = i µ[(0), (1) − √n ∇ k=0 and the (almost surely finite) random variable τ which is the first instant such that the rank Σn is maximal. The main theorem of this paper states (under the assumptions1,2,3 and4 given in Section1) that the Markov additive process is transient or recurrent (see Definition 1.6) accordingly as Z   X 1 1 ∗ −1 1n≥τ exp ∆ Σ ∆n dλ nd/2 −2 n n n≥1 Ω is finite or infinite provided τ has a finite expectation. Thus, the main term of the series involves the expectation of a functional of two random variables. The recurrence or transience behavior is interpreted as the result of the competition of this two random quantities. More precisely, the process is recurrent if locally it is sufficiently diffusive in order to balance the instantaneous trend. It turns out that ∆∗Σ−1∆ is in some circumstances a self-normalized martingale (see [13] for instance). Inspired by models introduced in [4], the main theorem is applied to a new class of examples in Section2. In facts, in these examples, depending on the moves of a simple random walk on Z and an (deterministic or random) environment εx x∈ 1, 1 Z, the random walk admits { } Z ∈ {− } a strong drift taking values in one of the quarter plane of Z2. In [4] are studied in particular the simple random walks on graphs represented on the figure 1 below. For the graph (a) of Figure1, denoted by L, a random walker choose at random one of the nearest neighbour among North, South or West if the ordinate of its current position is odd, and the nearest neighbour among North, South and East if it is even. It is shown in [4] that this random walk is recurrent. For the graph (b) of Figure1, denoted by H, the possible movements for a random walker are toward the North, South and West on the upper half-plane and North, South and East otherwise. The resulting random walk is shown to be transient. Beyong their own interests in the classical probability theory, it is worth noting the study of these models was initially motivated by natural questions arising in quantum theory as described in [5]. For modified random walks studied in Section2, very similar results are proved. In partic- ular, it is shown that recurrence is rather exceptional whereas transience is typical.

1 The type problem for Markov additive processes

In this section, we shall consider the restricted class of Markov additive processes, for short MAP, that are stochastic processes whose jumps, taking values in an Abelian group G, are

3 1 1 1 1 3 3 3 3

1 (0, j) 1 3 3

(0, 0) (0, 0)

1 3

1 1 1 3 3 3 (0, j) − 1 1 3 3

(a) Periodically directed graph. (b) Graph with two directed half- plane.

Figure 1: Directed graphs L et H de [4]. independent but with a distribution depending on a Markovian dynamics. We shall be even d d more restrictive by setting G = Z . For a non discrete Abelian group, say G = R or G compact, the Fourier analysis of continuous measure is sensibly different but remains irrelevant for the scope of this paper.

1.1 Markov additive processes In the sequel, we refer to [33] for the notion of Markov chain on general spaces. We shall consider a Markov kernel P˚ relatively to a measurable space (X, ) for which the σ-algebra is separable. X The space (X, ) can be endowed with a σ-finite measure m dominating mP˚. Without loss of generality, we mayX assume that the Markov chain induced by P˚ is m-irreducible and aperiodic. A stronger assumption is to suppose the Markov chain m-recurrent. In this case, the time m shift on XN preserves the possibly infinite, but σ-finite, Markov measure P . The probability ω measures µ , ω XN are supposed to depend only on the two first coordinates2 of ω XN and ∈ x,y ∈ shall be alternatively denoted by µ , x, y X. A Markov additive process with internal∈ Markov chain P˚ together with the system of prob- d ability measures µ is the Markov chain on the space X Z whose Markov operator is given, ∞ d × for f Lm (X Z ), by ∈ × Z ˚ x,y d P f(x, u) = P (x, dy)(δy µ )(dzdv)f(z, u + v), for (x, u) X Z . X×Zd ⊗ ∈ ×

d The Markov operator P is invariant by translations of the kind X Z (x, u) (x, u + v) d × 3 → ∈ X Z . × In this context, we may introduce the so called Fourier transform operator, denoted by t, d ∞ ∞ F for t R , acting as a contraction on Lm (X) and defined for f Lm (X) by ∈ ∈ Z ˚ x,y tf(x) = P (x, dy)µd (t)f(y), for m a.e. x X, (2) F X − ∈ 2Actually, we use here the traditional definition, though it is completely equivalent to suppose that the probabilities µω depends only on the first coordinate providing we consider the 2-order Markov chain induced by P˚.

4 x,y d where µdx,y is the standard Fourier transform of the probability measure µ defined, for t R , by ∈ X x,y iht,vi µdx,y(t) = µ (v)e , v∈Zd , standing for the standard inner product. h· ·i 1.2 Basic assumptions d Let σ be a bounded map from X to Z and define Z ˚ x,y d P σf(x, u) = P (x, dy)(δy µ )(dzdv)f(z, u + v σ(y) + σ(x)), for (x, u) X Z . X×Zd ⊗ − ∈ × Intuitively, it corresponds to a change of origin of each fiber parametrized by X. Such a function x,y x,y σ is called a change of section. The translated probability shall be denoted µσ = µ x,y ∗ δσ(y)−σ(x), or simply µ if no ambiguity. Because of the invariance of P under translations d of Z , the changes of section have no fundamental importance in the study of the recurrent or transient behavior of the Markov additive process. To keep notations light and readable, we shall adopt alternatively the ones related to dy- namical system (such as µω, Pm-a.e.,. . . ) or those ones proper to Markov additive processes (µx,y, m-a.e.,. . . ).

