Walking within growing domains: recurrence versus transience

For normally reflected Brownian motion and for simple random walk on independently growing in time d-dimensional domains, d>=3, we establish a sharp criterion for recurrence versus transience in terms of the growth rate.

positive f (t), t ≥ 0 (and B c ⊂ R d denotes an Euclidean ball of radius c, centered at the origin 0 ∈ Z d ).
In this context, we propose the following universality conjecture (namely, that only the asymptotic growth rate of t → f (t) matters for transience/recurrence of such srw).
Conjecture 1.2.Almost surely, the srw {Y t } on {D t } satisfying Assumption 1.1 and starting at Y 0 = 0, returns to the origin finitely often iff (1.1) Indeed, we show in Theorem 1.4 that under Assumption 1.1, having J f < ∞ implies that P(A) = 0 for A := { t I {y} (Y t ) = ∞} and any y ∈ Z d .For the more challenging part, namely we resort to connecting the srw {Y t } with a normally reflected Brownian motion (in short rbm), via an invariance principle (see Lemma 3.2).Thus, our approach yields sample-path recurrence results for reflected Brownian motion on growing domains in R d (in short rbmg, see Definition 1.13 and Theorem 1.15), which are of independent interest.This strategy comes however at a cost of imposing certain additional restrictions on t → D t .Specifically, when proving in part (b) of Theorem 1.4 the recurrence of the srw on growing domains D t in Z d , d ≥ 3, we further assume that D t = f (t)K ∩ Z d for some K regular enough, to which end we recall the following definition.
3. An open connected K ⊆ R d is called a uniform domain if there exists a constant C < ∞ such that for every x, y ∈ K there exists a rectifiable curve γ ⊆ K joining x and y, with length(γ) ≤ C|x − y| and min{|x − z|, |z − y|} ≤ Cdist(z, ∂K) for all z ∈ γ.
Dealing with a discrete time srw, we may consider without loss of generality only t → f (t) piecewise constant, that is, from the collection a l I [t l ,t l+1 ) (t), for t 1 = 0, {t l } ↑ ∞, 0 < a l ↑ ∞}.
(1.3)However, as seen in our main result below, for our proof of (1.2) we further require the following separation of scales (1.4) Theorem 1.4.Consider a srw {Y t } on {D t } satisfying Assumption 1.1, with Y 0 = 0. (a).Whenever J f < ∞, the srw {Y t } a.s.visits every y ∈ Z d finitely often.(b).Such srw {Y t } a.s.visits every y ∈ Z d infinitely often, in case (1.5) Remark 1.5.Requiring (a l − a l−1 ) ↑ ∞ results in l → a l super-linear, and hence in the series l a 2−d l log(1 + a l ) converging whenever d ≥ 4 (so the latter restriction on f ∈ F * is relevant only for d = 3).We need K to be a uniform domain only for the invariance principle of Lemma 3.2, and impose on K the star-shape condition of (1.5) merely to guarantee that the corresponding sub-graphs t → D t are non-decreasing.
One motivating example for our study is the srw {Y t } on the independently growing Internal Diffusion Limited Aggregation (idla) cluster D t , formed by particles injected at the origin according to a Poisson process of bounded away from zero intensity λ(t), and independently performing srw with jump-rate v.While the microscopic boundary of such idla cluster D t is rather involved, it is well known (see [LBG]), that M −1/d t D t → B κ , where M t denotes the number of particles reaching the idla cluster boundary by time t, and the value of κ = κ d is chosen such that B κ has volume one.Consequently, from part (a) of Theorem 1.4 we have that Corollary 1.6.The srw on such idla clusters is a.s.transient when the random variable J := ´∞ 1 M −1 t dt is finite.Further, our analysis (i.e.part (b) of Theorem 1.4), suggests the a.s.recurrence of the srw on such idla clusters, whenever J = ∞ (this is also a special case of Conjecture 1.2).
Remark 1.7.In our idla clusters example, let g(t) := ´t 0 λ(s)ds denote the mean of the Poisson number of particles N t injected at the origin by time t.Then, a.s.J < ∞ iff Indeed, clearly M t ≤ N t and for large t the Poisson variable N t is concentrated around g(t).
