School of Mathematics | School of Mathematics | University of Bristol

Exit times from cones for non-homogeneous random walk with asymptotically zero drift Iain M. MacPhee1

Mikhail V. Menshikov1

Andrew R. Wade∗,2

27th June 2008 1 2

Department of Mathematical Sciences, University of Durham, South Road, Durham DH1 3LE, UK. Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK.

Abstract We study the first exit time τ from an arbitrary cone with apex at the origin by a non-homogeneous random walk on Z2 with mean drift that is asymptotically zero. Assuming bounded jumps and a form of weak isotropy, we give conditions for τ to be almost surely finite, and for the existence and non-existence of moments E[τ p ], p > 0. Specifically, if the mean drift at x ∈ Z2 is of magnitude O(kxk−1 ), then τ < ∞ a.s. for any cone, and we give polynomial estimates on the tail of τ , while a mean drift of order kxk−β , β ∈ (0, 1), can lead to τ = ∞ with positive probability for any cone. For mean drift o(kxk−1 ) (such as in the driftless case) we essentially recover, amongst other things, the random walk analogue of a theorem for Brownian motion due to Spitzer.

Key words and phrases: Non-homogeneous random walk; asymptotically zero perturbation; recurrence/transience; passage-time moments; exit from cones; Lyapunov functions. AMS 2000 Mathematics Subject Classification: 60J10 (Primary) 60G40, 60J15, 60F15 (Secondary)

1

Introduction

The theory of time- and space-homogeneous random walks in Zd (d ≥ 2), i.e., sums of i.i.d. random integer-component vectors, is classical and extensive; see for example [8,20,35]. For random walks that are not spatially homogeneous the theory is less complete, since many techniques available for the study of homogeneous random walks can no longer be applied, or are considerably complicated; see, for instance, [19, 31]. In general non-homogeneous processes can be wild; thus we restrict ourselves to walks that have mean drift that tends to zero as the distance to the origin tends to infinity (but with no restriction on the direction of the drift) and satisfy some weak regularity conditions on the jumps. We do not require symmetry or uniform ellipticity, as are assumed, for example, ∗

Partially supported by the Heilbronn Institute for Mathematical Research

1

for the results on non-homogeneous random walks in [19, 31]. Random walk models are applied in many contexts. Often, simplifying assumptions of homogeneity are made in order to make such models tractable, whereas non-homogeneity is more realistic. Thus our nonhomogeneous model shares some motivation with random walks in random environments (see e.g. [39]); in such terms, our results deal with a particular class of ‘asymptotically zero drift’ environments. In the present paper we develop methods to study passage-times for certain sets for such non-homogeneous random walks. On an underlying probability space (Ω, F, P), let Ξ = (ξt )t∈Z+ be a discrete-time stochastic process, with state-space S ⊂ Rd . For A ⊆ S, set τ := min{t ∈ N : ξt ∈ A}. (Here and throughout the paper we adopt the conventions Z+ := {0, 1, 2, . . .}, N := {1, 2, 3, . . .}, and min ∅ := ∞.) Then the stopping time τ is the first passage time to A for the process Ξ. A fundamental question is whether or not P[τ < ∞] = 1. If τ is almost surely finite, the natural follow-up question is whether the expectation E[τ ] is finite, or, more generally, E[τ p ] for p > 0. Together, these questions constitute the passage-time problem that is the main focus of the present paper, in the context of non-homogeneous random walks in the plane. The theory of homogeneous zero-mean random walks stands hand-in-hand with the corresponding continuum theory for Brownian motion. Once the assumption of spatial homogeneity is removed, Brownian motion ceases to be a reliable analogy for the random walk problem. In the case of one dimension, this is exemplified by results on processes with asymptotically zero mean drifts; see e.g. [17, 18]. Our results in the present paper include conditions under which our non-homogeneous random walk does display essentially ‘Brownian’ behaviour; we also give results in which its behaviour is completely different. We now describe informally the type of non-homogeneous random walk studied in the present paper. Consider a Markovian random walk, homogeneous in time but not necessarily in space, with bounded jumps in the plane. Suppose that the walk has one-step mean drift that tends to zero as the distance from the origin tends to infinity. This asymptotically zero drift regime is the natural setting in which to probe the transition away from behaviour that is essentially ‘zero-drift’ in character. In one dimension, the corresponding regime is rather well understood, following fundamental work of Lamperti; see [17, 18, 26–28] and the Appendix in [2] (some analogous results in the continuous setting of Brownian motion with asymptotically zero drift are given in [11]). Problems in higher dimensions of a ‘radial’ nature can often be reduced to this one-dimensional case. The exit-from-cones problems that we consider in the present paper (which we describe below), on the other hand, are to a large extent ‘transverse’ (and inhomogeneous) in nature and so are truly two-dimensional. This two-dimensional case is the natural next point of analysis after the classical onedimensional work of Lamperti and others. The two-dimensional case is qualitatively different from the one-dimensional case (see Theorem 2.1 below), and it seems likely that this phenomenon persists in higher dimensions. We work in two dimensions rather than higher dimensions in generality to ease technical details, and since the two-dimensional case is the most explicit and well-studied case for Brownian motion and homogeneous random walks. The random walks that we consider are non-homogeneous, but some regularity assumptions are certainly required for our results. We assume a weak isotropy condition without 2

which highly degenerate behaviour is possible. In addition, we restrict our attention to random walks on Z2 with uniformly bounded jumps. We need some regularity conditions on the state-space of our walk and it is most convenient to take Z2 . We are confident that our proofs can be adapted for more general lattices. The condition of bounded jumps is imposed to ease technical difficulties and keep our arguments to a reasonable length. It is likely that this assumption can be relaxed to a higher moment assumption on the jumps at the expense of extra work; compare for instance [2] (Section 5) or [15]. To summarize, we believe that the specific assumptions that we make can be relaxed in several directions, and that the phenomena that our results reveal are more general; in order to present all these phenomena for non-homogeneous random walks in a single paper we inevitably need some simplifications. The particular aspects of the non-homogeneous random walk that we study are, still rather informally, as follows. For α ∈ (0, π] let W(α) be an open half-cone or ‘wedge’ in the plane, with apex at the origin and interior angle 2α. Starting the random walk from some point in the interior of the wedge, let τα be the first exit time from the wedge; for example, τπ/4 is the first exit time from a quadrant. We address the passage-time problem for τα , i.e. the questions: (i) is τα almost surely finite; and (ii) which moments of τα exist? We obtain a full spectrum of behaviour, depending on the nature of the walk: both finiteness and infiniteness of τα are possible, and we show both existence and non-existence of polynomial moments E[ταp ], p > 0. The non-homogeneous random walk with zero mean drift generalizes the homogeneous case. The relaxation to asymptotically zero drift can be viewed as a perturbation of the zero-drift case. In this paper we identify the critical magnitude of such a perturbation. Specifically, we identify and study three broad classes of perturbation: (i) subcritical, with drift of magnitude o(kxk−1 ) at x ∈ Z2 , where we show that the behaviour is similar qualitatively and quantitatively to the zero-drift case; (ii) supercritical, with drift of magnitude of order kxk−β , β ∈ (0, 1), where we show that the perturbation is large enough to lead to behaviour that is completely contrary to the zero-drift case; and (iii) critical, when the drift is of magnitude of order kxk−1 , and the exit behaviour is qualitatively comparable to the zero-drift case but quantitatively can be strikingly different. More specifically, our results demonstrate that if the perturbation is sufficiently strong (roughly speaking, of order kxk−β for some β ∈ (0, 1)), τα can be infinite with positive probability: in some circumstances, eventually the random walk remains in an arbitrarily thin wedge. Our results show that the critical perturbation size is of order kxk−1 : such a walk has τα < ∞ a.s. for any wedge. This result has some interesting consequences. For example, for a random walk with mean drift ckxk−1 in the outwards radial direction (c > 0), one-dimensional results of Lamperti [17] show that for c large enough this walk is transient in the plane. However, our results show that it will in fact leave any wedge: the transience has no limiting direction. We also show that for α ∈ (0, π/2), τα has polynomial tails provided that the mean drift is O(kxk−1 ). It is clear by comparison with one-dimensional results [2, 17, 18, 28] that even in this case where the mean drift is O(kxk−1 ), our non-homogeneous random walk model can behave very differently to Brownian motion; for instance, for any ε > 0 the drifts can conspire so that almost surely the walk remains at distance O(tε ) from its starting point for all t (compare Theorem 4.3 in [28]). We are confident that some of these results extend to cones in higher dimensions. A 3

random walk Ξ on Zd (d ≥ 3) exits from a cone contained in a half-space if its projection Ξ∗ onto any 2-dimensional subspace exits an appropriate wedge. If Ξ satisfies a 3-dimensional version of our regularity conditions and has mean drift O(kxk−1 ), then Ξ∗ satisfies a 2dimensional version of the same conditions, but is not necessarily Markov with respect to its natural filtration. We believe however that our results also hold for a process that is Markov with respect to a general filtration. (We did not pursue this generalization in the present paper for reasons of space.) This means that finiteness of τα and some of its moments in the d-dimensional case could be deduced from (the extended versions of) our Theorems 2.1 and 2.2 below. We briefly survey some relevant literature. In the homogeneous zero-drift setting, for the analogous continuous problem of planar Brownian motion in a wedge, a classical result due to Spitzer (Theorem 2 in [34]) says that E[ταp ] < ∞ if and only if p < π/(4α). A deep study of the existence and non-existence of passage-time moments for Brownian motion in Rd was carried out by Burkholder [7]; subsequent related results on Brownian motion in cones include, for example, those in [3, 10, 32]. We do not know of any results on exit from cones for non-homogeneous diffusions; analogues of our results in the present paper should hold in the continuous setting. The random walk problem has received less attention, even in the homogeneous zero-drift case. It is known however that the zero-drift homogeneous random walk on the quarter-lattice Z+ × Z+ with jumps that are bounded in second moment and uncorrelated in the horizontal and vertical directions has E[τπ/4 ] = ∞ (see [9, 16, 36]); this is a random walk version of the α = π/4 case of the non-existence part of Spitzer’s theorem for Brownian motion. In the particular case α = π, the problem is the study of the hitting time of a half-line; this is sometimes called the problem of a walk on a slit-plane, see e.g. [6]. For a homogeneous zero-drift random walk on Z2 , results of Lawler [20] (see also finer results of Fukai [15]) show that E[τπp ] < ∞ if and only if p < 1/4. This is a random walk analogue of the α = π case of Spitzer’s theorem for Brownian motion. A consequence of our results in the present paper is that Spitzer’s theorem for Brownian motion, and the homogeneous random walk results for α = π/4, α = π mentioned above, essentially extend, under some moderate regularity conditions, to non-homogeneous random walks with mean drifts that tend to zero sufficiently rapidly (in fact, as o(kxk−1 )). A related problem is the study of random walks (or Brownian motion) in a wedge or quadrant with reflection at the boundaries. This problem has received considerable attention over the years; see e.g. [1, 2, 13] and references therein. The existence and non-existence of moments of recurrence-times of a neighbourhood of the origin for reflecting zero-drift random walks in the quadrant was studied in [1,2]; reflecting Brownian motion in a wedge was studied in [37] and recurrence-time moments thereof in [29]. Random walks in quadrants (more generally, orthants) are of importance in applications, in particular in the theory of queues and stochastic networks, and so have been well studied: see for example [5, 8, 12, 13, 21–25, 30]. Our results for non-homogeneous random walks add to these previous studies. We briefly comment on the techniques that we use in the present paper. In general, we adopt the ideology of semimartingale (Lyapunov-type) criteria; for instance, we prove transience in the supercritical case by an argument based on results in [13]. Such methods are powerful and robust enough to cope with the non-homogeneity of the model. Given 4

the diverse nature of our results, it is not surprising that our proofs need several different technical ideas. Often it is possible to prove the existence of passage-time moments directly via semimartingale criteria such as those in [1, 2, 18] in the vein of Foster [14]. In the subcritical case for our perturbation, we have Lyapunov functions that are well-adapted to do this. In the critical case, the non-homogeneity forces us to adopt a more direct approach, where nevertheless martingale techniques are a primary tool. The situation is similar for the problem of non-existence of passage-time moments, although even in the subcritical case rather delicate estimates are required. In Section 3 below we collect a ‘tool-box’ of martingale-type results that we need, and that we believe will be useful in any investigation of non-homogeneous random walks. In the next section we formally define our model and state our main results. We also mention some possible directions for future research.

2

Model and results

In this section we describe more precisely the probabilistic model that is our object of study, and then give our main theorems. First we collect some notation. For x ∈ R2 , write x = (x1 , x2 ) in Cartesian coordinates. Let k · k denote the Euclidean norm on R2 . Write 0 := (0, 0) for the origin and e1 := (1, 0), e2 := (0, 1) for the unit vectors in the x1 , x2 directions. Let Ξ = (ξt )t∈Z+ be a discrete-time Markov process with state-space Z2 . The random walk Ξ will be time-homogeneous but not necessarily space-homogeneous, although we will impose some natural regularity assumptions on the jump distribution for our walk; we describe these now. We need to impose some form of regularity condition that ensures the walk cannot become trapped in lower-dimensional subspaces or finite sets. For example, if the only possible jumps of the walk are in the ±e1 directions, it will be trapped on a line. Then one-dimensional results (see e.g. [17]) imply that even for a mean drift of magnitude O(kxk−1 ) the process can be transient in the positive e1 direction, and so will with positive probability never leave any cone with principal axis in the e1 direction. To avoid this kind of situation, we will assume the following weak isotropy condition: (A1) There exist κ > 0, k1 , k2 ∈ N and n0 ∈ N such that min

x∈Z2 ; y∈{±k1 e1 ,±k2 e2 }

P[ξt+n0 − ξt = y | ξt = x] ≥ κ (t ∈ Z+ ).

Note that (A1) is an n0 -step regularity condition. In terms of one-step regularity, its implications are minimal: a simple consequence of (A1) is that P[ξt+1 = x | ξt = x] = (P[ξt+n0 = x, . . . , ξt+1 = x | ξt = x])1/n0 ≤ (1 − 4κ)1/n0 ≤ 1 − (4κ/n0 ) so that P[ξt+1 6= x | ξt = x] ≥ (4κ/n0 ) > 0 5

(2.1)

uniformly in x. Condition (A1) can be seen as a form of ellipticity, but is considerably weaker than uniform ellipticity (such as often assumed in the random walk in random environment literature, see e.g. [39]). For example, there can be sites x ∈ Z2 at which the walk only jumps in one direction. The assumptions (A1) and that the state-space of Ξ is Z2 are particularly important for our proofs of Theorems 2.1 and 2.2 below, where we make use of a decomposition for the random walk based upon the weak isotropy implied by (A1). However, we believe that the proofs can be adapted to more general situations if these assumptions are relaxed. The Z2 assumption is used rather less essentially in our proofs of the other theorems below. Let θt := ξt+1 − ξt denote the jump of Ξ at time t ∈ Z+ . We assume that the distribution of the random vector θt depends only upon the location ξt ∈ Z2 at time t. In other words, there exists a Z2 -valued random field θ = (θ(x))x∈Z2 such that for all t ∈ Z+ L(ξt+1 − ξt | ξt ) = L(θt | ξt ) = L(θ(ξt )), where L stands for ‘law’. The law of θ is the jump distribution of Ξ. We write θ(x) in components as (θ1 (x), θ2 (x)). Our second regularity condition is an assumption of uniformly bounded jumps, that is: (A2) There exists b ∈ (0, ∞) such that supx∈Z2 kθ(x)k ≤ b a.s.. As mentioned above, this condition could be replaced by higher-order moment assumptions at the expense of some technical work, but our arguments are clearer under the bounded jumps assumption. Under (A2), the moments of θt are well-defined. Denote the one-step mean drift vector µ(x) := E[θt | ξt = x] = E[θ(x)] for x ∈ Z2 , and write µ(x) = (µ1 (x), µ2 (x)) in components. We are primarily interested in the case where the random walk has asymptotically zero mean drift, i.e., limkxk→∞ kµ(x)k = 0. Similarly we denote the covariance matrices M = (Mij )i,j∈{1,2} of θ by M(x) := E[θt> θt | ξt = x] = E[θ(x)> θ(x)], for x ∈ Z2 , where θt is viewed as a row-vector. That is, Mii (x) = E[θi (x)2 ] and M12 (x) = M21 (x) = E[θ1 (x)θ2 (x)]. Note that when (A1) holds, (2.1) implies that M11 (x) + M22 (x) > 0 uniformly in x. For the problem of the exit of the random walk Ξ from a wedge, it is natural to introduce polar coordinates. For x = (x1 , x2 ) ∈ R2 we use polar coordinates (r, ϕ) relative to the ray Γ0 in the e1 direction starting at 0. Thus if r = kxk and ϕ ∈ (−π, π] is the angle, measuring anticlockwise, of the ray through 0 and x = (x1 , x2 ) from the ray Γ0 , we have x1 = r cos ϕ and x2 = r sin ϕ. We occasionally write ϕ as ϕ(x) for clarity. Let er (ϕ) = e1 cos ϕ + e2 sin ϕ, the radial unit vector, and e⊥ (ϕ) = −e1 sin ϕ + e2 cos ϕ, the transverse unit vector. For α ∈ (0, π), let W(α) be an open wedge in R2 with apex 0 and half-angle α: W(α) := {x ∈ R2 : r > 0, −α < ϕ < α}. 6

So, for example, W(π/4) = {(x1 , x2 ) : x1 > 0, |x2 | < x1 } is a quadrant and W(π/2) = {(x1 , x2 ) : x1 > 0} a half-plane. The case of α = π we will treat slightly differently: for s ≥ 0, define Hs := {(x1 , x2 ) : x1 ≤ 0, |x2 | ≤ s}; for s > 0 this is a thickened half-line. Then with the jump bound in (A2) b > 0 set W(π) := R2 \ Hb . (For convenience, we often call W(π) a ‘wedge’ also.) It will also be convenient to set W(α) := R2 for any α > π. The subject of this paper is the random walk’s first exit time from the wedge W(α) (starting from inside the wedge). Define the random time τα := min{t ∈ Z+ : ξt ∈ / W(α)}.