Definition 1.1 (Adaptation, aperiodicity, irreducibility). A MAP is respectively adapted, ape- riodic and irreducible if for any change of section σ

d m ω 1. there is no proper subgroup H Z such that P -a.e. µ (H) = 1 ; ⊂ d d m ω 2. there is no proper subgroup H Z and no a R such that P -a.e. µ (a + H) = 1. ⊂ ∈ d m ω 3. there is no half-space H R such that P -a.e. µ (H) = 1. ⊂ The properties of adaptation and aperiodicity can be read on the spectrum of the Fourier d ∗ transform t as shown in Corollary 1.3. Below Wd denotes the set [ π, π) and W the set F − d Wd 0 . \{ } ω m Proposition 1.2. A MAP is adapted if and only if µcσ (t) = 1, P -a.e. for any change of d ∗ 6 ω iθ m section σ : X Z and any t Wd.A MAP is aperiodic if and only if µcσ (t) = e , P -a.e., → ∈ ∗ 6 for any θ R, any change of section σ and any t W . ∈ ∈ d Proof. If a MAP is not aperiodic then there exist a change of section, a proper subgroup H and d ω m a R such that µσ (a + H) = 1, P -a.e.. It is well known that there exist integers n1, . . . , nr, ∈ 1 r d, such that the subgroup H is generated by a set n1ei1 , . . . , nreir for some indices ≤ ≤ th { d } 1 i1 ... ir d, where ei is the i vector of the canonical basis of Z . Suppose r < d, then ≤ ≤ ≤ ∗ ≤ ⊥ ω iht,ai for any t Wd (span H) it follows that µcσ (t) = e . If r = d, set t = (π/n1, . . . , π/nd) ∈ ω ∩ iht,ai ∗ d then again µcσ (t) = e and t obviously belongs to Wd unless n1 = = nd = 1, i.e. H = Z . ···∗ ω iθ Conversely, suppose there exists a change of section σ and t Wd such that µcσ (t) = e , m ω ∈ ω P -a.e., for some θ R. It means that µσ (t) is an extremal convex combination and since µσ ∈ d d is supported by a subset of Z , one can choose a Z such that θ = t, a . Then, for any n 1 ∈ h i ≥ n−1 Y k X n−1 eiht,nai = µ[T ω(t) = µω µT ω(w)eiht,wi, Pm a.e., σ ∗ · · · ∗ − k=0 w∈Zd

5 or equivalently,

X ω T n−1ω iht,wi m 1 = µ δ−a µ δ−a (w)e , P a.e.. { σ ∗ } ∗ · · · ∗ { σ ∗ } − w∈Zd Thus, the convex combination of points on the unit circle on the right hand side is extremal ω ω T n−1ω so that each w n = supp µσ δ−a µσ δ−a , n 1, satisfies t, w = 0 ∈ S { N ∗ ω } ∗ · ·i ·ht,a ∗i { ∗ } ≥ h i modulo 2π. Setting N = ω X : µcσ (t) = e and defining H as the smallest subgroups ω { ∈ 6 { } m ω containing the sets n , for all n 1 and ω N , it follows that, P -a.e., (µσ δ−a)(H) = 1, ω S ≥ ∈ ∗ i.e. µσ (a + H) = 1. If the MAP was aperiodic, the group H should not be a strict subgroup d ∗ of Z so that t, ei = 0 modulo 2π for 1 i d. Contradiction with the fact t Wd. The statementh i involving the adaptation≤ property≤ follows exactly the same lines∈ by setting a = 0 and θ = 0.

∗ Corollary 1.3. A MAP is adapted if and only if, for any t Wd there exists a closed t- invariant subspace E containing the constants such that one is not∈ an eigenvalue of the operatorF ∗ t acting on E . It is aperiodic if and only if, for all t W there exists a closed t-invariant F ∈ d F subspace E containing the constants such that the operator t, acting on E, has no eigenvalue of modulus one. F

Remark 1.1. The proof of this corollary requires that the bounded superharmonic functions are constant. A natural assumption to ensure this property is the m-recurrence of the Markov operator P˚ (see for instance [33, Proposition 3.13, p. 44]). In the terminology of [28], one might suppose P˚ conservative and ergodic.

∗ Proof. Let t Wd and suppose there exists f E, where E is any closed t-invariant subspace containing the∈ constants, such that ∈ F Z ˚ x,y iθ tf(x) = P (x, dy)µdσ (t)f(y) = e f(x), for some θ R. F X ∈

x,y By Jensen inequality and the fact that µdσ (t) 1, it follows that f P˚ f . Thus, the | | ≤ | | ≤ | | function f ∞ f is superharmonic and hence constant m-a.e. since P˚ is supposed m- recurrent.k Ask a− consequence, | | Z x,y 1 = µdσ (t) P˚(x, dy), m a.e., X | | −

m and µdx,y(t) is of modulus one, P -a.e.. By Proposition 1.2, the MAP can not be aperiodic. Conversely, if the MAP is periodic, the same proposition implies eiht,ai is an eigenvalue. The proof of the statement involving the adaptation property follows exactly the same lines.