Our claim thus follows immediately when v → ∞, for then one has further that M t ↑ N t ∼ g(t).More generally, for v finite and t ≫ 1, the variable M t is still concentrated, say around some non-random u(t), which is roughly comparable to N t−cu(t) 2/d for some c = c(v), and thereby also to g(t − cu(t) 2/d ).Solving for Next, considering part (b) of Theorem 1.4 for K = B 1 we see that (at least subject to our conditions about {a l }), Conjecture 1.2 is a consequence of the more general monotonicity conjecture: Conjecture 1.8.Suppose non-decreasing in t graphs G t , G ′ t of uniformly bounded degrees are such that G t ⊆ G ′ t for all t, and the srw {Y t } on {G t } is transient, i.e. its sample-path a.s.returns to Y 0 = y 0 finitely often.Then, the same holds for the sample path of the srw Remark 1.9.By Rayleigh monotonicity principle, Conjecture 1.8 trivially holds whenever G t and G ′ t do not depend on t.However, beware that it may fail when the graphs depend on t and unbounded degrees are allowed.For example, on G t = Z 3 the srw is transient, but we can force having a.s.infinitely many returns to 0 by adding to the edges of Z 3 , at times t k ↑ ∞ fast enough, edges in G ′ t , t ≥ t k , between 0 and each vertex in a wide enough annulus A k := {x ∈ Z 3 : x 2 ∈ [r k , R k )} (specifically, with r k ≪ R k suitably chosen to make sure the srw on G ′ t is at times t k in A k and thereby force at least one return to zero before exiting A k , while t k−1 ≪ t k gives separation of scales).
As shown for example in Theorem 1.4, when the srw on the limiting graph G ∞ is transient, one may still get recurrence by imposing slow enough growth on G t .In contrast, whenever the srw on G ∞ is recurrent, we have the following consequence of the Conjecture 1.8.
Conjecture 1.10.If srw {Y t } on a fixed graph G ∞ of uniformly bounded degrees is recurrent, then the same applies to srw on non-decreasing In particular, Conjecture 1.10 implies that the srw on any non-decreasing D t ⊆ Z 2 is recurrent.We note in passing that monotonicity of t → D t is necessary for the latter statement (hence for Conjectures 1.8 and 1.10).Indeed, with D t being Z 2 without edges (x, y) for Remark 1.11.Conjecture 1.10 was proposed to us by J. Ding and upon completing this manuscript we found a more general version of it in [ABGK].Specifically, [ABGK] conjecture that a random walk {Y t } on graph G ∞ with non-decreasing edge conductances {c t (e)} is recurrent as soon as the walk on (G ∞ , {c ∞ (e)}) is recurrent (Conjecture 1.10 is just its restriction to {0, 1}-valued conductances).This is proved for G ∞ a tree (by potential theory, see [ABGK,Theorem 5.1]).A weaker version of Conjecture 1.8 is also proposed there (and confirmed in [ABGK,Theorem 4.2] for G ∞ = Z + ), whereby the transience of the walk on (G ∞ , {c t (e)}) is conjectured to hold whenever the walk on (G ∞ , {c 0 (e)}) and the walk on (G ∞ , {c ∞ (e)}) are both transient.Finally, we note in passing that the zero-one law P(A) ∈ {0, 1} in Conjecture 1.2 is not at all obvious given [ABGK,Example 4.5], where 0 < P(A) < 1 for some random walk on Z with certain non-random, non-increasing c t (e) ∈ (0, 1]. Remark 1.12.Recall [GKZ] that the srw on the infinite cluster D 0 of Bernoulli bond percolation on Z d is a.s.recurrent for d = 2 and transient for any d ≥ 3. Hence, by Conjectures 1.8 and 1.10 the same should apply to the srw on any independently growing domains D t ⊇ D 0 .Whereas the latter is an open problem, by [Ke,Theorem 1.1] such conclusion trivially holds when D t is the set of vertices connected to the origin by time t in First-Passage Percolation with finite, non-negative i.i.d.passage times on Z d , subject only to the mild moment condition [Ke,(1.6)].
We consider also Brownian motions on growing domains, as defined next.
Definition 1.13.We call (W t , D t ) reflected Brownian motion on growing domains (rbmg), if the non-random, monotone non-decreasing D t ⊆ R d are such that the normally reflected Brownian motion W on the time-space domain D := {(t, x) ∈ R d+1 : x ∈ D t } is a welldefined strong Markov process solving the corresponding deterministic Skorohod problem.
That is, for any (s, x) ∈ D there is a unique pair of continuous processes (W, L) adapted to the minimal admissible filtration of Brownian motion {U t } t≥0 , with L non-decreasing, such that for any t ≥ s, both (t, W t ) ∈ D and where n(u, y) denotes the inward normal unit vector at y ∈ ∂D u .
As shown in [BCS,Theorem 2.1 and 2.5], Definition 1.13 applies when ∂D is C 3 -smooth with γ(t, x) • (0, n(t, x)) bounded away from zero uniformly on compact time intervals, where γ(t, x) denotes the inward normal unit vector at (t, x) ∈ ∂D.Focusing on D t = f (t)K this condition holds whenever both f (t) and ∂K are C 3 -smooth.Further, this construction easily extends to handle isolated jumps in t → f (t).