(2.2)

The notation τα suppresses the dependence on the starting point ξ0 . Thus when Ξ is started from W(α), τα is the first exit time of Ξ from the wedge W(α); particularly, τπ is the hitting time of a thickened half-line. In the latter case, we could instead list the jumps that cross the half-line H0 = {(x1 , 0) : x1 ≤ 0} and stop the walk after its first jump from this list, but we prefer to work with the thickened half-line. We introduce some terminology. If for x ∈ W(α) ∩ Z2 , P[τα < ∞ | ξ0 = x] = 1, then the set Z2 \ W(α) is recurrent for Ξ starting from x; otherwise Z2 \ W(α) is transient for Ξ from x. In the recurrent case, we are also interested in which moments of τα exist, and which do not. We now list some examples of random walks with which we will illustrate the results that follow. These examples are all rather special, and our results permit considerably more general examples. Let I denote the 2 × 2 identity matrix. (E1) Canonical random walk. Suppose that (A1) and (A2) hold, µ(x) ≡ 0, and M(x) ≡ σ 2 I for some σ 2 ∈ (0, ∞). (E2) Random walk with radial drift. Suppose that (A1) and (A2) hold, and that for some c ∈ R, β > 0, for x 6= 0, µ(x) = ckxk−β er (ϕ). (E3) Random walk with drift in the principal direction. Suppose that (A1) and (A2) hold, and that for some c ∈ R, β > 0, for x 6= 0, µ(x) = ckxk−β e1 . Our first result, Theorem 2.1 below, deals with the critical perturbation O(kxk−1 ). This critical case is most important and also the most difficult to study. It is a classical result (see e.g. [35], p. 83) that a genuinely two-dimensional spatially homogeneous random walk on Z2 with mean drift zero and bounded jumps is recurrent, and so leaves any wedge in almost surely finite time. Theorem 2.1 says that the latter is true in our non-homogeneous case, provided that the perturbation is O(kxk−1 ); the direction of µ is not important. In particular, the zero-drift case is included. This result contrasts sharply with the situation in one dimension [17, 18], where a drift of O(x−1 ) at x does not imply finiteness of the time of exit from a half-line. 7

Theorem 2.1 Suppose that (A1) and (A2) hold, and that for x ∈ Z2 as kxk → ∞, kµ(x)k = O(kxk−1 ).

(2.3)

Then for any α ∈ (0, π] and any x ∈ W(α) P[τα < ∞ | ξ0 = x] = 1. To illustrate this result, consider the case β = 1 of (E2). Results of Lamperti (see the discussion around (4.13) in [17]) imply that this ‘centrally biased random walk’ is transient in the plane for c > 0 large enough, and so eventually leaves every bounded region. On the other hand, Theorem 2.1 shows that such a walk will eventually leave any wedge. This gives information about how the walk winds around the origin: there is no direction of transience. (An early result on the winding number of planar Brownian motion is also contained in Spitzer’s paper [34]; a more recent reference, including corresponding results for homogeneous random walks, is [33].) Of course, (E1), and (E3) with β ≥ 1 also satisfy the conditions of Theorem 2.1. Note that even walks satisfying (E1) may be non-homogeneous, so that classical results such as those in [35] do not apply directly. Theorem 2.1 says that the walk in the critical or subcritical cases will eventually leave any wedge. The next result, Theorem 2.2, gives information on the tails of the exit time τα , α < π/2. In particular, it shows that even for this non-homogeneous walk, the tail behaviour is essentially polynomial in character, as in the zero-drift case: compare Theorem 2.6 below. However, the detailed nature of which moments exist (i.e., the exponent s0 in the statement of Theorem 2.2) will depend on the details of the walk: compare Theorems 2.3, 2.6, and 2.8 below. For a one-dimensional analogue of this result, see the Appendix in [2]. Theorem 2.2 Suppose that (A1) and (A2) hold, α ∈ (0, π/2), and that for x ∈ Z2 as kxk → ∞, (2.3) holds. Then there exists s0 ∈ (0, ∞) such that for any x ∈ W(α)  = ∞ if s > s0 s . E[τα | ξ0 = x] < ∞ if s < s0 It is an open problem to show that Theorem 2.2 holds with s0 > 0 for wedges with α ≥ π/2. Walks satisfying Theorem 2.2 can have radically different characteristics. For example, for small enough wedges a zero-drift walk will have E[τα ] < ∞ (see Theorem 2.6 below). On the other hand, the next result implies that for any α ∈ (0, π/2), for a suitably strong O(kxk−1 ) perturbation, E[τα ] = ∞. In fact, the theorem says that for any ε > 0, there exist walks satisfying the conditions of Theorem 2.2 for which (1/2) + ε moments of τα do not exit. An open question is to determine whether (1/2) − ε moments can be infinite under the conditions of Theorem 2.3. We take the random walk to have dominant drift in the principal direction. Specifically, we assume that there exists d > 0 for which lim (kxkµ1 (x)) = d,

lim (kxkµ2 (x)) = 0.

kxk→∞

kxk→∞

The β = 1, c > 0 case of (E3) is an example of a walk satisfying (2.4). 8

(2.4)

Theorem 2.3 Suppose that (A1) and (A2) hold, and α ∈ (0, π/2). Then for any s > 0, there exists d0 ∈ (0, ∞) such that if (2.4) holds for any d > d0 , then for all x ∈ W(α), E[τα(1/2)+s | ξ0 = x] = ∞. Now we move on to the supercritical case. Theorem 2.4(i) below says that if the mean drift is sufficiently strong in the outwards radial direction, the random walk is transient on any wedge, that is P[τα = ∞] > 0, and so eventually remains in some arbitrarily narrow wedge; in some sense the transience has a (random) limiting direction. This is in contrast to Theorem 2.1. On the other hand, when the drift is supercritical and in the inwards radial direction, Theorem 2.4(ii) says that all polynomial moments exist for τα . This is in contrast to Theorem 2.2. We use the notation ‘·’ for the scalar product of vectors in R2 . Note that the conditions on µ(x) · e⊥ (ϕ) in (2.6) and (2.8) are each satisfied when |µ(x) · e⊥ (ϕ)| = O(kxk−1 ), for instance. Theorem 2.4 Suppose that (A1) and (A2) hold, and α ∈ (0, π]. (i) Suppose that for some β ∈ (0, 1), c > 0, δ > 0, and A0 > 0, {kxkβ µ(x) · er (ϕ)} ≥ c, and

(2.5)

sgn(x1 x2 )kxkβ+δ µ(x) · e⊥ (ϕ) ≤ 0.

(2.6)

inf x∈W(α):kxk>A0

lim sup x∈W(α),kxk→∞

Then for any x ∈ W(α)

P[τα = ∞ | ξ0 = x] > 0.

(ii) Suppose that for some β ∈ (0, 1), c < 0, and A0 > 0, sup x∈W(α):kxk>A0

lim inf x∈W(α),kxk→∞

{kxkβ µ(x) · er (ϕ)} ≤ c, and

(2.7)

sgn(ϕ)kxkβ µ(x) · e⊥ (ϕ) ≥ 0.

(2.8)

Then for any s ∈ [0, ∞) and all x ∈ W(α) E[ταs | ξ0 = x] < ∞. For instance, in (E2) with β ∈ (0, 1), Theorem 2.4(i) applies when c > 0 and Theorem 2.4(ii) applies when c < 0. In the former case, the transience behaviour contrasts with the case (as mentioned after Theorem 2.1 above) when β = 1 and c > 0 is large enough, where the walk is transient in the plane but leaves any wedge. It is likely that using the techniques of the present paper and the results of [1] (see particularly Corollary 2 of [1]) existence of super-polynomial ‘moments’ for τα can be obtained under the conditions of part (ii) of Theorem 2.4. The next result will follow as a corollary to the preceding theorems, and deals with a particularly interesting example for α ≤ π/2, which should serve to clarify the nature of the preceding results. See Figure 1. 9

−1 Figure 1: Two examples of drift fields: cx−1 1 e1 (left) and ckxk er (ϕ) (right).

Theorem 2.5 Suppose that (A1) and (A2) hold. Let α ∈ (0, π/2]. (i) Suppose that µ(x) = xc1 e1 . Then for α < π/2, for any c ∈ R, and any x ∈ W(α), P[τα < ∞ | ξ0 = x] = 1. On the other hand, there exists c0 ∈ (0, ∞) such that for c > c0 and any x ∈ W(π/2), P[τπ/2 = ∞ | ξ0 = x] > 0. (ii) Suppose that µ(x) =

c e1 xβ 1

for c > 0 and β ∈ (0, 1). Then for any α ∈ (0, π/2] and any

x ∈ W(α), P[τα = ∞ | ξ0 = x] > 0.

In particular, Theorem 2.5(i) shows that when the mean drift is c/x1 in the x1 direction, the walk leaves a wedge of angle α < π/2, but, for c large enough, with positive probability does not leave the half-plane. Theorem 2.5 should be contrasted with (E3). For (E3) with α ≤ π/2, β = 1, the walk always leaves the wedge, by Theorem 2.1, even when α = π/2. The instance in Theorem 2.5 when α = π/2 demonstrates homogeneity in the x2 direction, and so is related to the one-dimensional so-called Lamperti problem named after [17, 18]. In the case α = π/2, (E3) demonstrates a more localized perturbation, since near the boundary of the half-plane we can have kxk  x1 . Our next result, Theorem 2.6, gives rather sharp tail asymptotics for τα when Ξ is a subcritical perturbation of a canonical zero-drift random walk. That is, under a stronger condition than those of Theorems 2.1, 2.2 we obtain sharper results; also all α ∈ (0, π] are covered. This result shows that the critical exponent for the moment problem depends only on the wedge angle and is the same in this random walk setting as in the Brownian motion case, where the result is due to Spitzer (Theorem 2 of [34]). In particular, Theorem 2.6 includes the case of a homogeneous random walk with zero drift; but it extends to nonhomogeneous random walks such as (E1), and (E2), (E3) with β > 1 and suitable M. We write o(1) for a 2 × 2 matrix each of whose entries is o(1). Theorem 2.6 Suppose that (A1) and (A2) hold, and there exists σ 2 ∈ (0, ∞) such that as kxk → ∞ kµ(x)k = o(kxk−1 ), and M(x) = σ 2 I + o(1). (2.9) Suppose α ∈ (0, π]. Then:

10

(i) for any s ∈ [0, π/(4α)) and any x ∈ W(α), E[ταs | ξ0 = x] < ∞; (ii) for any s > π/(4α) and any x ∈ W(α), E[ταs | ξ0 = x] = ∞. Certain cases of Theorem 2.6 extend results of Klein Haneveld and Pittenger [16] and Lawler [20] for homogeneous zero-drift random walks (that is, sums of i.i.d. mean-zero random variables) to non-homogeneous random walks satisfying the conditions of Theorem 2.6. First, in the case of hitting a half-line (α = π), Theorem 2.6 implies that 1/4-moments are critical, a result obtained for homogeneous zero-drift random walks by Lawler (see (2.35) in [20], also [15]). Second, in the case of a quadrant (α = π/4), Theorem 2.6(ii) implies that s E[τπ/4 ] = ∞ for s > 1, a result contained in Theorem 1.1 of [16] for a homogeneous zero-drift random walk with certain regularity conditions (see also Theorem 1.1 of [9] and p. 81 of [8]). s Of course, Theorem 2.6 also gives E[τπ/4 ] < ∞ for s < 1 in this case, a result that must be known for the simple random walk (and is clear by analogy with Brownian motion) but for which we could not find a reference. The next result is the analogue of Theorem 2.6(i) in the case of a random walk with skewed covariance matrix. It shows that higher moments exist than in the corresponding canonical case. Once more this generalizes known results for homogeneous random walks (see the remark following the theorem). One reason for interest in walks on Z2 with such skewed covariance matrices is that they can arise from transformations of walks on non-rectangular lattices. Theorem 2.7 Suppose that (A1) and (A2) hold. Let α ∈ (0, π/2). Suppose that there exist σ 2 ∈ (0, ∞) and λ ∈ [−1, 0) such that for x ∈ W(α) as kxk → ∞   1+λ 0 −1 2 + o(1). (2.10) kµ(x)k = o(kxk ), and M(x) = σ 0 1−λ Then there exists ε > 0 such that for all s ∈ [0, (1 + ε)π/(4α)) and any x ∈ W(α) E[ταs | ξ0 = x] < ∞. Remark. In the particular case of the quadrant, consider the change of coordinates from √ √ 0 0 0 0 (x1 , x2 ) to (x1 , x2 ) where x1 := (x1 + x2 )/ 2 and x2 := (x1 − x2 )/ 2. Then the boundary of W(π/4) lies along the axes x01 = 0, x02 = 0. In these new coordinates, condition (2.10) reads   1 λ 0 −1 0 2 kµ (x)k = o(kxk ), and M (x) = σ + o(1). λ 1 That is, λ < 0 corresponds to jumps of equal variance but negative correlation in the directions parallel to the two boundary lines. Then Theorem 2.7 implies in particular that E[τπ/4 ] < ∞ for λ < 0; this result in the case of a homogeneous zero-drift random walk on the quarter-lattice is given in [9, 16] (see also [36] and p. 81 of [8]). The last two results that we present below deal with the ‘critical’ perturbation of order kxk−1 . In particular they show that which moments of τα exist depend upon the nature 11

of the perturbation. We assume that there exists d ∈ R \ {0} such that (2.4) holds, and moreover we also assume that the covariance matrix M is asymptotically canonical. Theorem 2.8 shows that in the critical perturbation case, the drift can lead to the existence of higher moments than in the subcritical case (Theorem 2.6). In particular, in the 1+ε quadrant case (α = π/4), when d < 0, E[τπ/4 ] exists for some ε > 0. On the other hand, if 1+ε d = 0 in (2.4) then Theorem 2.6 applies, so that E[τπ/4 ] = ∞ for any ε > 0. Theorem 2.8 Suppose that (A1) and (A2) hold. Let α ∈ (0, π). Suppose that (2.4) holds for d < 0, and that M(x) = σ 2 I + o(1) for some σ 2 ∈ (0, ∞). Then there exists ε > 0 such that for all s ∈ [0, (1 + ε)π/(4α)) and any x ∈ W(α) E[ταs | ξ0 = x] < ∞. The next result says that for any given d > 0 and p > 0, we can choose α > 0 small enough so that E[ταp ] < ∞; i.e. arbitrarily high (polynomial) moments exist for sufficiently small wedges for any given d > 0. This contrasts with Theorem 2.3. For α ∈ (0, π) and σ 2 ∈ (0, ∞), define d∗ := d∗ (α, σ 2 ) :=

σ2 πσ 2 > > 0. 4α 4

(2.11)

Theorem 2.9 Suppose that (A1) and (A2) hold. Let α ∈ (0, π). Suppose that (2.4) holds and that M(x) = σ 2 I + o(1) for some σ 2 ∈ (0, ∞). Suppose that 0 ≤ d < d∗ as given by (2.11). Then for any x ∈ W(α) E[ταs | ξ0 = x] < ∞, for all s ≥ 0 with

s
0. The remainder of the results in this section are stated for one-dimensional, not necessarily Markov, stochastic processes. Let (Ft )t∈Z+ be a filtration on a probability space (Ω, F, P). Let (Yt )t∈Z+ be a discrete-time (Ft )t∈Z+ -adapted stochastic process taking values in [0, ∞). Typically, when we come to apply the following lemmas later on, we will have Yt = r(ξt ) for some r : R2 → [0, ∞). For m ∈ N let `m denote the mth upper passage time of (Yt )t∈Z+ , that is `m := min{t ∈ Z+ : Yt ≥ m}. Then `1 , `2 , . . . is a nondecreasing sequence of stopping times for the process (Yt )t∈Z+ ; under the condition lim supt→∞ Yt = +∞ a.s., `m < ∞ a.s. for every m. For real numbers a, b, we use the notation a∧b := min{a, b}. Then for m ∈ N, (Yt∧`m )t∈Z+ is the process stopped at `m . We have the following result, which is a ‘reverse Foster’s criterion’ (compare Theorem 2.1.1 in [13]). Lemma 3.2 Let (Yt )t∈Z+ be an (Ft )t∈Z+ -adapted process on [0, ∞), such that for some B ∈ (0, ∞), P[Yt+1 − Yt ≤ B] = 1.