ω d m Remark 1.2. Since the probability measures µ are supposed to be supported by Z , P -a.e., it follows that the operator valued map t t is 2π-periodic along the directions given by the d −→ F vectors of the standard basis of R . In fact, the aperiodicity means that it can not be periodic with shorter periods.

Definition 1.4. A MAP is said to satisfy condition (S) if for any t W∗ there exists a ∈ d closed t-invariant subspace E, containing the constants, for which the set of spectral values of F modulus 1 of t, operating on E, consists of eigenvalues. F

6 Remark 1.3. It is worth noting that the invariant space E appearing in the definition of condition (S) may vary with respect to t W∗. Though, it is not clear if this flexibility ∈ d is actually necessary. In a word, may we find an example of pertubations t for which the F ∗ condition (S) is satisfied but there is no common invariant space E, shared for every t Wd, such that the spectral values of modulus one are eigenvalues ? ∈ Remark 1.4. If the Markov operator P˚ is quasi-compact (as in most of the litterature) then the Fourier transform operator, as a continuous perturbation, is also quasi-compact for every t in a neighborhood of the origin. It is known that the set of spectral values of modulus one consists of isolated eigenvalues. Somehow, the condition (S) extends this property to large perturbations. Nonetheless, condition (S) can be satisfied without P˚ being quasi-compact which is the main motivation of this paper. In examples of Section2, the Markov operator P˚ is that one of a symmetric nearest neighbor random walk which is not quasi-compact (otherwise, it is well known, for instance see [28], that P˚ would admit an invariant probability measure).

For any closed subspace E, and any bounded operator Q we denote by Q E the subordi- k k nated norm restricted to E defined as Q E = sup Qf . Let t Wd, we denoted k k f∈E:kfk=1 k k ∈ by Et the intersection of all closed subspaces E such that tE E and 1 E. Obviously, F ⊂ ∈ the subspace Et is itself closed, invariant and contains the constant. Also, we may define the pseudo spectral radiusr ˜(t) of t by F r˜(t) = lim n 1/n. n→∞ kFt kEt Lemma 1.5. If a MAP is aperiodic and satisfy condition (S), then the pseudo spectral radius ∗ r˜(t) of t is strictly smaller than one for t W . F ∈ d Proof. As a matter of fact, the pseudo spectral radius can be defined alternatively as follows   n 1/n r˜(t) = inf , n 1,E closed subspace satisfying tE E and 1 E . kFt kE ≥ F ⊂ ∈

Moreover, let t0 Wd and E be a closed subspace such that t0 E E and 1 E. Then, by reverse triangle inequality,∈ F ⊂ ∈

n n n n n n E E E kFt0 k − kFt k ≤ kFt − Ft0 k ≤ kFt − Ft0 k which can be made arbitrarily small since t t is continuous. It follows thatr ˜ is the point- wise infimum of continuous functions, thusr ˜ is→ upper F semi-continuous. Furthermore, the pseudo spectral radius reaches its maximum on compact K Wd. Aperiodicity together with condition ⊂ (S) imply that maxt∈K r˜(t) < 1 excepted when 0 K. ∈ From now on, we shall assume the following. Assumptions 1. The MAP is adapted, aperiodic and irreducible. Assumptions 2. The condition (S) is fulfilled. Additionally, we make the following assumption on the system of probability measure µ where stands for the Euclidean norm. | · | Assumptions 3. Assume that the system of probability measures µ admits a uniform third order moment, that is

X u 3µ(u) < . | | ∞ ∞ u∈Zd

7 d Remark 1.5. Recall that a change of section is a bounded function σ : X Z , thus Assump- tion3 needs not to be stated relatively to any change of section. −→

d For n 1 and u, v Z , denote respectively the n-step transition probability and its Fourier transform≥ by ∈

n−1 n−1 Y k P ω (u, v) := µω µT ω(v u) and φω (t) = µ[T ω(t). (3) 0,n ∗ · · · ∗ − 0,n k=0 Introduce the quantities

n−1 1 X h ∗ T kω T kω ∗ T kω i Σn(ω) := µ[(0) ( µ[(0)) ( µ[(0)) −n ∇ ∇ − ∇ ∇ k=0 n−1 1 X T kω and ∆n(ω) := i µ[(0). (4) − √n ∇ k=0

d ∗ 2 Assumptions 4. There exists α > 0 such that for all t R , lim infn→∞ t Σnt α t , Pm-a.e.. ∈ ≥ | |

1.3 The type problem : a series criterion d Let A X and K Z be measurable subsets. We are interested in the mean time spent by ⊂ ⊂ d the Markov additive process — that is nothing but a Markov chain on X Z — in the product × d set A K. In fact, it is well known that this quantity, starting from (x, u) X Z , is actually given× by the Green operator defined by ∈ ×

X n G1A×K (x, u) = P 1A×K (x, u). n≥0

In the special case of A = X, using notation of Equation (3), the equation above rewrites Z X X x G1X×K (x, u) = 1X×K (x, u) + P0,n(u, v)1K (v)dP . d XN v∈Z n≥1 Definition 1.6 (Recurrence and Transience). A Markov additive process is said to be recurrent

( resp. transient) if, for any bounded change of section σ, G1X×{0}(x, 0) = , m-a.e. ( resp. G1 (x, 0) < , m-a.e.). ∞ X×{0} ∞ Remark 1.6. Since the changes of section are supposed bounded, a MAP is simultaneously recurrent or transient for every change of section.