In the context of R d -valued stochastic processes, we define recurrence as follows: Definition 1.14.The sample path x t of a stochastic process t → x t ∈ D t with x 0 = 0, is called recurrent, if it makes infinitely many excursions to B ǫ for any ǫ > 0, and is called transient otherwise.That is, recurrence amounts to the event A := ∩ ǫ>0 A ǫ , where Remark 1.16.In part (a) of Theorem 1.15 we implicitly assume that the rbmg (W t , D t ) is well defined, in the sense of Definition 1.13.Since J f = ∞ whenever f (t) is bounded, in which case part (b) trivially holds, we assume throughout that f (t) is unbounded.The condition ´∞ 0 f ′ (s) 2 ds < ∞ is needed in part (b) only for K = B r , and it holds for example whenever f (•) is piecewise constant, or in case f (s) = (c+s) α for some c > 0 and α ∈ [0, 1/2).
We prove Theorem 1.15 in Section 2 and Theorem 1.4 in Section 3, whereas in Section 4 we show that in the context of Conjecture 1.2, if recurrence/transience occurs a.s. with respect to the origin, then the same applies at any other point.

Proof of Theorem 1.15
Since the events A ǫ are non-decreasing in ǫ, it suffices for Theorem 1.15 to show that q ǫ := P(W ∈ A ǫ ) = I {J f =∞} for each fixed ǫ > 0. To this end we require the following three lemmas.

Since |W
(2) In particular, a normal reflection at ∂B g(t) reduces the norm, hence |W (1) η 2 , we repeat the above argument for [η 2 , τ 2 ], then for [η 3 , τ 3 ], etc.By construction, η n < τ n < η n+1 for all n.Moreover, a.s.τ n → ∞ when n → ∞.Indeed, assuming without loss of generality that η k < ∞, we have the stopping times , by Brownian scaling it follows that the sequence As n → ∞ the right-hand-side grows a.s. to infinity and so does the left-hand-side.(b).We follow the construction and reasoning of part (a), up to time η 1 , setting now (1) t for all t ≥ η 1 .Then, by the invariance to rotations of B g(t) and the fact that only normal reflections are used, we have that t → W (2) ), for which ψ t is non-negative.Lemma 2.2.Let P x denote the law of the rbm Z t on B a , starting at Z 0 = x.Consider the stopping times τ (a) := inf{s ≥ 0 : Z s ∈ ∂B a } and σ(a, r) := inf{s ≥ 0 : (2.1) Proof.In case the process starts at z ∈ ∂B r we use P re 1 to indicate probabilities of events which are invariant under any rotation of the sample path.Then, with U t denoting a standard Brownian motion, by Brownian scaling the left-hand side of (2.3) does not depend on a and is merely the positive probability P 0.5e 1 (|U s | < 1, ∀s ≤ 1).Further, by the Markov property, invariance to rotations and Brownian scaling, for x ∈ B a , with η = η d > 0, out of which we get (2.1).Proceeding similarly, we have for (2.2) that where Z denotes the rbm on B 1 .Further, Lemma 2.3.Let P x denote the law of the rbm Z t on B a , starting at Z 0 = x.Fixing ǫ, δ ∈ (0, 1/2), there exist finite M d (ǫ, δ) and C = C d (ǫ, δ) such that for all M, T, a and r with M ≥ M d (ǫ, δ), T ≥ Ma 2 log a and a − M ≥ r ≥ aδ, Proof.Starting at Z 0 = x ∈ ∂B r , let σ(a, ǫ) := inf{t ≥ 0 : |Z t | ≤ ǫ}, and setting σ 0 := 0, (2.7) Recall that conditional on their starting and ending positions, these excursions of the rbm on B a are mutually independent.Consequently, where b ǫ (x, w) is the probability of entering B ǫ in one such excursion.Elementary potential theory (e.g.see [MP,Theorem 3.18]), yields the formula Hence, applying the strong Markov property of Z • at the stopping time σ i where |Z σ i | = a/2, going from i = k − 1 backwards to i = 1 we deduce that It is easy to check that b ǫ ( a 2 e 1 ) = c 0 a 2−d (1+o(1/M)) for some finite, positive c 0 = c 0 (d, ǫ) and all a ≥ M ≥ 1/δ, whereas b ǫ (re 1 ) ≤ c ′ b ǫ ( a 2 e 1 ) for some finite c ′ = c ′ (d, δ), and all r ≥ aδ ≥ 1.Thus, setting k = [T a −2 κ ∓1 ] for some universal κ yet to be determined, we see that (2.11) Indeed, our assumption that T ≥ Ma 2 log a translates to ck ≥ cκ ∓1 M log a, so that for all large enough M ≥ M 0 (κ, c, d, C) we have that resulting by (2.7) and (2.10) in the claimed bounds.