Fix m ∈ N and Y0 = y0 ∈ [0, m). Suppose that there exists g : [0, ∞) → [0, ∞) a nondecreasing function for which, for some ε > 0, E[g(Y(t+1)∧`m ) − g(Yt∧`m ) | Ft ] ≥ ε1{`m >t} a.s. for all t ∈ Z+ . Then

E[`m ] ≤ ε−1 g(m + B). 13

(3.1)

Proof. Fix m ∈ N. Taking expectations in (3.1) we have for all s ∈ Z+ E[g(Y(s+1)∧`m )] − E[g(Ys∧`m )] ≥ εP[`m > s]. Summing for s from 0 up to t we obtain E[g(Y(t+1)∧`m )] − g(Y0 ) ≥ ε Since g(Y0 ) = g(y0 ) ≥ 0 we obtain E[`m ] = lim

t→∞

t X s=0

t X

P[`m > s].

s=0

P[`m > s] ≤ ε−1 lim sup E[g(Y(t+1)∧`m )] ≤ ε−1 g(m + B), t→∞

since, a.s., Yt∧`m ≤ m + B and g is nondecreasing.



Our next lemma is essentially a modification of the Doob submartingale inequality, which gives a maximal inequality for processes with increments that are bounded above in expectation. Lemma 3.3 Let (Yt )t∈Z+ be an (Ft )t∈Z+ -adapted process on [0, ∞) such that P[Y0 = y0 ] = 1 and for some B ∈ (0, ∞) and all t ∈ Z+ E[Yt+1 − Yt | Ft ] ≤ B a.s.. Then for any x > 0 and any t ∈ N   P max Ys ≥ x ≤ (Bt + y0 )x−1 . 0≤s≤t

Proof. Similarly to Doob’s decomposition (see e.g. [38], p. 120), set W0 := Y0 , and for s ∈ N − − − let Ws := Ys + A− s−1 + As−2 + · · · + A1 , where As = E[Ys+1 − Ys | Fs ], As = max{−As , 0} ≥ 0, + As = max{As , 0} ≥ 0. Then + − E[Ws+1 − Ws | Fs ] = E[Ys+1 − Ys + A− s | Fs ] = As + As = As ∈ [0, B]

so that (Ws )s∈Z+ is a nonnegative (Fs )s∈Z+ -submartingale with Ws ≥ Ys for all s, and E[Wt ] ≤ W0 +Bt = y0 +Bt. Consequently, by Doob’s submartingale inequality (see e.g. [38], p. 137)     P max Ys ≥ x ≤ P max Ws ≥ x ≤ x−1 E[Wt ] ≤ (Bt + y0 )x−1 , 0≤s≤t

as required.

0≤s≤t



Next we state the semimartingale criteria that we can in some cases apply to determine the existence and non-existence of generalized moments of τα . Following work of Lamperti [18], the primary results available for establishing the existence and non-existence of passagetime moments for a (not necessarily Markov) stochastic process are contained in [2]. For some of the applications in the present paper, we could not apply these general results and so have to use other techniques. First we state an existence result. The following lemma is contained in Theorem 1 of [2]. 14

Lemma 3.4 Let (Yt )t∈Z+ be an (Ft )t∈Z+ -adapted stochastic process taking values in an unbounded subset S of [0, ∞). Suppose B > 0. Set υB := min{t ∈ N : Yt ≤ B}. Suppose that there exist C ∈ (0, ∞), p0 > 0 such that for any t ∈ Z+ , Yt2p0 is integrable, and 2p0 E[Yt+1 − Yt2p0 | Ft ] ≤ −CE[Yt2p0 −2 ]

on {υB > t}.

Then for any p ∈ [0, p0 ), for any x ∈ S, E[υBp | Y0 = x] < ∞. The corresponding non-existence result that we will need is Corollary 1 in [2]: Lemma 3.5 With the notation of Lemma 3.4, suppose that there exist C, D ∈ (0, ∞), r > 1, and p0 > 0 such that for any t ∈ Z+ the following three conditions hold on {υB > t}: 2p0 E[Yt+1 − Yt2p0 | Ft ] ≥ 0; 2 E[Yt+1 − Yt2 | Ft ] ≥ −C; 2r E[Yt+1 − Yt2r | Ft ] ≤ DYt2r−2 .

(3.2) (3.3) (3.4)

Then for any p > p0 , for any x ∈ S large enough, E[υBp | Y0 = x] = ∞.

3.2

Lyapunov functions

In this section we introduce some Lyapunov functions that we will use to study our random walk, primarily in the case of a subcritical perturbation, and analyze their basic properties. These functions will be built upon standard harmonic functions in the plane. Such harmonic functions were employed by Burkholder [7] in his sharp analysis of the exit-from-cones problem for Brownian motion; it is natural that they are the correct tools when our random walk is sufficiently close to zero-drift, but only give partial results in the critical order kxk−1 case. First we need some further notation. Let Br (x) denote the closed Euclidean ball (a disk) of radius r > 0 centred at x ∈ R2 . For α ∈ (0, π] and s > 0 define the modified wedge Ws (α) := W(α) \ Bs (0) = {x ∈ W(α) : kxk > s}, which is W(α) with a disk-segment around the origin removed. During our proofs, we will often work with the exit time from the locally modified set WA (α) for some fixed (large) value of A > 0. Let W0 (α) := W(α) and for A ≥ 0, define τα,A := min{t ∈ Z+ : ξt ∈ / WA (α)}.

(3.5)

Then τα,A ≥ τα,B for B ≥ A, and τα,0 = τα with the notation of (2.2). We will use multi-index notation for partial derivatives on R2 . For σ = (σ1 , σ2 ) ∈ Z+ ×Z+ , Dσ will denote D1σ1 D2σ2 where Djk for k ∈ N is k-fold differentiation with respect to xj , and Dj0 is the identity operator. We also use the notation |σ| := σ1 + σ2 and xσ := xσ1 1 xσ2 2 . 15

With our polar coordinate system as defined in Section 2, we state the elementary facts (following from the chain rule) that for a differentiable function f : R2 → R in the above coordinate system ∂f 1 (r, ϕ) − sin ϕ · ∂r r 1 ∂f (r, ϕ) + cos ϕ · D2 f (r, ϕ) = sin ϕ · ∂r r

D1 f (r, ϕ) = cos ϕ ·

∂f (r, ϕ); ∂ϕ ∂f (r, ϕ). ∂ϕ

(3.6)

For R ⊂ R2 , let ∂R denote the (topological) boundary of R. For w > 0, define the function fw : R2 → R by fw (x) := fw (r, ϕ) := rw cos(wϕ).

(3.7)

Then differentiating using (3.6), we have that for any w > 0 D1 fw (r, ϕ) = wrw−1 cos((w − 1)ϕ); D2 fw (r, ϕ) = −wrw−1 sin((w − 1)ϕ), and D1 D2 fw (r, ϕ) = D2 D1 fw (r, ϕ) = w(w − 1)rw−2 sin((w − 2)ϕ).

(3.8) (3.9)

Moreover, D12 fw (r, ϕ) = w(w − 1)rw−2 cos((w − 2)ϕ) = −D22 fw (r, ϕ),

(3.10)

so that fw is harmonic in R2 . For w > 1/2, fw is positive in the interior of the wedge W(π/(2w)), and zero on the boundary ∂W(π/(2w)); f1/2 is positive on R2 \ H0 and zero on the half-line H0 . For w ∈ (0, 1/2), fw is positive throughout R2 . The asymptotic behaviour of fw (x) for x with kxk large will be crucial for us. It follows by repeated applications of (3.6) that fw and all of its derivatives Dσ fw are of the form rk u(ϕ) where u is bounded, and hence for any σ with |σ| = j there exists a constant C ∈ (0, ∞) such that for all x ∈ R2 −Crw−j < Dσ fw (x) < Crw−j .

(3.11)

As an example, we have that the harmonic function f2 (x) = r2 cos(2ϕ) = x21 − x22

(3.12)

is positive on the quadrant W(π/4) and zero on ∂W(π/4). The next result gives expressions for the first three moments of the jumps of fw (ξt ). Lemma 3.6 Suppose that (A2) holds. Then with fw defined at (3.7), for w > 0, there exists C ∈ (0, ∞) such that for any x ∈ Z2 P[|fw (ξt+1 ) − fw (ξt )| ≤ C + Ckξt kw−1 | ξt = x] = 1. Also, for any x ∈ Z2 as r = kxk → ∞, we have the following asymptotic expansions: E[fw (ξt+1 ) − fw (ξt ) | ξt = x] = wrw−1 (µ1 (x) cos((w − 1)ϕ) − µ2 (x) sin((w − 1)ϕ)) 16

(3.13)

1 (M11 (x) − M22 (x)) w(w − 1)rw−2 cos((w − 2)ϕ) 2 + M12 (x)w(w − 1)rw−2 sin((w − 2)ϕ) + O(rw−3 ); (3.14) +


E[(fw (ξt+1 ) − fw (ξt ))2 | ξt = x] = w2 r2w−2 M11 (x) cos2 ((w − 1)ϕ) + M22 (x) sin2 ((w − 1)ϕ) − M12 (x)w2 r2w−2 sin(2(w − 1)ϕ) + O(r2w−3 ); (3.15) 3 3w−3 E[(fw (ξt+1 ) − fw (ξt )) | ξt = x] = O(r ). (3.16) Proof. Since fw is smooth we can expand it in a disk of radius b using Taylor’s theorem with Cartesian coordinates and the Lagrange form for the remainder at any x ∈ R2 . We obtain X fw (x + y) = fw (x) + yj (Dj fw )(x + ηy), j

for some η = η(y) ∈ (0, 1), for any y = (y1 , y2 ) with kyk ≤ b. Taking y = ξt+1 − ξt = θt , conditioning on ξt = x, we then obtain, with (3.8), a.s., for some C ∈ (0, ∞), |fw (ξt+1 ) − fw (ξt )| ≤ Ckx + ηθ(x)kw−1 , for any y ∈ Z2 . Now (A2) implies (3.13). To obtain the moment estimates, we include more terms in the Taylor expansion to obtain fw (x + y) = fw (x) +

X

yj (Dj fw )(x) +

j

X 1X 2 2 yj (Dj fw )(x) + yi yj (Di Dj fw )(x) 2 j i 0,

since wα ∈ (0, π/2). Then (3.19) follows from (3.7). The statement (3.20) follows similarly, using the fact that for w ≥ k/2 and k ≥ 0 −w ≤ −

k ≤ w − k ≤ w, 2

so that inf cos((w − k)ϕ) ≥

x∈W(α)

This completes the proof.

inf

ϕ∈(−α,α)

cos(wϕ) = εα,w .



Lemma 3.8 Suppose that (A2) holds. Suppose that α ∈ (0, π], γ ∈ R, w ∈ (0, π/(2α)). Then for all x ∈ W(α) we have that as r = kxk → ∞ E[fw (ξt+1 )γ − fw (ξt )γ | ξt = x] = γfw (x)γ−1 wrw−1 (µ1 (x) cos((w − 1)ϕ) − µ2 (x) sin((w − 1)ϕ)) + γfw (x)γ−1 M12 (x)w(w − 1)rw−2 sin((w − 2)ϕ) 1 + γfw (x)γ−1 (M11 (x) − M22 (x)) w(w − 1)rw−2 cos((w − 2)ϕ) 2
1 + γ(γ − 1)fw (x)γ−2 w2 r2w−2 M11 (x) cos2 ((w − 1)ϕ) + M22 (x) sin2 ((w − 1)ϕ) 2 1 − γ(γ − 1)fw (x)γ−2 w(w − 1)r2w−2 M12 (x) sin(2(w − 1)ϕ) + O(fw (x)γ−3 r3w−3 ). 2 Proof. Let ∆ := fw (ξt+1 ) − fw (ξt ). Then for γ ∈ R and x ∈ W(α)  γ  ∆ γ γ γ E[fw (ξt+1 ) − fw (ξt ) | ξt = x] = fw (x) E 1+ − 1 ξt = x , fw (x)

and as long as ∆/fw (x) is not too large we can use the fact that for γ ∈ R and small x 1 (1 + x)γ = 1 + γx + γ(γ − 1)x2 + O(x3 ). 2

Under the conditions of the lemma, for x ∈ W(α) with r = kxk large enough we have that ∆ Crw−1 −1 fw (x) ≤ fw (x) = O(r ), using (3.13) and (3.19). Hence for γ ∈ R and all kxk large enough

E[fw (ξt+1 )γ − fw (ξt )γ | ξt = x] = γfw (x)γ−1 E[∆ | ξt = x]
1 + γ(γ − 1)fw (x)γ−2 E[∆2 | ξt = x] + O fw (x)γ−3 E[∆3 | ξt = x] . 2

Then from (3.21), Lemma 3.6, and (3.19) we obtain the desired result. We also will need the following straightforward result. 19



(3.21)

Lemma 3.9 Let h : R2 → R and R ⊂ R2 be such that h(x) ≤ 0 for all x ∈ R2 \ R. Set ˆ h(x) := h(x)1{x∈R} . Then for all x ∈ R and all t ∈ Z+ ˆ t+1 ) − h(ξ ˆ t ) ≥ h(ξt+1 ) − h(ξt ), on {ξt = x}. h(ξ Proof. For x ∈ R we have on {ξt = x} that ˆ t+1 ) − h(ξ ˆ t ) = h(ξt+1 ) − h(ξt ) − h(ξt+1 )1{ξ ∈R} h(ξ , t+1 / which yields the result given that h(x) ≤ 0 for x ∈ / R.

4 4.1



Finite exit times: critical case Decomposition

We show in this Section that Theorem 2.1 holds: under the conditions of the theorem, with probability 1 the random walk Ξ will leave a wedge W(α), α ∈ (0, π] after a finite time. It suffices to give the proof for α = π, i.e., hitting the thickened half-line Hb . A key step in the proof is a result on the exit from rectangles (Lemma 4.3 below) that says, loosely speaking, that if the walk starts somewhere near the centre of a rectangle, there is strictly positive probability (uniformly in the size of the rectangle) that the walk will first exit the rectangle via the top/bottom. Here the fact that kµ(x)k = O(kxk−1 ) is crucial. This result highlights the source of the crucial difference between the problem of exit from cones in two dimensions and the analogous problem of exit from a half-line in one-dimension, where O(x−1 ) does not always imply finiteness of the exit time. The one-dimensional analogue of our result for exit from rectangles is false: classical one-dimensional gambler’s ruin estimates imply that for a random walk on Z+ with mean-drift O(1/x) at x, the probabilities of hitting 0, 2M first, starting from M , are not necessarily bounded uniformly away from 0. Crucial to the proof of Lemma 4.3 will be a decomposition of the random walk Ξ based on the regularity condition (A1). We establish Theorem 2.1 by studying the behaviour of the walk on a set of seven overlapping quarter-planes that together span W(π) (the plane minus a thickened half-line). For this reason, we need to consider wedges like W(α) with several different principal axes. This requires some more notation. Define lattice vectors qi , i ∈ {1, . . . , 7} by q5 = −q1 = e1 + e2 ,

q6 = −q2 = e2 ,

q3 = −q7 = e1 − e2 ,

We also need notation for perpendiculars to the qi , specifically q⊥ i = qi+2 , i ∈ {1, 2, . . . , 5},

q⊥ 6 = −q4 ,

For the corresponding unit vectors, write ˆ i := kqi k−1 qi , q

⊥ −1 ⊥ ˆ⊥ q i := kqi k qi ;

20

q⊥ 7 = q1 .

q4 = e1 .