A simple computation gives rise to the identities Z Z X x X n−1 x G1 ×{0}(x, 0) = 1 + P0,n(0, 0)dP = 1 + lim r P0,n(0, 0)dP X r↑1 n≥1 XN n≥1 XN Z (5) 1 X n−1 n = 1 + lim Re r t 1(x)dt r↑1 (2π)d F Wd n≥1 where the latter equality is obtained by inverse Fourier transform and Fubini’s theorem.

8 Theorem 1.7 (Series criterion). Let d 2 and suppose that the assumptions1,2,3 and4 are ≥ satisfied. In addition, let τ be the stopping time defined by τ = inf n 1 : rk Σn = d and x { ≥ } assume E (τ) < for m-a.e. x X. Then, G1X×{0}(x, 0) is finite ( resp. infinite) if and only if ∞ ∈ Z   X 1 1 ∗ −1 x 1n≥τ exp ∆ Σ ∆n dP < ( resp. = ), nd/2 −2 n n ∞ ∞ n≥1 XN and the MAP is transient or recurrent accordingly.

The proof of this theorem is postponed to Section3. This section is ended with few com- ments.

1.4 Few comments on the main theorem In Theorem 1.7, there is a rather technical assumption on the integrability of the stopping time τ, the first time at which the matrix Σn is invertible so that the quantities I1,2(n) and I1,3(n) in Section3 are well-defined. Due to Assumption1 and Proposition 3.3, this stopping time τ is x P -a.s. finite for m-a.e. x X. The integrability of τ is required to deduce estimates in mean from almost-sure estimates.∈ For instance, for the simple random walk on a comb (a subgraph of Z2 with two parallel infinite half-lines joined by an edge at the origine), the related MAP should not enjoy this integrability condition. The formulation of the criterion may seem non conventional and, inspired by [10], one may think about a criterion stating a MAP is recurrent or transient depending on Z 1 −1 lim Re (I r t) 1(x) dt r↑1 d/2 (2π) Wd − F is finite or infinite (and even without the r-limit adapting the proof of [39] to our context at the cost of a long and rather technical proof involving potential theory arguments). The main concern is related to the analysis of the invert of (I t) without strong conditions giving informations about the spectrum in the neighborhood− of F the origin. Finally, in the equality (5), an r-limit is introduced allowing several exchanges of limits, integrals and summations in the computations. At some point in the proof of Theorem 1.7, the r-limit can be dropped using monotone convergence so that the final criterion obtained here is stated without this r-limit. In some circumstances, one may be interested in dropping this limit inside the computations in order to obtain other estimates that might be relevant for other problematics beyond the type problem.

2 Applications : Markov additive processes with strong local drift

In this section, the main result of this paper is applied to some examples inspired by [4]. More precisely, we consider the simple random walk Sn n≥ with i.i.d. increments whose common { } 0 distribution is given by the probability (δ−1 + δ1)/2. As this random walk is null recurrent, the corresponding Markov operator can not be quasi-compact (see [28]). Let ε = εx x∈Z be a random or deterministic sequence taking value in 1, 1 . { } {− } There is obviously many ways to define a MAP on Z2. Here, we consider simple examples x,y x,y ε ,sgn y−x for which the probability measures µ are of the form µ = µ x ( ), x, y Z for some p,q ∈ fixed measures µ p,q∈{−1,1}. In order to exhibit transient behavior, we follow the ideas in { } p,q [4] by supposing that µ is supported in the quarter plane pN qN. Thus, on one hand, ×

9 the vertical component of the Markov additive process will merely follow the moves of the underlying random walk in the sense they go north or south simultaneously. On the other hand, the horizontal component moves in the direction dictated by ε. From now on, it is supposed, for all p, q 1, 1 , that ∈ {− } 1. the mesures µp,q admit a third moment,

p,q p,q 2. µ are aperiodic in the sense µd(t) = 1 on Wd if and only if t = 0, | | 3. The map x εx 1, 1 is surjective. → ∈ {− } These three assumptions implies the related MAP satisfied assumptions1,3 and4. Besides, remark the random time τ of the theorem is deterministic equal to one. Finally, as it follows from the proposition below, the condition (S) is satisfied.

Proposition 2.1. The condition (S) is fullfilled.