The universal exponential tail bounds of (2.11), are a direct consequence of having control on the log-moment generating functions Λ k (θ) := log E[e θσ k ] for large k and small θ := θa 2 .Specifically, by Markov's exponential inequality (also known as Chernoff's bound), we get (2.11) as soon as we show that with {L i } mutually independent conditional on the values of {Z σ i }.Thus, proceeding in the same manner as done in (2.8), we have that for any θ ∈ R and k ∈ N, where m(θ, x, w) . By invariance of the joint law of {σ k } with respect to rotations of the rbm sample path t → Z t , the unconditional function m(θ, x) = E[m(θ, x, Z σ 1 )] = m(θ, |x|) depends only on |x|.Hence, exploiting once more the strong Markov property at the stopping times σ i where (2.14) Further, L 0 is the sum of two independent variables, having the laws of τ (a) and σ(a, a/2) of Lemma 2.2.Thus, the universal upper bounds (2.1) and (2.2) imply that for any 0 (2.15) Combining this with (2.14) we get (2.12) (with Step I.For f ∈ F * we set ∆T l := t l+1 − t l and p l := a 2−d l log(1 + a l ), so that l p l < ∞ and (2.16) Considering here D t = B f (t) for f ∈ F * , we proceed to prove the a.s.recurrence of the rbmg sample path in case J f = ∞.To this end, consider the events Γ l := {∃t ∈ [t l−1 , t l ) : |W t | < ǫ}, adapted to the filtration G l := σ{W s , s ≤ t l }.Fixing δ ∈ (0, 1/2) we set r l := (a l−1 + 1) ∨ δa l and further assume that ∆T l ≥ 2M d a 2 l log(1 + a l ) . (2.17) Then, since we have by (2.4) that Recall that J f of (2.16) is infinite, hence a.s.{ ∞ l=1 ζ l = ∞}, which implies that Γ l occurs infinitely often (by the conditional version of Borel-Cantelli II, see [Du,Theorem 5.3 By transience of the d ≥ 3 dimensional Brownian motion we can set k 1 = 1 and recursively pick u j := inf{t > s k j : |W t | > 1/2}, k j+1 := inf{k : s k > u j }, for j = 1, 2, . . ., thus yielding the event A ǫ .To remove the spurious condition (2.17) set ψ l := ∆T l /(a 2 l log(1 + a l )), so l ψ l p l = l a −d l ∆T l diverges by (2.16) whereas l p l is finite.Hence, l ψ l p l I {ψ l ≥2M d } = ∞, and the preceding argument is applicable even when restricted to {l k } ↑ ∞ such that ψ l k ≥ 2M d .
Step II.Still considering D t = B f (t) for f ∈ F * , we show next that P(A ǫ ) = 0 whenever J f of (2.16) is finite.To this end, note that for l = 1, 2, . . ., are a.s.finite and proceed to show that where Γ l := {∃t ∈ [τ l , τ l+1 ) : |W t | < ǫ}.Indeed, in this case by Borel-Cantelli I, a.s. the rbmg does not re-enter B ǫ during [τ l , ∞), for some l finite.In any finite time, even the rbm on B 1 a.s.makes only finitely many excursions between B ǫ and B c 1/2 , hence P(A ǫ ) = 0. Turning to prove (2.20), recall that t l ≤ τ l and t l+1 ≤ τ l+1 , so the interval [τ l , τ l+1 ) splits into [τ l , ξ l+1 ) and [ξ l+1 , τ l+1 ), where Restricted to t ∈ [τ l , ξ l+1 ), the process {W t } has the law of a rbm on B a l , and the length of [τ l , ξ l+1 ) is at most ∆T l plus the length of [t l+1 , ξ l+1 ).By (2.1), for some constant C = C d (δ) > 0, any l and all t, (2.21) Combining (2.5) with (2.21) for t = M log a l , M = M d ∨ 2 C , we have that with the first term on the right-hand-side summable in l iff J f < ∞ (the other two terms are summable for any f ∈ F * ).Further, restricted to t ∈ [ξ l+1 , τ l+1 ), the process {W t } has the law of Brownian motion {U t } (since r l+1 < a l+1 ), hence (2.23) Bounding P( Γ l ) by the sum of the left-hand-sides of (2.22) and (2.23), we thus conclude that Step III.Given non-decreasing, unbounded, positive t → f (t) (which without loss of generality we assume hereafter to be also right-continuous), let g ∈ F * with a l = 2 l−1 f (0) and t l := inf{t ≥ 0 : f (t) ≥ 2 l−1 f (0)}.Since g(t) ≤ f (t) ≤ 2g(t) for all t ≥ 0, we have by part (a) of Lemma 2.1, the coupling Step II, a.s.{ W t } enters B ǫ finitely often.Hence, P(A ǫ ) = 0, yielding the stated a.s.transience of the sample path for any such rbmg (W ′ t , D t ), thereby completing the proof of part (a).