√ note that kqi k = kq⊥ 2 for odd i. i k, which is 1 for even i and For β ∈ (0, π/2) and i ∈ {1, . . . , 7} let Wi (β) denote the wedge with apex 0, internal angle 2β, and principal direction qi ; that is Wi (β) := {x ∈ R2 : x · qi > 0, |x · q⊥ i | < (tan β)|x · qi |}.

(4.1)

With our existing notation, this means that W4 (α) is W(α); the other Wi (α) are rotations of W(α) through angles kπ/4, k ∈ {±1, ±2, ±3}. In the proof of Theorem 2.1 below we will need the quarter-planes Wi (π/4); when it comes to the proof of Theorem 2.2 we need Wi (α) for α ∈ (0, π/2). Thus we work in this generality for now. For each i, using the regularity condition (A1) we now decompose Ξ into a symmetric 2 + walk in the q⊥ i direction and a residual walk. For x, y ∈ Z , n ∈ N and t ∈ Z let " n−1 # X p(x, y; n) := P[ξt+n = y | ξt = x] = P θ(ξt+j ) = y − x | ξt = x . j=0

It follows from (A1) by considering finite combinations of jumps that for each i ∈ {1, 2, . . . , 7} there exist constants γi ∈ (0, 1/2), ni , ji ∈ N such that ⊥ min {p(x, x + ji q⊥ i ; ni ), p(x, x − ji qi ; ni )} ≥ γi .

(4.2)

x∈Qi

Now we fix i and consider the random walk at time spacing ni , i.e. the embedded process (ξtni )t∈Z+ . For notational convenience, set ξt∗ := ξtni ,

(t ∈ Z+ ).

Then Ξ∗ = (ξt∗ )t∈Z+ is a Markovian random walk on Z2 with transition probabilities ∗ P[ξt+1 = y | ξt∗ = x] = p(x, y; ni ),

and ξ0∗ = ξ0 . The walk Ξ∗ inherits regularity from Ξ: from (A2) we have that for t ∈ Z+ ∗ P[kξt+1 − ξt∗ k ≤ bni ] = 1,

(4.3)

and from (4.2) we have that for all x ∈ Z2

2 ∗ 2 ⊥ 2 ∗ ˆ⊥ E[|(ξt+1 − ξt∗ ) · q i | | ξt = x] ≥ 2ji kqi k γi > 0.

Moreover, it follows from (A2) that  P max

tni ≤s≤(t+1)ni

kξs − xk ≤ ni b |

ξt∗

(4.4)

 = x = 1,

and hence (t+1)ni −1 ∗ E[ξt+1

−

ξt∗

|

ξt∗

= x] =

X

s=tni

E[ξs+1 − ξs | kξs − xk ≤ ni b, ξtni = x].

Thus with (2.3) we have that for x ∈ Z2 , as kxk → ∞,

∗ kE[ξt+1 − ξt∗ | ξt∗ = x]k = O(kxk−1 ).

By (4.2), there exist sequences of random variables (Vt )t∈N and (ζt )t∈N such that: 21

(4.5)

(i) the (Vt )t∈N are i.i.d. with Vt ∈ {−1, 0, 1} and P[Vt = 0] = 1 − 2γi ,

P[Vt = −1] = P[Vt = +1] = γi ;

(ii) ζt+1 ∈ Z2 with P[ζt+1 = 0 | Vt 6= 0] = 1; and (iii) we can decompose the jumps of Ξ∗ via ∗ ξt+1

−

ξt∗

= ξ(t+1)ni − ξtni =

n i −1 X s=0

+ θ(ξtni +s ) = Vt+1 ji q⊥ i + ζt+1 . (t ∈ Z )

(4.6)

Note that given point (ii), (4.6) is equivalent to ∗ + ξt+1 − ξt∗ = Vt+1 ji q⊥ i 1{Vt+1 6=0} + ζt+1 1{Vt+1 =0} . (t ∈ Z )

Thus we decompose the jump of Ξ∗ at time t into a symmetric component in the perpendicular direction (Vt+1 ji q⊥ i ), and a residual component (ζt+1 ), such that at any time t only one of the two components is present in a particular realization. Then for t ∈ N, (4.6) yields a decomposition for ξt∗ as ξt∗

= ξ0 +

t X

(Vs ji q⊥ i + ζs ).

(4.7)

s=1

This decomposition is valid throughout Z2 , but for our purposes we will apply the decomposition involving q⊥ i in the wedge Wi (β) for appropriate β ∈ (0, π/2). Note that the residual jump components are given for t ∈ Z+ by
∗ ∗ ∗ ζt+1 = ξt+1 − ξt∗ − Vt+1 ji q⊥ (4.8) i 1{Vt+1 =0} = (ξt+1 − ξt )1{Vt+1 =0} , and ζ1 , ζ2 , . . . are not in general independent; however, they do inherit regularity properties from Ξ. From (4.8) and (4.3) we have ∗ kζt+1 k ≤ kξt+1 − ξt∗ k ≤ bni a.s.;

(4.9)

also conditioning on ξt∗ = x, taking expectations in (4.6) and noting that E[Vt+1 | ξt∗ = x] = 0, ∗ E[ζt+1 | ξt∗ = x] = E[ξt+1 − ξt∗ | ξt∗ = x].

4.2

(4.10)

Exit from rectangles

We will eventually use this decomposition to establish (in Lemma 4.3 below) how the walk exits from sufficiently large rectangles aligned in the qi , q⊥ i directions. First we need two lemmas that deal in turn with the two parts of the decomposition. The rough outline of the proof of Lemma 4.3 below is as follows. In time bεN 2 c, we show that the process driven by V1 , V2 , . . . will with positive probability attain distance sufficient to take it well beyond the top/bottom of the rectangle; this is Lemma 4.1 below. On the other hand, we show that in time bεN 2 c, for small enough ε > 0, the residual process does 22

not stray very far from its initial point with good probability, regardless of the realization of V1 , V2 , . . .; this is Lemma 4.2 below. Together, these two results will enable us to conclude that with good probability the walk will leave a rectangle via the top/bottom. First we need some more notation. ˆ⊥ Set Y0 := ξ0 · q i and for t ∈ N Yt := Y0 +

ji kq⊥ i k

t X

Vs .

(4.11)

s=1

Then Yt is the displacement of the symmetric part of the decomposition for ξt∗ in the q⊥ i direction. The process (Yt )t∈Z+ is a symmetric, homogeneous random walk on kq⊥ kZ with i P[Yt = Yt−1 ] = P[Vt = 0] = 1 − 2γi < 1 and jumps of size kq⊥ i kji . For h ∈ (0, ∞), let  τh⊥ := min t ∈ Z+ : |Yt | ≥ d3hN ekq⊥ (4.12) i k , the time when |Yt | first reaches value d3hN ekq⊥ i k.

Lemma 4.1 Let h ∈ (0, ∞). For any ε > 0, there exist δ > 0 and N1 ∈ N such that for any N ≥ N1 and any y ∈ Z with |y| ≤ 2hN P[τh⊥ ≤ bεN 2 c | Y0 = kq⊥ i ky] ≥ 2δ. Proof. Fix h ∈ (0, ∞). Suppose Y0 = kq⊥ i ky, |y| ≤ 2hN . If y 6= 0 then couple a copy of the ⊥ walk Yt started from kqi ky with another Y˜t started from 0 which has jumps in the opposite ˜ direction to Yt until |Yt − Y˜t | ≤ kq⊥ i kji for the first time, from which time on Yt , Yt jump in ⊥ ˜ the same direction. Then when |Yt | ≥ K we have |Yt | ≥ K − kqi kji , and with probability γi the next jump will take |Yt+1 | ≥ K. It follows that for any ε > 0 ⊥ 2 P[τh⊥ ≤ bεN 2 c | Y0 = kq⊥ i ky] ≥ γi P[τh ≤ bεN c − 1 | Y0 = 0]

≥ γi P[τh⊥ ≤ bε0 N 2 c | Y0 = 0],

for any ε0 ∈ (0, ε) and all N large enough. Hence it suffices to prove the lemma in the case y = 0. Suppose now that y = 0. The process (Yt )t∈Z+ is a symmetric random walk on kq⊥ i kZ 2 ⊥ 2 2 with independent, bounded jumps and E[|Yt+1 − Yt | ] = 2γi kqi k ji > 0. Thus it follows from standard central limit theorem estimations that for any ε > 0 there exists δ > 0 such that for all N sufficiently large     ⊥ P YbεN 2 c > d3hN ekq⊥ i k | Y0 = 0 ≥ δ, and P YbεN 2 c < −d3hN ekqi k | Y0 = 0 ≥ δ. Each of the (disjoint) events in the previous display implies that τh⊥ ≤ bεN 2 c. completes the proof. 

This

ˆ i )ˆ Let Z0 := (ξ0 · q qi and for t ∈ N let Zt := Z0 +

t X s=1

23

ζs .

(4.13)

Thus (Zt )t∈Z+ is the residual part of the process (ξt∗ )t∈Z+ after the symmetric perpendicular process (Yt )t∈Z+ has been extracted. Indeed, with Yt , Zt as defined at (4.11), (4.13) we have ˆ⊥ ξ0∗ = ξ0 = Y0 q i + Z0 and also from (4.7) that for t ∈ N ˆ⊥ ξt∗ = Yt q i + Zt .

(4.14)

We next show that with good probability the residual process (Zt )t∈Z+ does not exit from a suitable ball around its initial point by time bεN 2 c. By construction the process (Zt )t∈Z+ depends upon (Vt )t∈N because the distribution of ζt+1 depends upon the value of ξtni . For t ∈ N, let ΩV (t) := {−1, 0, 1}t and let ωV ∈ ΩV (t) denote a generic realization of the sequence (V1 , . . . , Vt ). Lemma 4.2 Let r ∈ (0, 1/2]. There exist N2 ∈ N and ε > 0 such that for all N ≥ N2 , all z ∈ Z with |z| ≤ b, and all ωV ∈ ΩV (bεN 2 c)   1 P max 2 kZt − Z0 k ≤ rN | (V1 , . . . , VbεN 2 c ) = ωV , Z0 = (N + z)qi ≥ . 0≤t≤bεN c 2 Proof. Let r ∈ (0, 1/2]. Suppose that Z0 = (N + z)qi with z ∈ Z, |z| ≤ b. For the duration of this proof, define the stopping time τ0 := min{t ∈ Z+ : kZt − Z0 k > rN }. Observe that on {t < τ0 } we have ˆi ≥ Z − 0 · q ˆ i − kZt − Z0 k ≥ (N − b)kqi k − rN ≥ N/3 Zt · q

(4.15)

for all N large enough. Let Ft = σ(ξ0∗ , ξ1∗ , . . . , ξt∗ ), the σ-algebra generated by the history of Ξ∗ up to time t. Then Ft and (V1 , . . . , Vt ) ∈ ΩV (t) specify ζ1 , . . . , ζt and Z0 , Z1 , . . . , Zt . Consider the stopped square-deviation process defined for t ∈ Z+ by Wt := kZt∧τ0 − Z0 k2 . Suppose that t ≤ bεN 2 c, and let ωV ∈ Ω(bεN 2 c). On the event {τ0 > t} we have that Wt+1 − Wt = kZt+1 − Z0 k2 − kZt − Z0 k2 = kZt+1 − Zt k2 + 2(Zt+1 − Zt ) · (Zt − Z0 ) = kζt+1 k2 + 2ζt+1 · (Zt − Z0 ), while on {τ0 ≤ t}, Wt+1 − Wt = 0. So conditioning on Ft and the realization ωV of (V1 , . . . , VbεN 2 c ), and taking expectations, we obtain E[Wt+1 − Wt | ωV , Ft ]


= E[kζt+1 k2 | ωV , Ft ] + 2E[ζt+1 · (Zt − Z0 ) | ωV , Ft ] 1{t r | ξ0 = x] ≤ Cr−γ .

(4.23)

Proof. Suppose that ξ0 ∈ Wi (β), β ∈ (0, π/2). Take h = tan β ∈ (0, ∞). Let ˆ i , 2k ≥ N0 , 2k ≥ b}, k0 := min{k ∈ N : 2k kqi k ≥ ξ0 · q where N0 is as in Lemma 4.3 and b is as in (A2). Consider the sequence of rectangles S(2k ) where k ∈ Z+ , as defined at (4.21), with h = tan β. Set σ0 := 0 and define the hitting times ˆ i ≥ 2k kqi k}. σk := min{t ∈ Z+ : ξt · q Suppose that Ξ has not exited Wi (β) by the time σk for some k ≥ k0 , i.e., τi (β) > σk . Using the Markov property at time σk , we consider the walk started anew at ξσk . Then, ˆ i ≤ 2k kqi k + b and on {τi (β) > σk }, from (4.1), using (A2), 2k kqi k ≤ ξσk · q ˆ⊥ ˆ i | ≤ 2k hkqi k + hb ≤ 2 · 2k hkqi k, |ξσk · q i | < h|ξσk · q for all k ≥ k0 . Hence we can apply Lemma 4.3 to the walk started at ξσk , with N = 2k ≥ N0 for k ≥ k0 . Then, with B1 (N ), B2 (N ) as defined in (4.22), we obtain, for all k ≥ k0 , P[(ξt )t≥σk hits B2 (2k ) before B1 (2k ) | τi (β) > σk ] ≥ δ > 0.