Proof. The quantity   p,q 2 s(t) = max µd(t) , (p, q) 1, 1 , t Wd | | ∈ {− } ∈

p,q is continuous and s(t) = 1 if and only if t = 0 in Wd since µ , p, q 1, 1 , are aperiodic. ∞ ∈ {− } Besides, the Fourier transform operator acts as a contraction on ` (Z), moreover, for all x Z and f `∞ : ∈ ∈ 1 1 ε ,1 ε ,−1 tf(x) = µ[x (t)f(x + 1) + µ\x (t)f(x 1). F 2 2 − ∗ As a matter of fact, t ∞ s(t) so that r( t) s(t). Thus for t Wd, the operator t has no spectral values ofkF modulusk ≤ 1. In particular,F it≤ satisfies condition∈ (S). F

p,q Let Tp,q be a random variable with distribution µ . For the sake of simplicity, we impose d some symmetries to the problem : assume for all p, q 1, 1 that Tp,q = (pTh, qTv) for some ∈ {− } 2 (non necessarily uncorrelated) random variables Th and Tv taking values in N . Then, the expectation and the variance matrix of Tp,q are respectively given by     pE(Th) V(Th) pqCov(Th,Tv) E(Tp,q) = and V(Tp,q) = , qE(Tv) pqCov(Th,Tv) V(Tv) where Cov(Th,Tv) denotes the covariance between Th and Tv. If we assume in addition that Th and Tv are uncorrelated, then the off diagonal coefficients of the variance matrix are null and the resulting Markov additive process is then recurrent or transient accordingly as

 2 Z 2 n−1 !  2  X 1 E(Th) X E(Tv) 2 x exp  εSk  exp (Sn S0) dP (6) n N −2nV(Th) −2nV(Tv) − n≥1 Z k=0 is finite or infinite for some (any) x Z. ∈ x Proposition 2.2. Set εx = ( 1) for all x Z. Then, the resulting Markov additive process is recurrent. − ∈

10 Proof. First, remark that

n−1 !2 X 1 + ( 1)n−1 εS = − , n 1. k 2 ≥ k=0 Thus,  2 2 n−1 ! E(Th) X lim exp  εSk  = 1. n→∞ −2nV(Th) k=0

2 2 Secondly, note that (Sn S ) /n converges in distribution to N where N is a standard − 0 gaussian random variable. Since x exp( x) is continuous and bounded on R+, it follows that → − Z  2  Z  2 2  2 E(Tv) 2 x E(Tv) y y dy lim exp (Sn S0) dP exp exp , n→∞ ZN −2nV(Tv) − → R − 2V(Tv) − 2 2π and the series in (6) is infinite.

Proposition 2.3. If, for all x Z, εx = sgn x, then the resulting Markov additive process is transient. ∈

In the sequel, it is useful to make the follwing transformation

n−1 X X εSk = Nn(x)εx (7) k=0 x∈Z where Nn(x), x Z and n 1 is the local time of Sn n≥0 up to time n 1 that is Nn(x) = Pn−1 ∈ ≥ { } − k=0 1{x}(Sk). Proof. As a matter of facts, an obvious majoration in (6) implies it suffices to prove that

 2 Z 2 n−1 ! X 1 E(Th) X x exp  εSk  dP (8) n 2 −2nV(Th) n≥1 Z k=0 is finite for all x Z. By Markov property and Z2-invariance on the Markov additive process, one can suppose∈ without loss of generality x = 0 and forget the exponent in Px. Using the transformation (7) and remarking that ε = sgn , it follows that

n−1 X εx = 2Nn(Z+) n. − k=0 Now, estimate the summand in (8) " ( )#   E(T )2  E(T )2  N ( ) 2 h 2 h n Z+ E exp (2Nn(Z+) n) E exp n 2 1 −2nV(Th) − ≤ − 2V(Th) n −

  E(T )2  h 2 E exp n (2Γ 1) − − 2V(Th) −   E(T )2  + E exp n h (2Γ 1)2 , − 2V(Th) −

11 where Γ is distributed as an arcsine law supported by [0, 1]. It turns out that for all n 1, the function ≥  nE(T )2  [0, ) x exp h (2x 1)2 (9) ∞ 3 → − 2V (Th) −

q 2 nE(Th) is kn-Lipschitz with constant kn = . Consequently, it follows a majoration of the first eV (Th) term on the right handside of (9) by kndW ( (Nn(Z+)/n), (Γ)) where dW denotes the Wasser- L L stein metric on probability measure. Namely, for any measure µ and ν on R, the Wasserstein metric is defined as Z Z

dW (µ, ν) = sup h dµ h dν , h∈H − where stands for the set of 1-Lipschitz continuous functions. From [15], we deduce that this term isH a O(n−1/2) for all n 0 large enough. Besides, the second term≥ on the right handside vanishes at a sub-exponential rate which ends the proofs of the proposition.