Step IV.Returning to Step V. We next extend the a.s.recurrence of the rbmg (W t , D t ) sample path to D t = f (t)K with J f = ∞, K from K of (1.5), such that ∂K and f (t) are both C 3 -smooth, and ´∞ 0 f ′ (s) 2 ds < ∞.To this end, we assume without loss of generality that B 1 ⊆ K ⊆ B c and note that t → ´t 0 1 f (u) dL u =: L t increases only when X t := 1 f (t) W t is at ∂K.Hence, applying Ito's formula to the C 1,2 -function v(t, x) = 1 f (t) x (with v xx = 0), and the semi-martingale {W t } of (1.7), we get that (X, L) is the strong Markov process solving the deterministic Skorohod problem corresponding for (s, x) ∈ R + × K to where n(x) denotes the inward unit normal vector at x ∈ ∂K and has uniformly in t bounded exponential moments.That is, for any κ > 1, ) is a uniformly integrable continuous martingale (see [RY,Proposition VIII.1.15]).The same applies for Z −1 t = exp( M t − 1 2 M t ) and the martingale M t = − ´t 0 f ′ (s)X s dB s under the measure Q such that {B t , t ∈ [0, ∞)} is a standard Brownian motion in R d .Hence, by Girsanov's theorem, restricted to the completion of the canonical Brownian filtration, the measure Q equivalent to P (see [RY,Proposition VIII.1.1]).Moreover, under Q the process {X t } is a normally reflected time changed Brownian motion (in short tcrbm), on K for the deterministic time change (2.27) Applying the same procedure for the rbmg (W ′ t , f (t)B c ), such that W ′ 0 = W 0 , yields another probability measure H, likewise equivalent to P, under which and similarly to Definition 1.14 we have that out of which we deduce that P(W • ∈ A) = 1 as well.The key implication, marked by (a), is a consequence of the proof of [Pa,Theorem 5.4].This theorem is a comparison result about Neumann heat kernels over domains smooth boundary, and some ball B centered at 0, such that for any x ∈ D 2 , the line segment from 0 to x is in D 2 .Its proof in [Pa] is by constructing a (mirror) coupling between the rbm X on D 2 and the rbm Y on D 1 , such that | X s | ≤ | Y s | for all s ≥ 0 and any common starting point x ∈ D 2 .We use it here for D 2 = K ⊆ B c = D 1 and note that the monotonicity of the radial component under this coupling extends to the tcrbm-s X s (under Q), and Y s (under H), thereby assuring that Step VI.We proceed to show that the conclusion of Step V holds in case t → f (t) has jumps ∆ j > 0 at isolated jump points with a C 3 -smooth function f c (•) and piecewise constant f d (t) = j ∆ j I t≥t j .Setting t 0 = 0 and re-using the notations of Step V, upon applying Ito's formula we get that X t (and Y t ) solve the corresponding deterministic Skorohod problem (2.24)-(2.25)within each interval [t i−1 , t i ), and B t is again defined via (2.26) except for f ′ c (t) replacing f ′ (t).In addition, 2 ds is finite, as in Step V we have measures Q and H, both equivalent to P, under which within each interval [t i−1 , t i ] the processes X t and Y t are tcrbm-s on K and B c , respectively, for the same time change τ (•).With J f = ∞, we already saw in Step IV that P(W ′ • ∈ A) = 1.Following the argument of Step V this would yield that P(W • ∈ A) = 1, provided we suitably extend the scope of the implication (a).That is, suffices to show the existence of coupling between rbm-s X on K and Y on B c , such that | X s | ≤ | Y s | for all s ≥ 0, in the setting where at a sequence of isolated times s i = τ (t i ) one applies the common shrinkage by η i ∈ (0, 1) to both X • and Y • .To achieve this, starting at Y 0 = Y ′ 0 = X 0 = x, we produce inductively for i = 0, 1, . . .