(4.24)

But by definition of B1 (N ), B2 (N ), and (4.1), we have that if Ξ hits B2 (N ) before B1 (N ), then Ξ leaves the wedge Wi (β). Moreover, if Ξ has not hit B1 (2k ) by time τi (β), then ˆ i < 2k+1 kqi k, max ξt · q

0≤t≤τi (β)

so that τi (β) < σk+1 . Hence the inequality (4.24) can now be expressed as P[τi (β) ≤ σk+1 | τi (β) > σk ] ≥ δ > 0 for all k ≥ k0 . Hence for all k > k0 P[τi (β) > σk ] =

k Y

j=k0 +1

P[τi (β) > σj | τi (β) > σj−1 ] · P[τi (β) > σk0 ] ≤ C(1 − δ)k ,

(4.25)

for some C = C(k0 , δ) ∈ (0, ∞) that does not depend on k. We now need to estimate the tail behaviour of the times σk . It is most convenient to work once again via the embedded walk Ξ∗ . Set ˆ i ≥ 2k kqi k}. σk∗ := min{t ∈ Z+ : ξt∗ · q ˆ i . Let A, C > 0 and consider For t ∈ Z+ , for the remainder of this proof write Xt := ξt∗ · q A the process ((C + Xt ) )t∈Z+ . We show that for A, C sufficiently large, this process is a strict submartingale so that we can apply Lemma 3.2 to obtain a bound for E[σk∗ ]. 27

Note that Taylor’s theorem implies that for any x ≥ 0 and any y ∈ R with |y| < B # " A y 1+ −1 (C + x + y)A − (C + x)A = (C + x)A C +x   Ay A(A − 1)y 2 A −3 = (C + x) + + O((C + x) ) . C +x (C + x)2 ∗ ˆ i and − ξt∗ . By (4.3) we may apply the last displayed equation with x = x · q Set θt∗ = ξt+1 ∗ ˆ i to obtain y = θt · q

E[(C + Xt+1 )A − (C + Xt )A | ξt∗ = x]   ˆ i | ξt∗ = x] A(A − 1) E[(θt∗ · q ˆ i )2 | ξt∗ = x] E[θt∗ · q A −3 ˆi) A ˆi) ) . =(C + x · q + + O((C + x · q ˆi ˆ i )2 C +x·q 2 (C + x · q Now using (4.4) and (4.5) we see that we can choose A, C sufficiently large so that E[(C + Xt+1 )A − (C + Xt )A | ξt∗ = x] ≥ ε > 0 ∗ for all x. Moreover, from (4.3) we have |Xt+1 − Xt | ≤ kξt+1 − ξt∗ k ≤ ni b. Hence we can apply Lemma 3.2 to Xt with function g(x) = (C + x)A to obtain, for all k ≥ k0

E[σk∗ ] ≤ E[min{t ∈ Z+ : Xt ≥ 2k+1 }] ≤ ε−1 g(2k+1 + ni b) = ε−1 (C + 2k+1 + ni b)A . By definition of ξt∗ , we have that σk ≤ ni σk∗ a.s., hence there exists C ∈ (0, ∞) such that E[σk ] ≤ 2kC , for all k ≥ k0 . Applying Markov’s inequality with M = C + 1 this implies that for k ≥ k0 P[σk > 2kM ] ≤ 2−kM E[σk ] ≤ 2kC · 2−kM = 2−k . Combining this with (4.25) and the fact that for any k P[τi (β) > 2kM ] ≤ P[τi (β) > σk ] + P[σk > 2kM ], we have that P[τi (β) > 2kM ] ≤ C(1 − δ)k + 2−k for all k ≥ k0 . It follows that there exist constants M, γ 0 ∈ (0, ∞), not depending on x, and C ∈ (0, ∞), which does depend on x, such that for all k ≥ k0 , 0 P[τi (β) > 2kM | ξ0 = x] ≤ C2−γ k . (4.26) (4.23) follows for k ≥ k0 . Clearly the result extends to all k ∈ Z+ for a suitable choice of C in (4.26), depending on k0 and so also on ξ0 . For any r > 0, we have that r ∈ [2kM , 2(k+1)M ) for some k ∈ Z+ . Then given ξ0 = x we have from (4.26) that 0

0

P[τi (β) > r] ≤ P[τi (β) > 2kM ] ≤ C2−γ k ≤ C(2r−1/M )γ = C 0 r−γ , for some C 0 , γ ∈ (0, ∞), not depending on r, where moreover γ does not depend on x. This completes the proof. 

28

We are now ready to give a proof of Theorem 2.1, i.e., to show that Ξ exits from any wedge W(α), α ≤ π, in a.s. finite time. Proof of Theorem 2.1. For notational ease let Qi := Wi (π/4), τi := τi (π/4) for i ∈ {1, . . . , 7} so that the Qi form a set of overlapping quarter-planes and the τi are the corresponding first exit times. Also write Q8 := Hb , the thickened half-line. Suppose that ξ0 ∈ Qi . It follows immediately from Lemma 4.4 that P[τi < ∞] = 1, and so Ξ almost surely exits the initial quadrant Qi . By the bounded jumps assumption (A2) and the definition (4.1), we see that at time τi , ξτi is within distance b of the principal axis of either Qi+1 or Qi−1 , working mod 8 for the indices of the Qj s, or ξτi ∈ Q8 (has hit the thickened half-line). As Ξ exits any quadrant in finite time, this process repeats indefinitely until Ξ hits Hb . Thus we have a sequence of ‘home quadrants’ (Hn )n∈Z+ , where H0 = i, H1 is either i + 1 or i − 1, and so on, and the sequence terminates (at 8) if Ξ hits Hb . Moreover, for each Hn 6= 8, n ≥ 1, Ξ starts within distance b of the principal axis of the quadrant QHn . In Lemma 4.1 the process Yt in the q⊥ i direction is symmetric and it follows from the proof of the lemma that provided |y| ≤ b there, for each i ∈ {1, 2, . . . , 7} there exists δi > 0 such that for each n ∈ N P[Hn+1 = i + 1 | Hn = i] ≥ δi ,

P[Hn+1 = i − 1 | Hn = i] ≥ δi .

The transitions to state 8 represent exit from the wedge W(π), i.e., hitting the thickened half-line Hb . With Hn = i for any n and any i ∈ {1, . . . , 7} we see that such a transition occurs with strictly positive probability in the next four transitions, and hence Ξ exits W(π) in finite time almost surely. 

5 5.1

Exit time moments: critical case An almost-linear Lyapunov function

To prove Theorem 2.2, it suffices to show that E[ταp ] = ∞ and E[ταq ] < ∞ for some p, q with 0 < q < p < ∞. The existence part of the proof will follow from Lemma 4.4 of the last section. The main part of the present section is thus devoted to our technique for establishing the non-existence part of the proof of Theorem 2.2, which will also enable us to give a proof of Theorem 2.3. We work in the wedge W(α), α ∈ (0, π/2). For its regularity properties, as in Section 4 we work with the embedded walk given by ξt∗ = ξtni where in this case we can take ni = n0 as in (A1). We first aim to show that for any wedge W(α), α ∈ (0, π/2), there exists p ∈ (0, ∞) such that E[ταp ] = ∞. The outline of our approach is roughly speaking as follows. We consider a one-dimensional process (Yt )t∈Z+ where Yt = g(ξt∗ ) for a suitably chosen g and apply Lemma 3.5. More specifically, we construct an almost linear or ε-linear (in the sense of Malyshev [24], see also Chapter 3 of [13]) function g to enable us to apply the generalized form [2] of “Lamperti’s conditions” [18] in Lemma 3.5. The idea is to construct g so that its level curves are horizontal translates of ∂W(α) but with the apex replaced by a circle arc. Fix α ∈ (0, π/2). During the remainder of this section, set s := sin α ∈ (0, 1), c := cos α ∈ (0, 1). We now construct the function g : R2 → [0, ∞). Set g(x) = 0 for x ∈ R2 \ W(α). 29

x2 g(x) = k k

k

k

α k/s

2k/s

x1

Figure 2: Level curve of the function g. sc For x = (x1 , x2 ) ∈ W(α) such that |x2 | ≥ 1+c 2 x1 let g(x) = sx1 − c|x2 |. For x ∈ W(α) with sc |x2 | ≤ 1+c2 x1 , set g(x) = k ∈ [0, ∞) on the minor arc of the circle

((2k/s) − x1 )2 + x22 = k 2

(5.1)

between (k(1 + c2 )/s, kc) and (k(1 + c2 )/s, −kc). Then g is specified on W(α) by its level curves g(x) = k, k ≥ 0, each of which is ∂W(α) translated so that the apex is at (k/s, 0) and the tip of the wedge smoothed to a circular arc. See Figure 2. We now state some properties of the function g. Observe that for x ∈ R2 g(x) ≤ kxk. For x ∈ W(α) with |x2 | ≥ sc |x2 | ≤ 1+c 2 x1 we have

sc x 1+c2 1

(5.2)

we have ∇g(x) = (s, ±c) and k∇g(x)k = 1, while for

1 1 ∇g(x) = (−((2g(x)/s) − x1 ), x2 ) = D(x) D(x)

  q 2 2 − g(x) − x2 , x2 ,

(5.3)

from (5.1), where When |x2 | ≤

sc x, 1+c2 1

D(x) = g(x) + (2/s)(x1 − (2g(x)/s)).

so that the level curve of g is a circular arc, we have g(x)((2/s) − 1) ≤ x1 ≤ g(x)((2/s) − s),

(5.4)

−g(x)((2/s) − 1) ≤ D(x) ≤ −g(x).

(5.5)

and hence sc It follows from (5.3) that for x ∈ W(α) with |x2 | ≤ 1+c 2 x1 , q −1 k∇g(x)k = |D(x)| g(x)2 − x22 + x22 = |D(x)|−1 g(x).

Hence from (5.5) we have that for all x ∈ W(α) k∇g(x)k ≥

s s ≥ ; 2−s 2

30

(5.6)

and for all x ∈ R2

k∇g(x)k ≤ 1.

(5.7)

To obtain our non-existence of moments result for τα , we aim to apply Lemma 3.5 to the process given by Yt = g(ξt∗ ). The next lemma presents some further properties of the function g that we will need here and in the proof of Theorem 2.3 later. Lemma 5.1 For x ∈ W(α) with |x2 | ≤

sc x 1+c2 1

we have

s g(x) ≥ kxk. 2

(5.8)

Also, there exists ε > 0 such that for all x ∈ W(α) D1 g(x) ≥ ε.

(5.9)

Finally, there exists C ∈ (0, ∞) such that for all x ∈ R2 and all i, j ∈ {1, 2} |Dij g(x)| ≤ Ckxk−1 .

(5.10)

sc Proof. To obtain (5.8), we observe that for |x2 | ≤ 1+c 2 x1 , from (5.4) # " 2 sc kxk2 = x21 + x22 ≤ ((2/s) − s) + ((2/s) − s)2 g(x)2 1 + c2   = c2 + ((2/s) − s)2 g(x)2 = ((4/s2 ) − 3)g(x)2 ,

and (5.8) follows. Consider (5.9). It suffices to suppose |x2 | ≤

sc x. 1+c2 1

Now from (5.3)   2g(x) − sx1 Sx1 D1 g(x) = =R 1+ , ((4/s) − s)g(x) − 2x1 g(x) − Rx1

where we have set R=

2 , (4/s) − s

S = R − (s/2).

(5.11)

(5.12)

Here we have from (5.12) and (5.4) that

(s2 /4)g(x) ≤ g(x) − Rx1 ≤ (s/2)g(x). Also, since s ∈ (0, 1), we have

2s s ≤R≤ . 2 3

(5.13)

(5.14)

Similarly, using the fact that s S= 2 we note for later use that



   1 s2 /4 s −1 = , 1 − (s2 /4) 2 1 − (s2 /4) s3 s3 ≤S≤ . 8 6 31

(5.15)

Then it follows from (5.11) and (5.13) that D1 g(x) ≥ R, and so with (5.14) we obtain (5.9). sc Now consider (5.10). We have that Dij g(x) = 0 unless x ∈ W(α) with |x2 | ≤ 1+c 2 x1 , so it suffices to consider that case. First we consider D11 g(x). Now differentiating in (5.11) we obtain D11 g(x) =

RSx1 RS − (D1 g(x) − R) g(x) − Rx1 (g(x) − Rx1 )2
RS = (g(x) − Rx1 )2 − RSx21 , 3 (g(x) − Rx1 )

(5.16)

using (5.11) once more. Then from (5.16), using the bounds in (5.14), (5.15), (5.13), and (5.4), we obtain " # 2    3   2 s 2 (s/2)(s3 /8) s2 2s 2 g(x) − − 1 g(x) D11 g(x) ≥ ((s/2)g(x))3 4 3 6 s   s3 /2 s/2 s2 s4 4 − · 2 ≥ ((1/4) − (1/2)), ≥ g(x) 4 9 s g(x) which together with (5.8) yields the lower bound in (5.10) in the case i = j = 1; the upper bound follows from (5.16) similarly. The other cases of (5.10) follow by analogous but tedious calculations, which we omit. 

5.2

Proofs of Theorems 2.2 and 2.3

Next we give some basic properties of the process (g(ξt∗ ))t∈Z+ . Lemma 5.2 Suppose that (A1), (A2) and (2.3) hold. Then for some B ∈ (0, ∞) ∗ P[|g(ξt+1 ) − g(ξt∗ )| ≤ B] = 1.

(5.17)

Moreover, there exists C ∈ (0, ∞) such that for all x ∈ W(α) ∗ |E[g(ξt+1 ) − g(ξt∗ ) | ξt∗ = x]| < Ckxk−1 .

(5.18)

Finally, there exists ε > 0 such that for all x ∈ W(α) ∗ E[(g(ξt+1 ) − g(ξt∗ ))2 | ξt∗ = x] ≥ ε.

(5.19)

Proof. Taylor’s theorem implies that for x ∈ W(α), for any y g(x + y) − g(x) = y · ∇g(x + ηy) ∗ for some η = η(y) ∈ (0, 1). Taking x = ξt∗ and y = ξt+1 − ξt∗ we then obtain ∗ ∗ g(ξt+1 ) − g(ξt∗ ) = (ξt+1 − ξt∗ ) · ∇g(z), ∗ where z = ξt∗ + η(ξt+1 − ξt∗ ). Then from (5.20) we have ∗ ∗ |g(ξt+1 ) − g(ξt∗ )| ≤ kξt+1 − ξt∗ k

32

(5.20)

a.s., using (5.7). Then from (4.3) we obtain (5.17). Similarly, from (5.20) with (5.7) we obtain ∗ ∗ |E[g(ξt+1 ) − g(ξt∗ ) | ξt∗ = x]| ≤ 2kE[ξt+1 − ξt∗ | ξt∗ = x]k,

and then (4.5) implies (5.18). Finally, using (A1) we have from (5.20) that for x ∈ W(α) ∗ E[(g(ξt+1 ) − g(ξt∗ ))2 | ξt∗ = x] ≥ κ[k1 e1 · ∇g(z1 )]2 ,

where z1 = x + η1 k1 e1 , for some η1 ∈ (0, 1), so z1 ∈ W(α). Then it follows from (5.9) that ∗ E[(g(ξt+1 ) − g(ξt∗ ))2 | ξt∗ = x] ≥ κk12 ε2 > 0,

completing the proof.



The previous lemma will enable us to verify that g(ξt∗ ) satisfies the conditions of Lemma 3.5. This is the next result. Lemma 5.3 Suppose that (A1), (A2) and (2.3) hold. For A > 0 large enough there exist C, D ∈ (0, ∞), r > 1, and p0 > 0 such that for any t ∈ Z+ , on {υA > t}, (3.2), (3.3), and (3.4) hold for Yt = g(ξt∗ ). 2r Proof. Let Yt = g(ξt∗ ), t ∈ Z+ . We need to estimate E[Yt+1 − Yt2r | ξt∗ = x] for r ≥ 1. We will repeatedly use the Taylor expansion of y 2r for r ≥ 1. Let r ≥ 1. For y > 0 and δ with |δ| ≤ B we have that there exists η = η(r, y) for which

(y + δ)2r − y 2r = 2rδy 2r−1 + r(2r − 1)δ 2 (y + ηδ)2r−2 = 2rδy 2r−1 + r(2r − 1)δ 2 y 2r−2 + o(y 2r−2 ).

(5.21)

We now establish (3.3). Let r = 1 in (5.21) to obtain 2 E[Yt+1 − Yt2 | ξt∗ = x] ≥ 2g(x)E[Yt+1 − Yt | ξt∗ = x] ≥ −2Cg(x)kxk−1 ,

by (5.18). Then (5.2) completes the proof of (3.3). Next we consider (3.4). From the r > 1 case of (5.21) we have that 2r − Yt2r | ξt∗ = x] E[Yt+1 ≤ 2rg(x)2r−1 E[Yt+1 − Yt | ξt∗ = x] + r(2r − 1)E[(Yt+1 − Yt )2 | ξt∗ = x](g(x) + B)2r−2 ≤ rg(x)2r−1 (2Ckxk−1 + (2r − 1)g(x)−1 (1 + Bg(x)−1 )2r−2 ) ≤ rg(x)2r−2 (2C + (2r − 1)(1 + Bg(x)−1 )2r−2 ),

using (5.2). Thus (3.4) is satisfied for r > 1 provided that g(x) remains bounded away from 0, which is indeed the case on {υA > t}, A > 0. Finally we establish (3.2). From (5.19) we have that E[(Yt+1 − Yt )2 | ξt∗ = x] ≥ ε > 0. Applying (5.21) we have 2r E[Yt+1 − Yt2r | ξt∗ = x] ≥ rg(x)2r−2 (−2Cg(x)kxk−1 + (2r − 1)ε) + o(g(x)2r−2 ),

33

which is non-negative on {υA > t}, for A sufficiently large provided that r > Cε−1 + (1/2).