We finish this series of examples by considering now that ε = y y∈ is an i.i.d. sequence { } Z of Z-valued centered random variables uniformly bounded. In addition, suppose its common distribution is not supported by any proper subgroup of Z. In the sequel, we denote by Q the distribution of y y∈Z. { } x,y The notations used in the previous examples extends easily. In fact, µ , x, y Z, is the ∈ distribution of the vector (εxTh, sgn (y x)Tv) where Th and Tv are N-valued random variable − independent with ε. If we assume in addition that Th and Tv are uncorrelated, one may check that

n−1  2  n−1   1 X εS V(Th) 0 i X εSk E(Th) Σn = k and ∆n = . −n 0 V(Tv) −√n sgn (Sk+1 Sk)E(Tv) k=0 k=0 − Thus, the problem reduces to the study of

 2  2 Pn−1  Z  E(Th) k εSk   2  X 1  =0  E(Tv) 2 x exp exp (Sn S0) dP . (10)  n−1  n ZN − 2V(T ) P ε2  −2nV(Tv) − n≥1  h k=0 Sk 

Proposition 2.4. Suppose that there exists M > 0 such that for all x Z, εx MQ-a.s.. ∈ | | ≤ For Q-a.e. sequences y y∈ , the random walk M is transient. { } Z Proof. Using Markov inequality and an obvious upper bound of the second factor in (10), it suffices to prove that the following series is finite for any x Z ∈   2   E(T )2 Pn−1 ε  X 1 Z  h k=0 Sk  exp dQ Px.  n−1  n ZZ×ZN − 2V(T ) P ε2  ⊗ n≥1  h k=0 Sk 

The proof follows four steps. Step 1: As a matter of facts, by assumption, one get

n−1 X ε2 nM 2 n1+δM 2, Sk ≤ ≤ k=0

12 for any δ > 0. Thus, it follows that  2   2  2 Pn−1  2 Pn−1   E(Th) k=0 εSk   E(Th) k=0 εSk  exp exp (11)   2 1+δ − 2V(T ) Pn−1 ε2  ≤ − 2V(Th)M n   h k=0 Sk   

Then, using (7), and setting for n 1 and x R ≥ ∈ ( ) E(T )2n1/2−δ f (x) = exp h x2 , n 2 − 2V(Th)M the right handside of (11) rewrites ! −3/4 X fn n Nn(x)εx . x∈Z −1 1/4−δ/2 Step 2: It turns out that for each n 1, the function is kn-Lipschitz with kn = kM n where k is a deterministic constant independent≥ of n. Now, from [25], it follows that

−3/4 X D n Nn(x)εx = ∆ , ⇒ 1 x∈Z where, for t 0, ≥ Z ∞ ∆t = Lt(x)dZ(x). −∞ Here, Lt denotes the local time, up to time t, of a one dimensional Brownian motion independent of the bilateral one dimensional Brownian motion Z. Decompose the right handside of (11) as follows    2  2 Pn−1 " !#  E(Th) k εSk   =0  −3/4 X E x exp  E x f n N (x)ε E[f (∆ )] Q⊗P 2 1+δ Q⊗P n n x n 1  − 2V(Th)M n  ≤ −   x∈Z

+ E[fn(∆1)]. (12) Step 3: Now, one may estimate the first term in the upper bound in (12). First remark that the stochastic process ∆t t≥0 is self-similar of index 3/4 (see [25]). That implies that for each n 1 { } ≥ −3/4 E[fn(∆1)] = E[fn(n ∆n)]. Using the strong embedding given in [12], it turns there exists a coupling such that for any η > 0

X 5/8+η sup Nk(x)εx ∆k = o(n ). 0≤k≤n − x∈Z In particular, using this coupling and renormalizing by n3/4, we get the first term of the right −1/8+η handside (12) is equal to o(knn ). Now, choose δ = 3/8, η = 1/32, it gives the first term is a o(n−1/32). Step 4: Since ∆1 is positive almost-surely (see [6] for an expression of the density), applying the theorem of dominated convergence, the second term vanishes at a higher rate than any polynomial.

13 3 Proof of the main theorem

This section is devoted to the proof of Theorem 1.7. We shall be interested in the asymptotics as n goes to infinity of Z 1 Z P (u, v)dPx = n1(x)e−iht,v−uidt 0,n (2π)d t XN Wd F and more specifically when u = v. Similarly to the context of classical random walks, let δ > 0 and split Z Z Z n −iht,v−ui n −iht,v−ui n −iht,v−ui t 1(x)e dt = t 1(x)e dt + t 1(x)e dt . (13) d d Wd F (−δ,δ) F Wd\(−δ,δ) F | {z } | {z } I1(n) I2(n)

3.1 Estimates far from the origin Proposition 3.1. Under Assumptions1 and2, there exist constants κ (0, 1) and C > 0 such d ∈ that for all v, u Z and all δ > 0 sufficiently small ∈

Z n −iht,v−ui n t 1( )e dt Cκ . W \ −δ,δ d F · ≤ d ( ) ∞ ∗ d Proof. By Lemma 1.5, under Assumptions1 and2, setting K := Wd ( δ, δ) , it follows that \ − −n n M := max r˜(t): t K < 1. Set κ := (M + 1)/2. It is a matter of fact that κ t Et { ∈ } −n n kF k vanishes as n goes to infinity so it is for κ 1 ∞ and the result follows. kFt k Remark 3.1. In the literature, it is usually considered that the whole family t t∈K shares the same invariant space E. Allowing different spaces for different points t K{Fimproves} the estimate of Proposition 3.1 while the pseudo spectral radius r˜ still statisfies the∈ nice property of upper semi-continuity for continuous perturbations of operators.