Indeed, as explained in Step V, employing [Pa,Theorem 5.4] separately within each interval [s i , s i+1 ) yields a (mirror) coupling of Y ′ and X that maintains the stated relation we have the latter inequality at i = 0.Then, for i ≥ 1 we have by induction, upon utilizing our coupling on [ after the common shrinkage by factor η i ), as needed for concluding the proof.

Proof of Theorem 1.4
Hereafter we denote the inner boundary of a discrete set G by ∂G and fix K from the collection K of (1.5), scaled by a constant factor so as to have K ⊇ B 2 and hence (B a ∩ Z d ) ∩ ∂(aK ∩ Z d ) = ∅ for all a ≥ a d large enough.We then have the following srw analog of Lemma 2.2.Lemma 3.1.Let P x denote the law of srw {Z t , t ≥ 0} on aK ∩ Z d , d ≥ 3, starting at In proving Lemma 3.1 we rely on the following invariance principle in bounded uniform domains, which allows us to transform hitting probabilities of srw to the corresponding probabilities for an rbm.
where W is the rbm on D starting from x, time changed by constant κ.
Proof: Lemma 3.2 merely adapts facts from [CCK, Theorem 3.17 and Section 4.2] to our context (alternatively, it also follows by strengthening [BC,Theorem 3.6] as suggested in [BC,Remark 3.7]).The original result presented in [CCK] is for variable-speed and constantspeed random walks (vsrw,csrw) on bounded uniform domain with random conductances uniformly bounded up and below.We are in a special case where all edges in nD ∩ Z d are present and have equal non-random conductance.Hence, here the csrw is merely a continuous-time srw Z t of unit jump rate on nD ∩ Z d and further the invariance principle holds for Z n t := n −1 Z n 2 t and any choice of x ∈ D. Indeed, while rbm W t constructed via Dirichlet forms is typically well defined only for a quasi-everywhere starting point in D, here this can be refined to every starting point.This is because in a uniform domain, such rbm admits a jointly-continuous transition density p(t, x, y) on R + × D × D of Aronson's type (see [GS,Theorem 3.10]), thereby eliminating the exceptional set in [FOT,Theorem 4.5.4].
It remains only to infer the invariance principle for the discrete-time srw {Y n t } out of the invariance principle for {Z n t }.To this end, recall the representation Y n t = Z n n −2 L(n 2 t) for L(t) := inf{s ≥ 0 : N(s) = ⌊t⌋} and the independent Poisson process N(t) of intensity one.Now, fixing T finite, by the functional strong law of large numbers for Poisson processes, Further, by [CCK,Proposition 3.10 and Section 4.2], for any r > 0, lim δ→0 lim sup n→∞ P nxn sup that is induced by the discrete-time srw on a n D ∩ Z d and any fixed a n ↑ ∞ (just note that the conditions laid out in [CCK, first paragraph, Page 13] hold with a n replacing n).
Proof of Lemma 3.1.Consider the rbm W • on K ⊇ B 2 and the rescaled discrete time srw Z a t := a −1 Z ⌊a 2 t⌋ .Starting with the proof of (3.1), for a > 0 and y ∈ K, let q rw (a, y) Then, by the Markov property of the srw, for any a, t > 0 and x ∈ B a ∩ Z d , An rbm on uniform domain admits jointly continuous, positive transition density ([GS, Theorem 3.10]), and in particular m bm = 1 − 2η for some η ∈ (0, 1/2).As we show in the sequel, setting ξ := 1−η 1−2η > 1, is finite.It then follows from (3.4) that for some positive C, all a > a d and t > 0, sup To complete the proof of (3.1), suppose to the contrary that a d = ∞ in (3.5), namely m rw (a l ) > ξm bm for some a l ↑ ∞.Taking the uniformly bounded y l ∈ B 1 ∩ (a −1 l Z) d such that q rw (a l , y l ) = m rw (a l ), we pass to a sub-sequence {l n } along which y ln → x ∈ B 1 .Then, considering Remark 3.3 for the sequence a ln , we deduce that as n → ∞, m rw (a ln ) = q rw (a ln , y ln ) → q bm (x) ≤ m bm , in contradiction with our assumption that m rw (a ln ) > ξm bm for some ξ > 1 and all n.