Proof of Theorem 2.2. Suppose α ∈ (0, π/2). First we show existence of some polynomial moment of τα . We apply Lemma 4.4 in the case i = 4, β = α, so that W4 (β) = W(α) and τi (β) = τα . Then given ξ0 = x we have from Lemma 4.4 that for some γ, C ∈ (0, ∞), where γ does not depend on x, Z ∞ Z ∞ Z ∞ s s 1/s E[τ | ξ0 = x] = P[τ > r]dr = P[τ > r ]dr ≤ C r−γ/s dr < ∞, 0

0

0

0

provided s ∈ (0, γ ). Finally, Lemma 5.3 implies that we can apply Lemma 3.5 with Yt = g(ξt∗ ). It follows that for some A, p ∈ (0, ∞) we have E[υAp | ξ0 = x] = ∞ for all x ∈ W(α) with kxk (hence g(x), by (5.2)) sufficiently large. But by definition of g and Ξ∗ we have that, a.s., τα ≥ n0 (υA − 1) for any A > 0. Hence E[ταp | ξ0 = x] = ∞ for all x ∈ W(α) with kxk sufficiently large. Then (A1) extends this to all x ∈ W(α).  Remark. The difficulty with extending the overlapping quadrant argument of Section 4 to show existence of moments for α ≥ π/2 (i.e., the half-plane or greater) is that the constant C in Lemma 4.4 depends upon x, and so some control is required over the location of Ξ on its exit from each quadrant Qi . While we believe that such an argument is possible using similar techniques to those employed here, for reasons of space we have not attempted to write down the details here. To prove Theorem 2.3, we use essentially the same argument as in the non-existence part of Theorem 2.2, but need more explicit estimates. In particular, we obtain a refinement of the lower bound in (5.18) that depends explicitly upon the constant d in (2.4). In order to do this (in Lemma 5.4 below) we replace the first-order Taylor expansion used in the proof of Lemma 5.2 with a second-order expansion. Also, we work with ξt itself rather than ξt∗ . Lemma 5.4 Suppose that (A1) and (A2) hold, and that α ∈ (0, π/2). Suppose that (2.4) holds for some d > 0. Then there exist ε, C ∈ (0, ∞), not depending on d, such that for all x ∈ W(α) E[g(ξt+1 ) − g(ξt ) | ξt = x] ≥ kxk−1 (εd − C).

Proof. Recall that we write ξt+1 − ξt = (θ1 (ξt ), θ2 (ξt )) in Cartesian components. Then conditional on ξt = x, Taylor’s theorem gives X 1X g(ξt+1 ) − g(ξt ) = θi (x)Di g(x) + θi (x)θj (x)Dij g(z) 2 i,j i

for some z ∈ R2 . Then taking expectations and using (2.4), (5.10), and (A2), we have E[g(ξt+1 ) − g(ξt ) | ξt = x] ≥ Then (5.9) completes the proof.

d + o(1) C D1 g(x) − . kxk kxk



Proof of Theorem 2.3. Now by Lemma 5.4, we can apply Lemma 3.5 to Yt = g(ξt ), analogously to the proof of the non-existence part of Theorem 2.2.  34

6

Infinite exit times: supercritical case

The main aim of this section is to prove Theorem 2.4(i). To do this we demonstrate transience, that is, P[τα = ∞] > 0. At the end of the section we also give a proof of Theorem 2.5. For the moment we restrict our attention to the quadrant W(π/4). Recall (3.12). We use the Lyapunov function defined for ε > 0 and all x ∈ R2 by hε (x) = hε (r, ϕ) := r

−2ε

−1

(cos(2ϕ))

2−2ε

= kxk

−1

(f2 (x))

(x21 + x22 )1−ε = . x21 − x22

(6.1)

Then hε is positive in the interior of the wedge W(π/4) and blows up on the boundary ∂W(π/4). For s > 0 define the unbounded region Γε (s) := {x ∈ R2 : 0 < hε (x) ≤ s}. Then for t ≥ s > 0, Γε (s) ⊆ Γε (t) ⊆ W(π/4), and for any s > 0, x1 → ∞ and x2 → ∞ as kxk → ∞ along any path in Γε (s). Also, given x ∈ Γε (s) we have from (6.1) that kxk−2 ≤ (f2 (x))−1 ≤ skxk2ε−2 .

(6.2)

We first show that when (2.5) holds, (hε (ξt ))t∈Z+ is a supermartingale on Γε (s) for any suitably small ε, s > 0. This is the next result. Lemma 6.1 Suppose that (A2) holds. Suppose that for some β ∈ (0, 1), c > 0, δ > 0, and A0 > 0, (2.5) holds. Then there exist ε > 0, s > 0 such that for all x ∈ Γε (s) E[hε (ξt+1 ) − hε (ξt ) | ξt = x] ≤ 0. Proof. Taylor’s theorem with Lagrange form for the remainder implies that for x ∈ Γε (s), hε (x + y) = hε (x) +

X

yj (Dj hε )(x) +

j

1 X σ y (Dσ hε )(x + ηy), 2

(6.3)

σ:|σ|=2

for some η = η(y) ∈ (0, 1), for any y with kyk ≤ b. Directly from (6.1) we obtain 2(1 − ε)x1 (x21 + x22 )−ε 2x1 (x21 + x22 )1−ε , and − x21 − x22 (x21 − x22 )2 2(1 − ε)x2 (x21 + x22 )−ε 2x2 (x21 + x22 )1−ε . + D2 hε (x) = x21 − x22 (x21 − x22 )2 D1 hε (x) =

(6.4)

Since (recalling that er (ϕ), e⊥ (ϕ) are the radial, transverse unit vectors at x) µ1 (x) = (µ(x) · er (ϕ)) cos ϕ − (µ(x) · e⊥ (ϕ)) sin ϕ, and µ2 (x) = (µ(x) · er (ϕ)) sin ϕ + (µ(x) · e⊥ (ϕ)) cos ϕ, it follows from (6.4) that µ1 (x)D1 hε (x) + µ2 (x)D2 hε (x) 35

(6.5)

4x1 x2 (x21 + x22 )(1/2)−ε 2ε(x21 + x22 )(1/2)−ε (µ(x) · e (ϕ)) + (µ(x) · e⊥ (ϕ)) r x21 − x22 (x21 − x22 )2
2ε(x21 + x22 )(1/2)−ε −1 −1 µ(x) · e (ϕ) − 2ε x x f (x) µ(x) · e (ϕ) . =− r 1 2 2 ⊥ x21 − x22 =−

(6.6)

Also, differentiating in (6.4) and using (6.2) we obtain that for any x ∈ Γε (s), as kxk → ∞ sup Dσ hε (x) = O(hε (x)kxk−2+4ε ).

(6.7)

σ:|σ|=2

Note that by (6.2), |x1 x2 f2 (x)−1 | = O(kxk2ε ) for x ∈ Γε (s). Now taking y = ξt+1 − ξt in (6.3), conditioning on ξt = x, and taking expectations, we obtain from (6.6) and (6.7) that for x ∈ Γε (s), as kxk → ∞, E[hε (ξt+1 ) − hε (ξt ) | ξt = x] ≤−


2ε(x21 + x22 )(1/2)−ε µ(x) · er (ϕ) − Csgn(x1 x2 )kxk2ε µ(x) · e⊥ (ϕ) + O(hε (x)kxk−2+4ε ). 2 2 x1 − x2 (6.8)

Now from (2.5) we have that for x ∈ Γε (s), sgn(x1 x2 )kxk2ε µ(x) · e⊥ (ϕ) ≤ o(kxk2ε−β−δ ). Using this bound in (6.8), with (2.5) again, we have E[hε (ξt+1 ) − hε (ξt ) | ξt = x]


≤ −2εhε (x)kxk−1 (c + o(1))kxk−β + o(kxk2ε−β−δ ) + O(kxk4ε−1 ) .

In particular, since β ∈ (0, 1) and δ > 0, we can take ε > 0 small enough so that E[hε (ξt+1 ) − hε (ξt ) | ξt = x] ≤ −2(c + o(1))εhε (x)kxk−1−β ≤ 0, for all x ∈ Γε (s) with kxk large enough. Also, for x ∈ Γε (s) we have from (6.2) that kxk ≥ s−1/(2ε) . So taking s small enough, the result follows.  Proof of Theorem 2.4(i). First we prove part (i) of the theorem for the quadrant case, α = π/4. In this case, Lemma 6.1 shows that hε is a supermartingale in Γε (s) for ε, s small enough. Also, by the boundedness of the jumps of Ξ, P[ξt+1 ∈ W(π/4) | ξt ∈ Γε (s)] = 1, for s small enough. Moreover, inf

x∈W(π/4)\Γε (s)

hε (x) ≥ s,

by definition of Γε , while for x ∈ Γε , hε (x) can be arbitrarily small (along the line x2 = 0 for instance). So Lemma 3.1 implies that for any x ∈ Γε (s), P[min{t ∈ Z+ : ξt ∈ / Γε (s)} = ∞ | ξ0 = x] > 0. 36

This in turn implies that for any x ∈ Γε (s) P[τπ/4 = ∞ | ξ0 = x] > 0.

(6.9)

Now we extend this argument to other angles α. For α ∈ (0, π], let Lα denote the linear transformation of R2 defined by   cos α 0 Lα = . 0 sin α Then Lα W(π/4) = W(α). So consider the random walk Ξ in wedge W(α). Given that condition (2.5) holds in 2 −1 W(α), the same condition also holds for the walk L−1 α Ξ on W(π/4). Under Lα , Z will be deformed, but an inspection of the argument in this section shows that it did not rely on the underlying state-space being Z2 : any countable state-space suffices. Hence the argument for (6.9) implies that for small enough ε, s and for any x ∈ Lα Γε (s) P[τα = ∞ | ξ0 = x] > 0. It remains to extend this result to all x ∈ W(α), not just x ∈ Lα Γε (s). However, condition (A1) provides this extension.  Proof of Theorem 2.5. For part (i), suppose that µ(x) = (c/x1 )e1 . For α ∈ (0, π/2), this implies that µ(x) = O(kxk−1 ) for x ∈ W(α). Hence Theorem 2.1 applies. For α = π/2, however, results of Lamperti [17] applied to the horizontal component of Ξ show that it is transient for c > 0 sufficiently large. Part (ii) follows from Theorem 2.4 when α ∈ (0, π/2), and again from [17] when α = π/2. 

7 7.1

Existence of moments: subcritical case and explicit exponents Preliminaries

In this section, we deal with the existence of moments of τα , the first exit time from the wedge W(α), primarily in the subcritical case. The technique in Section 4, used to prove the existence of moments in Theorem 2.2, does not give sharp exponents, since γ in Lemma 4.4 depends on the δ in Lemma 4.1, which depends upon the ε in Lemma 4.2, and these results assume very general conditions on Ξ. In addition, the method in Section 4 works only for α < π/2. Thus in this section we use a sharper technique, based on Lemma 3.4, with the harmonic Lyapunov functions fw defined at (3.7). The disadvantage of this technique is that it is best adapted to the subcritical case, or when there are additional conditions imposed on the drift field.

37

For a given α, we fix w ∈ (0, π/(2α)). Then W(α) sits inside the larger wedge W(π/(2w)). ˜ = (ξ˜t )t∈Z+ identical to Ξ on W(α) but from x ∈ Define the modified random walk Ξ / W(α) jumps directly to 0 and remains there, i.e. ξ˜t := ξt 1{t≤τα } ; thus ξ˜t = 0 for t ≥ τα + 1. Introduce the nonnegative process (Xt )t∈Z+ defined for t ∈ Z+ by Xt := fw (ξ˜t )1/w . For B ∈ (0, ∞) set τ˜α,B := min{t ∈ Z+ : Xt ≤ B}. The next result will be the basis for our results in this section. Lemma 7.1 Suppose that (A1) and (A2) hold. Fix α ∈ (0, π] and w ∈ (0, π/(2α)). Suppose that there exist p0 > 0, A0 , C ∈ (0, ∞) such that E[fw (ξt+1 )2p0 /w − fw (ξt )2p0 /w | ξt = x] ≤ −Cfw (x)(2p0 −2)/w ,

(7.1)

for all x ∈ W(α) with kxk ≥ A0 . Then for any p ∈ [0, p0 ) and any x ∈ W(α), E[ταp | ξ0 = x] < ∞. Proof. Let w ∈ (0, π/(2α)). Suppose that there exist p0 > 0, A0 , C ∈ (0, ∞) such that (7.1) holds for all x ∈ W(α) with kxk ≥ A0 . First observe that ξ˜τα +1 = 0 so that Xτα +1 = 0; hence, for any B > 0, τ˜α,B ≤ τα + 1 a.s.. In other words, for any B > 0 and t ∈ Z+ {˜ τα,B > t} ⊆ {τα ≥ t}. Thus we obtain {˜ τα,B > t} ⊆ {τα > t} ∪ {τα = t} = {τα > t, ξt ∈ W(α)} ∪ {τα = t, ξt ∈ W(π/(2w)) \ W(α)},

(7.2)

for all B sufficiently large, using (A2) and the fact that by definition {˜ τα,B > t} ⊆ {kξt k > B}. We consider the two events in the disjoint union in (7.2) in turn. On {τα > t} we have ξt = ξ˜t and ξt+1 = ξ˜t+1 . Thus it follows from (7.1) that there exists C 0 ∈ (0, ∞) such that for all x ∈ W(α) with kxk ≥ A0 , on {τα > t} 2p0 E[Xt+1 − Xt2p0 | ξt = x] ≤ −C 0 Xt2p0 −2 .

(7.3)

˜ that for any p0 On {τα = t}, we have by the definition of the process Ξ 2p0 E[Xt+1 − Xt2p0 | ξt = x] = −Xt2p0 ≤ −B 2 Xt2p0 −2 ,

on {˜ τα,B > t} for any x ∈ W(π/(2w)) \ W(α), since τ˜α,B > t implies that Xt2 ≥ B 2 . Thus we conclude that for some C 0 ∈ (0, ∞), (7.3) holds for all x ∈ W(π/(2w)) on {˜ τα,A0 > t}. Hence we apply Lemma 3.4 with Yt = Xt to deduce existence of moments for τ˜α,A , and we obtain p E[˜ τα,A | ξ0 = x] < ∞,

38

(7.4)

for any p ∈ [0, p0 ), A ≥ A0 , and x ∈ W(α). It remains to deduce the corresponding result for τα . On {τα ≥ t}, ξ˜t = ξt and so by (3.19) kξt k ≥ Xt ≥ ε1/w α,w kξt k.

(7.5)

Recall that τ˜α,B ≤ τα + 1. Thus either τ˜α,B = τα + 1 or τ˜α,B ≤ τα . On {˜ τα,B ≤ τα } we have from (7.5) that −1/w −1/w kξτ˜α,B k ≤ εα,w Xτ˜α,B ≤ εα,w B.

On the other hand, on {˜ τα,B = τα + 1}, clearly τα ≤ τ˜α,B . Recalling the definition of τα,A −1/w from (3.5), it follows that for all A ≥ Bεα,w , a.s., τα,A ≤ τ˜α,B .

Then with (7.4) we obtain that for all x ∈ W(α) and all A sufficiently large p E[τα,A | ξ0 = x] < ∞. Condition (A1) then extends the result to τα . This completes the proof.  In the remainder of this section we verify (7.1) for x ∈ WA (α) for various values of α, p0 and appropriately large A > 0.

7.2

Proof of Theorem 2.6(i)

The next result, with Lemma 7.1, will enable us to deduce Theorem 2.6(i). Lemma 7.2 Suppose that (A2) holds. Let α ∈ (0, π]. Suppose that for some σ 2 ∈ (0, ∞), for x ∈ W(α) as kxk → ∞ kµ(x)k = o(kxk−1 );

M12 (x) = o(1);

M11 (x) = σ 2 + o(1);

M22 (x) = σ 2 + o(1). (7.6)

Then for any w ∈ (0, π/(2α)) and any γ ∈ (0, 1), there exist constants A, C ∈ (0, ∞) for which for all x ∈ WA (α) E[fw (ξt+1 )γ − fw (ξt )γ | ξt = x] ≤ −Cfw (x)γ−(2/w) .

(7.7)

Proof. Let w ∈ (0, π/(2α)). For x ∈ W(α), we have that (3.19) holds. Then by Lemma 3.8 with (7.6) we have that for γ ∈ R 1 E[fw (ξt+1 )γ − fw (ξt )γ | ξt = x] = γ(γ − 1)w2 σ 2 fw (x)γ−2 r2w−2 (1 + o(1)), 2

(7.8)

for all x ∈ W(α), as kxk → ∞. From (3.19) we have that for x ∈ W(α) w −1 ε−1 ≥ 1. α,w ≥ r fw (x)

(7.9)

It follows from (7.8) and (7.9) that for γ ∈ (0, 1) and some C ∈ (0, ∞), for x ∈ W(α) with kxk sufficiently large E[fw (ξt+1 )γ − fw (ξt )γ | ξt = x] ≤ −Cfw (x)γ−2 r2w−2 ≤ −C 0 fw (x)γ−(2/w) , 39

using (3.19), as required.