3.2 Estimates in the neighborhood of the origin

√u At this level, it only remains to estimate the first integral term of Equation (13). Setting t = n , it is given by Z √ 1 n −ihu/ n,v−ui I (n) = u 1(x)e du. 1 d/2 √ √ √ n (−δ n,δ n)d F n The expected estimate shall be obtained by integrating an almost-sure estimate exploiting the following expression Z Z √ −d/2 −ihu/ n,v−ui x I1(n) = n √ φ0,n(u/√n)e du dP . (14) XN |u|≤δ n

| ω{z } I1 (n) Proposition 3.2. Under Assumption 3, there exists a deterministic δ > 0 such that for t δ, | | ≤ the quantity log φ0,1(t) is well defined. In addition, the following approximation formula holds Pm-a.e.

1 ∗ ∗ ∗ log φ , (t) = φ , (0)t + t [ φ , (0) φ , (0) φ , (0)] t + R(t), (15) 0 1 ∇ 0 1 2 ∇ ∇ 0 1 − ∇ 0 1 ∇ 0 1 where the remaining term R satisfies for t δ and for any  [0, 1) : | | ≤ ∈ 1− 2+ R(t) ∞ δ K t with K 0. k k ≤ | | ≥ 14 m Proof. Under Assumption3, the function φ0,1 is P -a.e. three times continuously differentiable. Therefore, the following majoration holds Pm-a.e.

φ , (t) 1 t φ , (0) ∞. | 0 1 − | ≤ | |k∇ 0 1 k ω Thus, there exists a deterministic δ > 0 such that for all t δ the function t log φ0,1(t) is well defined. In addition, for t δ, the Taylor formula yields| | ≤ Pm-a.e. → | | ≤ 1 ∗ ∗ ∗ log φ , (t) = φ , (0)t + t [ φ , (0) φ , (0) φ , (0)] t + R(t), 0 1 ∇ 0 1 2 ∇ ∇ 0 1 − ∇ 0 1 ∇ 0 1 where the remaining term R satisfies for t δ and any  [0, 1) | | ≤ ∈

1− 2+ X 3 R(t) ∞ δ K t , with K = u µ(u) . k k ≤ k k k k ∞ u∈Zd

This proposition implies that for δ > 0 sufficiently small and u δ√n | | ≤   1 ∗ φ ,n(t) = exp i ∆n, t t Σnt + Rn(t) , 0 h i − 2 with the notation n−1 ω X T kω Rn(t) := Rn(t) = R (t/√n). k=0 Proposition 3.3. Under Assumption3 the following properties hold

m 1. for all n 1, the matrix Σn is real positive symmetric P -a.e., ≥ 2. the sequence (Σn)n≥0 remains bounded in the following sense

sup Σn ∞ < , n≥0 k k k k ∞

m 3. P -a.e., the rank rk Σn is non decreasing with n 1, ≥ m 4. in addition, under Assumption1, limn→∞ rk Σn = d, P -a.e..

Proof. 1. Under Assumption3, nΣn is a sum of covariance matrices so is positive semidefinite real symmetric.

2. As a matter of facts, the sequence of matrices Σn satisfies

ess sup Σn ess sup Σ1 < . ω∈Ω k k ≤ ω∈Ω k k ∞

3. As a sum of positive semidefinite real symmetric matrices, the kernel of Σn is given by

n−1 \ h ∗ T kω T kω ∗ T kω i ker Σn = ker φ (0) ( φ (0)) ( φ (0)) , ∇ ∇ 0,1 − ∇ 0,1 ∇ 0,1 k=0 and the result follows.

15 m 4. Since rk Σn is a non decreasing discrete bounded sequence P -a.e., it suffices to show that   x x [ P (lim inf rk Σn = d ) = P  rk Σn = d  = 1, for m a.e. x X. (16) { } { } − ∈ n≥1

Moreover, the operator P is supposed aperiodic and recurrent so that we only need the asymptotic event in (16) holds with positive probability. Thus suppose on the contrary

x \ P (N) = 1 with N = rk Σn d 1 . { ≤ − } n≥1

d ω T n−1ω Then the subgroup H of Z generated by the supports supp µ µ , n 1, x ω ∗ · ·m · ∗ ≥ ω N, is P -a.s. independent of ω XN and satisfies µ (H) = 1, P -a.e.. Assumption ∈ d ∈ 1 yields H = Z which contradicts the non maximality of the asymptotic rank.

m Recall τ = inf n 1 : rk Σn = d and remark it is finite P -a.e. by Proposition 3.3. ω { ≥ } Rewrite I1 (n) as the sum of three terms I11,I12 and I13 defined by

Z 1 ∗ √  √  ω −d/2 − 2 t Σnt iht,∆ni−iht,(v−u)/ ni Rn(t/ n) I11(n) = n √ e e e 1 dt, |t|≤δ n −