Equipped with Lemma 3.1 we can now establish the following srw analog of Lemma 2.3.Proof. (a).We adapt the proof of Lemma 2.3 to the current setting of discrete time srw Z t on aK∩Z d , by taking throughout ǫ = 0 and re-defining the excursions of length L k := σ k+1 −σ k , k ≥ 0, to be determined now by the stopping times σ 0 = 0 and Since the laws of increments of srw are not invariant to rotations, x → m(θ, x) = E x [e θL 0 ] is not a radial function.However, replacing Lemma 2.2 (which we used when bounding m(θ, x) in case of Brownian motion), by the universal bounds of Lemma 3.1, yields (2.12) and (2.13) for the srw case considered here.Thereby, applying the discrete analogue of (2.9) b(x) := P x ( inf (3.9) where 0 < c d < ∞ is a dimensional constant (see [La,Proposition 1.5.9]), at x ∈ ∂(rK ∩ Z d ) and x ∈ ∂(B a/2 ∩ Z d ), yields the srw analog of (2.10), out of which the stated conclusions follow.(b).Let I k := [σ k , τ k+1 ), k ≥ 0. Our assumptions that Z 0 ∈ B c aδ and D t ⊇ B a+1 ∩ Z d result in {Z t , t ∈ I k } having for each k ≥ 0 the same conditional law given Z σ k , as in part (a).Since the event |Z t | = 0 can only occur for t ∈ ∪ k I k , the derivation leading to the srw analog of (2.10) applies here as well.Further, conditional on Z σ k = x, each L k , k ≥ 1, stochastically dominates the random variable τ (a) of Lemma 3.1 starting at same point x.Consequently and utilizing the uniform in x and a control on the r.h.s.due to (3.1), establishes yet again the analog of (2.13).Examining the proof of (2.5) in Lemma 2.3 we see that this suffices for re-producing the corresponding uniform upper bound (3.8).
Proof of Theorem 1.4.(a).Fix f (t) such that J f < ∞ and consider the srw {Y t } on D t ⊆ Z d , d ≥ 3 for which Assumption 1.1 holds.Similarly to Step II of the proof of Theorem 1.15, for a l := (c + 1) l , l ≥ 1, define With f (•) unbounded, for any l eventually D s ⊇ B a l ∩ Z d and by the transience of the srw on Z d , necessarily τ l are a.s.finite.Thus, by Borel-Cantelli I, (3.10) Turning to bound P( Γ l ), note that τ l+1 ≥ t l and D t l ⊇ B 1+a l ∩ Z d (by Assumption 1.1 and the choice of a l ).Hence, by (3.1), we have that for some constants C d > 0 and l d < ∞, all l ≥ l d and t ≥ 0, (3.11) Let ∆T l := (t l − t l−1 ) and for δ = 1/(c + 1) < 1/2 and M d = M d (δ) of Lemma 3.4, set T * l := M d log a l and T l = ∆T l + T * l a 2 l .Since τ l ≥ t l−1 the length of [τ l , τ l+1 ) is at most ∆T l plus the length of [t l , τ l+1 ), which by (3.11) is with high probability under T * l a 2 l .Further, a l , hence from part (b) of Lemma 3.4 we have that, With our choice of a l growing exponentially in l, the terms e −C d T * l and a 2−d l T * l in the bound (3.12) are summable over l ∈ N. Hence, the left-hand-side of (3.10) is finite whenever l a −d l−1 ∆T l is finite.Further, Assumption 1.1 and our definition of t l imply that f (t l − 1) ≤ 1 + a l+1 .Thus, Consequently, finite J f results in P 0 (Y t = 0 f.o.) = 1, which by Proposition 4.2 extends to P 0 (Y t = y f.o.) = 1 for all y ∈ Z d , as claimed.
(b).Fix f ∈ F * such that J f = ∞ and K ∈ K. Since J f /r = ∞ for any r > 0 and D t = (f (t)/r)(rK) ∩ Z d , taking r large enough we have with no loss of generality that K ⊇ B 2 .Then, considering the srw on D t , upon replacing (2.4) by (3.7), the argument we have used in Step I of the proof of Theorem 1.15 applies here as well, apart from the obvious notational changes (of replacing B a l and B a l−1 in (2.18) by a l K ∩ Z d and the collection of all x ∈ Z d within distance one of a l−1 K, respectively).

On recurrence probability independence of target states
The following, xy-recurrence property, generalizes Definition 1.14 to arbitrary starting and target locations, x, y ∈ R d , respectively.
x and the event A(y) := ∩ ǫ>0 A ǫ (y) occurs, where is the rbmg of Definition 1.13 with D 0 open connected set and D t ↑ R d .Then, the probability q xy of xy-recurrence does not depend on y.In case of rbmg, if q zy ∈ {0, 1} for some z ∈ D 0 then q xy = q zy for all x ∈ D 0 , whereas in case of srw, if q zy = 0 for some z ∈ D 0 then q xy = 0 whenever x − z 1 is even.