Proof of Theorem 2.6(i). Let w ∈ (0, π/(2α)). For γ ∈ (0, 1), take p0 = γw/2. Then on {t < τα }, Xt2p0 = fw (ξt )γ and Xt2p0 −2 = fw (ξt )γ−(2/w) . Thus Lemma 7.2 says that for x ∈ W(α) with kxk sufficiently large (7.1) holds for γ ∈ (0, 1) and w ∈ (0, π/(2α)). Then Lemma 7.1 implies that for any x ∈ W(α), E[ταp | ξ0 = x] < ∞ for all p ∈ [0, p0 ). Since γ ∈ (0, 1) can be taken as close to 1 as we like, and w < π/(2α) as close to its upper bound, it follows that p can be taken as close to π/(4α) as we like. 

7.3

Proof of Theorem 2.7

Lemma 7.3 Suppose that (A2) holds. Let α ∈ (0, π/2). Suppose that there exist σ 2 ∈ (0, ∞) and λ < 0 for which (2.10) holds. Then for any w ∈ (1, π/(2α)) there exist γ > 1 and A, C ∈ (0, ∞) for which (7.7) holds for all x ∈ WA (α). Proof. For α < π/2 we can choose w ∈ (1, π/(2α)). Suppose that γ > 1. Now from Lemma 3.8 with (7.9) we have that if (2.10) holds, for x ∈ W(α), as kxk → ∞ E[fw (ξt+1 )γ − fw (ξt )γ | ξt = x]   γ−1 2 γ−1 w−2 −1 w = γσ fw (x) wr fw (x) r w(1 + λ cos(2(w − 1)ϕ)) λ(w − 1) cos((w − 2)ϕ) + 2 + o(rγw−2 ). Now since w > 1, γ > 1, and λ ∈ [−1, 0), we obtain from (7.9) and (3.20) that   E[fw (ξt+1 )γ − fw (ξt )γ | ξt = x] ≤ γσ 2 fw (x)γ−1 wrw−2 λ(w − 1)εα,w + (γ − 1)wε−1 α,w + o(1) .

Hence provided that γ < 1 + |λ|w−1 (w − 1)ε2α,w , the term in square brackets in the last display is negative for all x ∈ W(α) with kxk sufficiently large. This proves the lemma.  Proof of Theorem 2.7. This time we apply Lemma 7.1 with Lemma 7.3 and p0 = γw/2 (where γ > 1 is as in Lemma 7.3), where we can take w ∈ (0, π/(2α)) arbitrarily close to π/(2α), and so we can take p0 = (π/(4α)) + ε for some ε > 0. 

7.4

Proof of Theorem 2.8

Lemma 7.4 Suppose that (A2) holds. Let α ∈ (0, π). Suppose that (2.4) holds for d < 0, and that M(x) = σ 2 I + o(1) for some σ 2 ∈ (0, ∞). Then for any w ∈ [1/2, π/(2α)) there exist γ > 1 and A, C ∈ (0, ∞) for which (7.7) holds for all x ∈ WA (α). Proof. For α ∈ (0, π) we can choose w ∈ [1/2, π/(2α)). From Lemma 3.8, under the conditions of the lemma we obtain that for γ ∈ R and x ∈ W(α), as kxk → ∞, E[fw (ξt+1 )γ − fw (ξt )γ | ξt = x]   1 −1 w 2 γ−1 w−2 = γfw (x) wr d cos((w − 1)ϕ) + (γ − 1)fw (x) r wσ + o(1) . 2 40

(7.10)

Suppose γ > 1. Then since d < 0, we obtain from (7.10) with (3.20) and (7.9) that   1 γ γ γ−1 w−2 −1 2 dεα,w + (γ − 1)εα,w wσ + o(1) . E[fw (ξt+1 ) − fw (ξt ) | ξt = x] ≤ γfw (x) wr 2 In particular, if we take γ < 1 + 2|d|ε2α,w w−1 σ −2 the term in square brackets in the last display is negative for kxk sufficiently large. This proves the lemma.  Proof of Theorem 2.8. This time we apply Lemma 7.1 with Lemma 7.4 and p0 = γ(w/2) (where γ > 1 is as in Lemma 7.4), where we can take w ∈ (0, π/(2α)) arbitrarily close to π/(2α), and so we can take p0 = (π/(4α)) + ε0 for some ε0 > 0. 

7.5

Proof of Theorem 2.9

Lemma 7.5 Suppose that (A2) holds. Let α ∈ (0, π). Suppose that (2.4) holds for d > 0 and that M(x) = σ 2 I + o(1) for some σ 2 ∈ (0, ∞). Suppose that w ∈ [1/2, π/(2α)) and γ ∈ (0, 1) are such that 2d (7.11) 1−γ > 2 . σ w Then there exist constants A, C ∈ (0, ∞) for which (7.7) holds for all x ∈ WA (α). Proof. Let w ∈ (0, π/(2α)). We have that (7.10) holds, this time with d > 0. In the case γ ∈ (0, 1), we obtain from (7.10) with (7.9) that   1 γ γ γ−1 w−2 2 d + (γ − 1)wσ + o(1) . E[fw (ξt+1 ) − fw (ξt ) | ξt = x] ≤ γfw (x) wr 2 Thus if (7.11) holds, the term in square brackets in the last display is negative for kxk sufficiently large. This completes the proof.  Proof of Theorem 2.9. Again we apply Lemma 7.1. For γ ∈ (0, 1), take p0 = γw/2. This time Lemma 7.5 says that (7.1) holds for x ∈ W(α) with kxk sufficiently large, provided that   w d 2d (w/2) = − 2 . 0 < p0 = γw/2 < 1 − 2 σ w 2 σ Taking w ∈ (0, π/(2α)), we can achieve any p0 with 0 < p0
0, (2.7) holds. Then for any w ∈ (0, π/(2α)) and any γ ∈ (0, ∞) there exist A, C ∈ (0, ∞) for which for all x ∈ WA (α) E[fw (ξt+1 )γ − fw (ξt )γ | ξt = x] ≤ −Cfw (x)γ−((1+β)/w) . Proof. Let w ∈ (0, π/(2α)). From Lemma 3.8 and (6.5), we have that for γ > 0, for x ∈ W(α), as kxk → ∞, E[fw (ξt+1 )γ − fw (ξt )γ | ξt = x] = γfw (x)γ−1 wrw−1 (cos(wϕ)µ(x) · er (ϕ) − sin(wϕ)µ(x) · e⊥ (ϕ)) + O(rγw−2 ) ≤ γfw (x)γ−1 wrw−1−β [dεα,w + o(1)],

using (2.7) and (3.19). Hence the coefficient of fw (x)γ−1 rw−1−β is negative for d < 0, γ ∈ (0, ∞) and kxk large enough. Then (7.9) completes the proof.  Proof of Theorem 2.4(ii). Fix w ∈ (0, π/(2α)). We use an argument analogous to that of Lemma 7.1, using a different definition of Xt . In particular, take Xt = fw (ξ˜t )(1+β)/(2w) . Setting p0 = γw/(1 + β), we have Xt2p0 = fw (ξ˜t )γ and Xt2p0 −2 = fw (ξ˜t )γ−((1+β)/w) . Then, given Lemma 7.6, the argument as in Lemma 7.1 implies that E[ταp | ξ0 = x] < ∞ for any x ∈ W(α) and p < p0 . Since γ ∈ (0, ∞) is arbitrary, the result follows. 

8 8.1

Non-existence of moments: subcritical case Preliminaries

The aim of this section is to prove the non-existence part of Theorem 2.6. The method used in Section 5 to prove the non-existence of moments in Theorem 2.2 is not sharp enough to produce the correct exponents that we require for Theorem 2.6. Also, it required α < π/2. Instead, we work once more with the harmonic Lyapunov functions of Section 3.2. Let α ∈ (0, π]. Throughout this section we will take w = π/(2α). We will again be interested in fw (ξt )γ , γ ∈ R, this time in the wedge W(α). Due to difficulties with estimating the behaviour of fw (ξt )γ near the boundary of the wedge W(α) (cf Lemma 3.8), we cannot apply the non-existence theorems from [2] (such as Lemma 3.5 above). Thus we need a different approach. A key step in this section is a good-probability lower-bound on the time taken to leave a wedge; this is Lemma 8.3 below. A similar approach is used in [4], where Lemma 6.2 deals with a special case of a random walk in a quarter-plane. In any case, to show non-existence of moments something like Lemma 8.3 is required; analogous lemmas are needed for the general results of [2, 18]. In fact we use the Lyapunov function fˆw where for x ∈ R2 fˆw (x) := fw (x)1{x∈W(α)} . 42

The first task of this section is to show that in the subcritical case fˆw (ξt )γ is a submartingale for any γ > 1. We recall that for w = π/(2α), Lemma 3.8 applies for fw only in a wedge smaller than W(α). The next result will allow us to overcome this obstacle. For K > 0 we use the notation  (8.1) W K (α) := x ∈ W(α) : fw (x) ≥ K −1 kxkw−1 . Lemma 8.1 Let α ∈ (0, π] and w = π/(2α). Suppose that (A2) holds, and that for some v ∈ (0, ∞), as kxk → ∞ in W(α), M12 (x) = o(1);

M11 (x) ≥ v + o(1);

M22 (x) ≥ v + o(1).

(8.2)

Suppose that kµ(x)k → 0 as kxk → ∞ in W(α). Then there exist A, K ∈ (0, ∞) such that E[fˆw (ξt+1 ) − fˆw (ξt ) | ξt = x] ≥ 0 for all x ∈ WA (α) \ W K (α). Proof. For some K > 0, take x ∈ W(α) \ W K (α). Then by (8.1) we have that fw (x) ≤ K −1 rw−1 and hence cos(wϕ) ≤ K −1 r−1 . Thus x is close to the boundary ∂W(α). In order to estimate the expected change in fw on a jump of Ξ started from x, we introduce the notation U (x) := {y ∈ W(α) : fw (y) ≥ fw (x)}. ˆ = fˆw (ξt+1 ) − fˆw (ξt ). Then we decompose We use the shorthand ∆ ˆ | ξt = x] = E[∆1 ˆ {ξ ∈U (x)} | ξt = x] + E[∆1 ˆ {ξ ∈U E[∆ t+1 t+1 / (x)} | ξt = x].

(8.3)

We now estimate the two terms in (8.3) separately. First, since fˆw (ξt+1 ) ≥ 0 a.s., we have −1 w−1 ˆ {ξ ∈U E[∆1 r , t+1 / (x)} | ξt = x] ≥ −fw (x) ≥ −K

(8.4)

by (8.1). We now proceed to bound the first term on the right-hand side of (8.3). For a random variable X with P[|X| < m] = 1, we have that P[m|X| > X 2 ] = 1 and so E|X| ≥ m−1 E[X 2 ]. Moreover, we have that E[X1{X≥0} ] = (E[X] + E|X|)/2. So we conclude that E[X1{X≥0} ] ≥ (E[X] + m−1 E[X 2 ])/2. (8.5)

Now write ∆ = fw (ξt+1 ) − fw (ξt ). Then {∆ ≥ 0, ξt = x} = {ξt+1 ∈ U (x), ξt = x}. Hence applying the elementary inequality (8.5) with X = ∆ and using the bound (3.13) gives, for some C ∈ (0, ∞) and all x ∈ W(α) with kxk large enough E[∆1{ξt+1 ∈U (x)} | ξt = x] 1 ≥ E[fw (ξt+1 ) − fw (ξt ) | ξt = x] + Ckxk1−w E[(fw (ξt+1 ) − fw (ξt ))2 | ξt = x]. 2

Using the fact that kµ(x)k = o(1) and (8.2), we then obtain from (3.14) and (3.15) that there exists C > 0, not depending on K, such that E[∆1{ξt+1 ∈U (x)} | ξt = x] ≥ Ckxkw−1 , 43

(8.6)

for all x ∈ W(α) with kxk large enough. Now it follows from Lemma 3.9 that we can ˆ in (8.6). Then the claimed result follows from (8.3), with (8.4) and (8.6), replace ∆ by ∆ by taking K large enough.  Now we are ready to prove that fˆw (ξt )γ is a submartingale for γ > 1. Lemma 8.2 Suppose that (A2) holds. Suppose that for some σ 2 ∈ (0, ∞), for x ∈ Z2 , as kxk → ∞, (2.9) holds. Let α ∈ (0, π]. Then for w = π/(2α) and any γ > 1, there exists A ∈ (0, ∞) for which, for all x ∈ WA (α), E[fˆw (ξt+1 )γ − fˆw (ξt )γ | ξt = x] ≥ 0.

(8.7)

Proof. It suffices to suppose γ ∈ (1, 2]. For x ∈ WA (α) \ W K (α), the result (8.7) follows directly from Lemma 8.1 for K large enough. So it remains to consider x ∈ WA (α)∩W K (α). ˆ = fˆw (ξt+1 ) − fˆw (ξt ), we have that conditional on ξt = x Now, writing ∆ " !γ # ˆ ∆ ˆ γ − fˆw (x)γ = fw (x)γ 1+ fˆw (ξt+1 )γ − fˆw (ξt )γ = (fˆw (x) + ∆) −1 . (8.8) fˆw (x) To obtain a lower bound, we will use the inequality for γ ∈ (1, 2] and L ∈ (0, ∞) 1 (1 + x)γ ≥ 1 + γx + (1 + L)γ−2 γ(γ − 1)x2 2

(8.9)

ˆ ≤ Lfˆw (x). The first for x ∈ [−1, L]. To apply (8.9) in (8.8) we need that −fˆw (x) ≤ ∆ ˆ inequality here is automatically satisfied by definition since fw (ξt+1 ) ≥ 0 a.s.. For the second inequality, we have for x ∈ W K (α) from (3.13) and (8.1) that on {ξt = x} ˆ ≤ Ckxkw−1 ≤ CKfw (x) = CK fˆw (x). k∆k So taking L > CK shows that we can indeed apply (8.9) in (8.8) to obtain, for some C ∈ (0, ∞), for any x ∈ WA (α) ∩ W K (α), conditional on ξt = x, ˆ + Cfw (x)γ−2 ∆ ˆ 2. fˆw (ξt+1 )γ − fˆw (ξt )γ ≥ γfw (x)γ−1 ∆ ˆ and so by Lemma 3.9 we can The right-hand side of the last display is increasing in ∆, ˆ replace ∆ by ∆ and then take expectations to obtain E[fˆw (ξt+1 )γ − fˆw (ξt )γ | ξt = x] ≥ γfw (x)γ−1 E[fw (ξt+1 ) − fw (ξt ) | ξt = x] +Cfw (x)γ−2 E[(fw (ξt+1 ) − fw (ξt ))2 | ξt = x], for some C > 0 and any x ∈ WA (α) ∩ W K (α). Now from Lemma 3.6 and the conditions on µ(x) and M(x) it follows that, for some C > 0, as kxk → ∞   E[fˆw (ξt+1 )γ − fˆw (ξt )γ | ξt = x] ≥ fw (x)γ−1 Cfw (x)−1 r2w−2 + o(rw−2 ) ,

for any x ∈ WA (α) ∩ W K (α). Then the result follows since fw (x)−1 ≥ r−w . 44



8.2

Key estimate

Now we state our key lemma for this section. As mentioned above, the idea is analogous to that used (in a simpler setting) for Lemma 6.2 in [4]. Lemma 8.3 Suppose that (A1) and (A2) hold. Suppose that for some σ 2 ∈ (0, ∞), as kxk → ∞, (2.9) holds. Let α ∈ (0, π] and w = π/(2α). Let A ∈ (0, ∞). Then there exist ε1 , ε2 > 0 such that for all x ∈ W(α) with kxk > A P[τα,A > ε1 kxk2 | ξ0 = x] ≥ ε2 cos(wϕ). To prove this result, we will make repeated use of the processes (Yt (x))t∈Z+ defined for x ∈ Z2 by Yt (x) := kξt − xk. (8.10) First note that the triangle inequality implies that kξt+1 − xk − kξt − xk ≤ kξt+1 − ξt k = kθ(ξt )k ≤ b, a.s. by (A2): the process (Yt (x))t∈Z+ has uniformly bounded jumps. The next lemma gives more information about these jumps. For notational ease, for x ∈ Z2 and C ∈ (1, ∞) write S(x; C) := {y ∈ Z2 : C −1 kxk ≤ kyk ≤ Ckxk};

R(x; C) := {y ∈ Z2 : ky − xk ≥ C −1 kxk}.