Z 1 ∗ √ ω −d/2 − t Σnt iht,∆ni−iht,(v−u)/ ni for n τ, I12(n) = n e 2 e dt, ≥ Rd and, Z 1 ∗ √ ω −d/2 − 2 t Σnt iht,∆ni−iht,(v−u)/ ni for n τ, I13(n) = n √ e e dt. ≥ − |t|>δ n Proposition 3.4. Under Assumptions1 and3 for all n τ the following holds Pm-a.e. for d ≥ u, v Z : ∈ d/2   (2π) 1 ∗ −1
1. I12(n) = d/2 1/2 exp [∆n (v u)/√n] Σn [∆n (v u)/√n] . n det (Σn) − 2 − − − − If additionally Assumption4 is fulfilled, uniformly in ω Ω, ∈ −d/ 1
2. there exists α > 0 such that I (n) O n 2 exp αδ√n , | 13 | ≤ {− 2 }  − d+  3. then there exists a deterministic δ > 0 such that I (n) = O n 2 for any  [0, 1). | 11 | ∈ Proof. First, set v = u. 1. Under Assumptions1 and3, Proposition 3.3 implies τ < Pm-a.e. and for n τ, there ∞ ≥ exist orthogonal matrices Pn and diagonal matrices Dn such that

−1 2 2 Σn = PnDnPn and Dn = diag(α1(n), . . . , αd(n))

2 with αi (n) > 0 for all i = 1, . . . , d and n τ. Setting t = Pnu, we obtain as the Fourier transform of Gaussian vectors ≥

Z 1 ∗ ∗ d/2 − u Dnu ihu,P ∆ni n I12(n) = e 2 e n du Rd 1 ∗ −1 −1/2 d/2 − ∆ Σn ∆n = det (Σn) (2π) e 2 n .

16 2. For the term I13(n) we can proceed analogously and we get the following (not sharp) upper bound for n τ : ≥

Z 1 ∗ d/2 − 2 u Dnu n I13(n) √ e du | | ≤ |u|>δ n    1 = O exp αδ√n , − 2 } for α > 0 of Assumption4.

3. Because of the point (2) of Proposition 3.3, the eigenvalues of Σn remain bounded uni- formly for n 0. Thus, with Assumption4, we deduce the following bound for I (n) ≥ 11 Z d/2 − 1 t∗Σ t −1/2 n I (n) e 2 n exp R (tn ) 1 dt 11 √ n | | ≤ |t|≤δ n { } − Z − 1 t∗Σ t −1/2 −1/2 e 2 n R (tn ) exp R (tn ) dt √ n n ≤ |t|≤δ n {| |} 1− Z Kδ − 1 t∗Σ t 2+ |t|2δK e 2 n t e dt. /2 √ ≤ n |t|≤δ n | |

The last estimates comes from Proposition 3.2 and holds for any  [0, 1). We conclude by choosing δ > 0 such that δK α/4 (where α is given by Assumption∈ 4). Consequently the integral is convergent and the≤ whole term goes to zero at rate, up to a constant, n−/2.

Setting ∆˜ n = ∆n (v u)/√n, the result follows for the general case. − − Proof of Theorem 1.7. Let r (0, 1), and compute ∈ Z X X Z Re rn−1 n1(x)dt rn−1 1 I (n)dPx Ft − {n≥τ} 12 Wd n≥1 n≥1 XN Z X n−1   x = r 1{n≥τ} I11(n) + I13(n) dP n≥1 XN X n−1 + r I2(n) n≥1 Z X n−1 x + r 1{n<τ}I1(n)dP n≥1 XN .

Taking absolute values on both side, under Assumptions1,2,3 and4, using Proposition 3.1, Proposition 3.4 with a suitable δ > 0, it follows that, for some K 0, C > 0, κ (0, 1) and any  (0, 1), ≥ ∈ ∈ Z Z   X X 1{n≥τ} 1 Re rn−1 n1(x)dt rn−1 exp ∆∗ Σ−1∆ dPx t d/2 n n n F − N det (Σn)n −2 Wd n≥1 n≥1 X X K X K  1  rn−1 + rn−1 exp αδ√n ≤ n(d+)/2 nd/2 − 2 n≥1 n≥1 X X + C κn−1 + (2π)d rn−1Px(τ > n). n≥1 n≥1

17 d m The latter summation follows from the fact I1(n) (2π) , P -a.e.. Then, letting r 1, the ∞ | | ≤ ↑ x right handside remains bounded in Lm (X) since d 2 and τ is integrable with respect to P , ≥ m-a.e. x X. The result follows by letting r 1 on the left hand-side and remarking that ∈ ↑ X Z 1  1  lim rn−1 1 exp ∆∗ Σ−1∆ dPx n≥τ d/2 n n n r↑1 N det (Σn)n −2 n≥1 X Z   X 1n≥τ 1 = exp ∆∗ Σ−1∆ dPx [0, ] d/2 n n n N det (Σn)n −2 ∈ ∞ n≥1 X m-a.e. by monotone convergence.

Acknowledgements The author thanks the referee for its careful reading of the paper and its useful remarks improving the readability. The author acknowledges the financial support from the Swiss National Foundation Grant FN 200021-138242/1. This work was mainly done while the author holds a position at the University of Burgundy, the University of Strasbourg and at the Ensai.

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