Lemma 8.4 Suppose that (A1) and (A2) hold. Suppose that for some σ 2 ∈ (0, ∞), as kxk → ∞, (2.9) holds. Then for any x ∈ Z2 and any C ∈ (1, ∞) we have that as kxk → ∞ sup E[Yt+1 (x)2 − Yt (x)2 | ξt = y] − 2σ 2 = o(1), (8.11) y∈S(x;C) 1 2 −1 −1 σ ky − xk (8.12) sup E[Y (x) − Y (x) | ξ = y] − t+1 t t = o(kxk ), 2 y∈S(x;C)∩R(x;C) E[(Yt+1 (x) − Yt (x))2 | ξt = y] − σ 2 = o(1). sup (8.13) y∈S(x;C)∩R(x;C)

Proof. Conditional on ξt = y ∈ Z2 we have that

L(Yt+1 (x) | ξt = y) = L((ky − xk2 + kθ(y)k2 + 2(y − x) · θ(y))1/2 ).

(8.14)

Now we obtain from (8.14) with (2.9) that E[Yt+1 (x)2 − Yt (x)2 | ξt = y] = E[kθ(y)k2 ] + 2E[(y − x) · θ(y)] = 2σ 2 + o(1) + o(ky − xkkyk−1 ) = 2σ 2 + o(1), for all y with C −1 kxk ≤ kyk ≤ Ckxk. This proves (8.11). Similarly, by (8.14) # " 1/2 kθ(y)k2 + 2(y − x) · θ(y) − 1 . (8.15) E[Yt+1 (x) − Yt (x) | ξt = y] = ky − xkE 1 + ky − xk2 45

The term in square brackets on the right-hand side of (8.15) can be expanded using Taylor’s theorem to obtain 1 kθ(y)k2 + 2(y − x) · θ(y) 1 4((y − x) · θ(y))2 − + O(kxk−3 ), 2 ky − xk2 8 ky − xk4 using (A2), provided that C −1 kxk ≤ ky − xk and C −1 kxk ≤ kyk ≤ Ckxk. Taking expectations of this last expression yields 1 ky − xk−2 E[kθ(y)k2 + 2(y − x) · θ(y)] 2

1 − ky − xk−4 E[(y1 − x1 )2 θ1 (y)2 + (y2 − x2 )2 θ2 (y)2 ] 2

1 − ky − xk−4 E[2(y1 − x1 )(y2 − x2 )θ1 (y)θ2 (y)] + O(kxk−3 ) 2 1 1 = ky − xk−2 (2σ 2 + o(1)) − ky − xk−2 (σ 2 + o(1)), 2 2 using (2.9), which with (8.15) gives (8.12). Finally observe that conditional on ξt = y (Yt+1 (x) − Yt (x))2 = (Yt+1 (x)2 − Yt (x)2 ) − 2Yt (x)(Yt+1 (x) − Yt (x)) = (Yt+1 (x)2 − Yt (x)2 ) − 2ky − xk(Yt+1 (x) − Yt (x)). So from (8.12) and (8.11) we obtain (8.13). This completes the proof.



Proof of Lemma 8.3. Suppose that ξ0 = x ∈ W(α). We may suppose that kxk > A0 for some arbitrary A0 ∈ (A, ∞), since if we prove that the result holds for all such x, the full result follows by (A1). Fix α0 ∈ (0, α), which we will take close to α. First suppose that x ∈ W(α0 ), so that the walk does not start too close to the boundary of the wedge W(α). Note that the shortest distance from x ∈ W(α) to the wedge boundary ∂W(α) is at least kxk sin(α − |ϕ|), and that for all x ∈ W(α0 ), ϕ ∈ (−α0 , α0 ) so this distance is at least ε0 kxk, where ε0 := sin(α−α0 ) > 0. Suppose that y ∈ Bε0 kxk/2 (x) ⊂ W(α). Note that for y ∈ Bε0 kxk/2 (x) we have ky − xk ≤ (ε0 /2)kxk, kyk ≤ (1 + (ε0 /2))kxk, and kyk ≥ (1 − (ε0 /2))kxk.

(8.16)

It then follows from (8.11) and (8.16) that for y ∈ Bε0 kxk/2 (x) E[Yt+1 (x)2 − Yt (x)2 | ξt = y] = 2σ 2 + o(1),

(8.17)

as kxk → ∞. For the rest of this proof, let κ = min{t ∈ Z+ : kξt − xk ≥ ε0 kxk/2}; so κ is the first exit time of Ξ from Bε0 kxk/2 (x). It follows from (8.17) that for all x ∈ W(α0 ) with kxk large enough, Yt∧κ (x)2 is a nonnegative submartingale with respect to the natural filtration for Ξ, and there exists C ∈ (0, ∞) such that for all x ∈ W(α0 ) with kxk sufficiently large and for all t ∈ Z+ , E[Yt∧κ (x)2 | ξ0 = x] ≤ Ct ∧ κ ≤ Ct. 46

Then Doob’s submartingale inequality implies that there exists C ∈ (0, ∞) such that for any x ∈ W(α0 ) with kxk sufficiently large, any t ∈ Z+ , and any x > 0,   2 P max Ys∧κ (x) ≥ x ξ0 = x ≤ Ct/x. 0≤s≤t

So in time t = x/(2C), there is probability at least 1/2 that max0≤s≤t Ys∧κ (x) ≤ x1/2 . Noting that Yκ (x) ≥ ε0 kxk/2 a.s., and taking x = ε20 kxk2 /9, we conclude that   P max kξs − xk ≤ ε0 kxk/3 ξ0 = x ≥ 1/2. 2 0≤s≤ε0 kxk2 /(18C)

The event in the last displayed probability implies that Ξ remains in Bε0 kxk/2 (x) ⊂ W(α) for all times up until ε20 kxk2 /(18C). In other words, for any x ∈ W(α0 ) with kxk sufficiently large,   ε20 2 P τα,A ≥ kxk ξ0 = x ≥ 1/2. (8.18) 18C

This yields the statement in the lemma for x ∈ W(α0 ), for any α0 ∈ (0, α) and kxk sufficiently large. Now we need to deal with the case x ∈ W(α) \ W(α0 ). We take α0 < α but close to α, so that ε0 = sin(α − α0 ) is small. Suppose that x ∈ W(α) \ W(α0 ), and, without loss of generality, that ϕ > 0; then ϕ ∈ [α0 , α). Set ( 1 if α ≥ π/2 R := R(α; x) := , 1 ∧ [(tan α)(cos(α − ϕ))] if α ∈ (0, π/2) and then define c(x) := er (α)kxk cos(α − ϕ) − R(α; x)e⊥ (α)kxk.

When R < 1, this means c(x) = e1 kxk cos(α − ϕ) sec α lies on the principal axis of the wedge. (See Figure 3.) Note that
1/2 kc(x)k = kxk R2 + cos2 (α − ϕ) ,

(8.19)

kx − c(x)k = (R − sin(α − ϕ))kxk ≥ ε1 kxk,

(8.20)

and also x − c(x) = (R − sin(α − ϕ))kxke⊥ (α) so that

for some ε1 > 0 and all x ∈ W(α) \ W(α0 ) provided that α0 is close enough to α. Also from (8.19) we have that for some ε2 > 0 and all x ∈ W(α) \ W(α0 ) √ (8.21) ε2 kxk ≤ kc(x)k ≤ 2kxk. Now consider the concentric disks D1 (x) := BRkxk/2 (c(x)),

D2 (x) := BRkxk (c(x)).

Figure 3 shows a typical situation. Observe that the shortest distance of c(x) from the line 47

kxk cos(α − ϕ) tan(α − α0 ) x

D1 (x) kxk cos(α − ϕ)

α − α0

α0

ϕ

c(x)

α

Figure 3: The geometrical construction of c(x) and D1 (x). through 0 in the er (α0 ) direction is, when R = 1, kxk cos(α − α0 ) − kxk sin(α − α0 ) cos(α − ϕ) ≥ (1 − ε0 )kxk cos(α − α0 ), for all x ∈ W(α) \ W(α0 ). When R < 1, the corresponding distance is kxk cos(α − ϕ) sec α sin α0 . In either case, choosing α0 close enough to α, it follows that D1 (x) ⊂ W(α0 ) for all x ∈ W(α) \ W(α0 ). Moreover, for ε0 small enough, for any y ∈ D2 (x),
kyk ≥ kc(x)k − Rkxk ≥ (R2 + 1 − ε20 )1/2 − R kxk ≥ ε0 kxk, (8.22)

by (8.19). We now aim to show that there exists ε0 > 0 such that for all x ∈ W(α) \ W(α0 ) with kxk sufficiently large,   p(x) := P Ξ visits D1 (x) before R2 \ D2 (x) | ξ0 = x ≥ ε0 cos(wϕ). (8.23)

From the geometrical argument leading up to (8.22), and the jumps bound (A2), it follows that if the event in (8.23) occurs, Ξ visits a region of W(α0 ) at distance at least ε0 kxk from 0 before leaving W(α). Hence given (8.23), (8.18) yields the statement in the lemma in this case also. Thus it remains to prove (8.23). With the notation defined at (8.10), we now consider Yt (c(x)) = kξt − c(x)k for ξt in the annular region D2 (x) \ D1 (x). For any y ∈ D2 (x) \ D1 (x) we have Rkxk/2 ≤ ky − c(x)k ≤ Rkxk, so that kyk ≤ kxk + kc(x)k. This together with (8.22) and (8.21) implies that for α0 close enough to α there exist C1 , C2 ∈ (0, ∞) such that for any x ∈ W(α) \ W(α0 ) and any y ∈ D2 (x) \ D1 (x) C1 kxk ≥ kyk ≥ C2 kxk, 48

ky − c(x)k ≥ C2 kxk.

(8.24)

Hence by (8.24) and (8.21), the estimates (8.12) and (8.13) are valid for Yt (c(x)) and y ∈ D2 (x) \ D1 (x), as kxk → ∞. Thus we have that there exists δ > 0 such that for x ∈ W(α) \ W(α0 ) with kxk large enough and all y ∈ D2 (x) \ D1 (x) E[Yt+1 (c(x)) − Yt (c(x)) | ξt = y] = O(kxk−1 ), E[(Yt+1 (c(x)) − Yt (c(x)))2 | ξt = y] > δ > 0.

(8.25) (8.26)

For C ∈ (0, ∞) consider now the process (Zt )t∈Z+ defined for t ∈ Z+ by    Yt (c(x)) ; Zt := exp C R(α; x) − kxk

then by (8.20), Z0 = exp{C sin(α − ϕ)}. Then we have for t ∈ Z+ , y ∈ Z2

E[Zt+1 − Zt | ξt = y]        C ky − c(x)k E exp − (Yt+1 (c(x)) − Yt (c(x))) − 1 ξt = y . = exp C R − kxk kxk

With the fact that there exist positive constants C1 , C2 such that e−x − 1 ≥ −x + C1 x2 for all x with |x| < C2 , we then obtain that for any x ∈ W(α) \ W(α0 ) and any y ∈ D2 (x) \ D1 (x)     C E exp − (Yt+1 (c(x)) − Yt (c(x))) − 1 ξt = y kxk   C C 2 ≥ E −(Yt+1 (c(x)) − Yt (c(x))) + C1 (Yt+1 (c(x)) − Yt (c(x))) ξt = y . kxk kxk Thus by (8.25), (8.26) we can choose C large enough such that for ξ0 = x ∈ W(α) \ W(α0 ) E[Zt+1 − Zt | ξt = y] ≥ 0,

(8.27)

for all y ∈ D2 (x) \ D1 (x) with kxk large enough, and all t ∈ Z+ . Now to estimate p(x) as in (8.23), we make the sets D1 (x) and R2 \ D2 (x) absorbing. Then (using (A2)) Zt is bounded for this modified random walk, and (using (A1)) Ξ leaves D2 (x) \ D1 (x) in almost surely finite time. Thus as t → ∞, Zt converges almost surely and in L1 to some limit Z∞ and E[Z∞ | ξ0 = x] ≤ p(x) exp{CR/2} + (1 − p(x)), while by (8.27) we also have that E[Z∞ | ξ0 = x] ≥ E[Z0 ] = exp{C sin(α − ϕ)}.

Hence we obtain that there exists C ∈ (0, ∞) such that for all x ∈ W(α) \ W(α0 ) with kxk large enough C exp{C sin(α − ϕ)} − 1 ≥ CR/2 sin(α − ϕ). p(x) ≥ exp{CR/2} − 1 e −1 Now for x ∈ W(α) \ W(α0 ) we have that α − ϕ < α − α0 , where α − α0 is small. Using the fact that for a > 0, sin(ax) → a as x → 0, it follows that there exists some ε0 > 0 such that sin(x) C

sin(α − ϕ) ≥ ε0 sin(w(α − ϕ)) = ε0 cos(wϕ). −1 This proves (8.23), and so the proof of the lemma is complete.  eCR/2

49

8.3

Proof of Theorem 2.6(ii)

Now we are ready to complete the proof of Theorem 2.6(ii). Proof of Theorem 2.6(ii). Let α ∈ (0, π] and w = π/(2α). We first show that for A (w/2)+ε sufficiently large, any ε > 0, and any x ∈ WA (α), E[τα | ξ0 = x] = ∞. We proceed in a similar way to the proof of Theorem 6.1 in [4]. Let A ∈ (0, ∞) be as in Lemma 8.2. For the duration of this proof, to ease notation, set τ := τα,A . Let x ∈ WA (α) be such that fw (x) > Aw . Suppose, for the purpose of deriving a contradiction, that E[τ (w/2)+ε | ξ0 = x] < ∞ for some ε > 0. Let Ξ0 = (ξt0 )t∈Z+ be an independent copy of Ξ, and let τ 0 be the corresponding independent copy of τ . Then for any t ∈ N, by conditioning on ξt and using the Markov property, we have     E[τ (w/2)+ε | ξ0 = x] ≥ E E (t + τ 0 )(w/2)+ε | ξ00 = ξt 1{τ ≥t} | ξ0 = x . Hence by Lemma 8.3

E[τ (w/2)+ε | ξ0 = x] ≥ ε2 cos(wϕ(ξt ))E[(t + ε1 kξt k2 )(w/2)+ε 1{τ ≥t} | ξ0 = x] h i ≥ CE (fˆw (ξt∧τ ))1+(2/w)ε | ξ0 = x − Aw ,

for some C ∈ (0, ∞), any t ∈ N, where we have used the fact that fˆw (ξτ ) ≤ kξτ kw ≤ Aw a.s.. We then obtain that there exists ε0 > 0 such that 0 ∞ > E[τ (w/2)+ε | ξ0 = x] ≥ E[(fˆw (ξt∧τ ))1+ε | ξ0 = x], 0 for any t ∈ N; that is, (fˆw (ξt∧τ ))1+ε is uniformly bounded in L1 . Hence we have shown that under the hypothesis E[τ (w/2)+ε | ξ0 = x] < ∞, we have that for some ε00 > 0 the process 00 (fˆw (ξt∧τ ))1+ε is uniformly integrable. Then (since by hypothesis τ < ∞ a.s.) it follows that 00 00 00 as t → ∞, E[(fˆw (ξt∧τ ))1+ε | ξ0 = x] → E[(fˆw (ξτ ))1+ε | ξ0 = x] ≤ Aw(1+ε ) . However, by the 00 00 00 submartingale property (8.7), we have E[(fˆw (ξt∧τ ))1+ε | ξ0 = x] ≥ (fˆw (x))1+ε > Aw(1+ε ) for all t ∈ N, given our condition on x. Thus we have the desired contradiction, and E[τ (w/2)+ε | ξ0 = x] = ∞ for any x ∈ WA (α) with fw (x) > Aw . (w/2)+ε Since τα ≥ τ a.s., this implies that E[τα | ξ0 = x] = ∞ for any x ∈ WA (α) with fw (x) > Aw . Now condition (A1) extends the conclusion to any x ∈ W(α), and the proof of the theorem is complete. 

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