Remark 4.3.Adapting the approach we use for the rbmg, it is not hard to show that for continuous time srw (on growing domains D t ↑ Z d ), having q zz ∈ {0, 1} for some z ∈ D 0 results in q xy = q zz for all x ∈ D 0 and y ∈ Z d .This approach is based on the equivalence of hitting measures of suitable sets when starting the process at nearby initial states.This however does not apply for discrete time srw, hence our limited conclusion in that case.
Proof.This proof consists of the following four steps.Starting with the srw we show in Step I that q xy does not depend on y, then for x, z ∈ D 0 with x − z 1 even, we prove in Step II that q zz = 0 implies q xx = 0.In case of the rbmg we have that q ǫ xy := P x (A ǫ (y)) ↓ q xy and deduce the stated claims upon showing in Step III that if q zz ∈ {0, 1} then q xz = q zz for any x, z ∈ D 0 , then conclude in Step IV that q ǫ xy = q ǫ xx for any fixed ǫ > 0 and all y ∈ R d (even when 0 < q xx < 1).
Step I.For the srw X t on D t ⊆ Z d and fixed s ∈ N we denote by P s x (•) the law of srw X t on the shifted-domains D t+s starting at X 0 = x.Then, for any x, y ∈ Z d and s ≥ 0, q xy (s) := P(X t = y i.o.| X s = x) = P s x (X t = y i.o.) , with q xy := q xy (0).Since D t ↑ Z d , clearly any y, w ∈ Z d are also in D t provided t ≥ t 0 (y, w) is large enough, with some non-self-intersecting path in D t 0 connecting y and w.Setting F X t := σ{X s , s ≤ t} ↑ F ∞ and events Γ s,t,z,w := {X s = z, X u = w some u > t}, we thus have η = η(y, w) > 0 such that for any starting point x, all z, s and t ≥ t 0 ∨ s, P x (Γ s,t,z,w |F X t ) ≥ ηI {Xs=z,Xt=y} .Further as t → ∞ we have that Γ s,t,z,w ↓ Γ s,z,w := {X s = z and X u = w i.o. in u} .
Clearly, Γ s,z,w ∈ F ∞ so it follows by Lévy's upward theorem (and dominated convergence, see [Du,Theorem 5.5.9]), that for any x, a.s.The same applies with the roles of y and w exchanged and consequently, a.s.Γ s,z,y = Γ s,z,w for all z, y, w ∈ Z d and s ≥ 0. In particular, q zy (s) = P(Γ s,z,y |X s = z) is thus independent of y, for any z and s ≥ 0.
Step II.Assuming now that q zz = q zz (0) = 0 for some z ∈ D 0 , we have from Step I that q zx = 0.As explained before (in Step I), s 0 := inf{t : P z (X t = x) > 0} is a finite integer and clearly P z (X 2s+s 0 = x) > 0 for any s ≥ 0. By the Markov property at time 2s + s 0 , 0 = q zx ≥ P z (X 2s+s 0 = x, X t = x i.o.) = P z (X 2s+s 0 = x)q xx (2s + s 0 ) .
Starting at X 0 = x, the event {X t = x} is possible only at t even.Since x − z 1 is even, so is the value of s 0 and from the preceding we know that P x -a.s.any visit of x at even integer larger than s 0 results in only finitely many visits to x.Since there can be only finitely many visits of x up to time s 0 , we conclude that q xx = 0.
Step III.Dealing hereafter with the rbmg, recall that A ǫ (y) ↓ A(y) for A ǫ (y) = {∃s k , u k ↑ ∞ : |X s k − y| < ǫ, |X u k − y| > 1/2, u k ∈ (s k , s k+1 )}.Let P s x (•) stand for the law of the rbmg {X t } on shifted-domains D t+s starting at X 0 = x, and q ǫ xy (s) := P s x (A ǫ (y)) with q ǫ xy = q ǫ xy (0), so that q ǫ xy ↓ q 0 xy = q xy when ǫ ↓ 0. We first prove that if q zz ∈ {0, 1} for some z ∈ D 0 then q xz = q zz for any x ∈ D 0 such that x+z 2 + B α ⊆ D 0 for some α > |x − z|/2.Indeed, with P x,α denoting the joint law of (X τα , τ α ) for the first exit time τ α := inf{s ≥ 0 : X s / ∈ x+z 2 + B α } and X 0 = x, we have that q ǫ xz = ˆqǫ x ′ z (γ)dP x,α (x ′ , γ) , (4.1) for any fixed ǫ > 0. By dominated convergence this identity extends to ǫ = 0 and considering it for x = z (and ǫ = 0), we deduce that q x ′ z (γ) = q zz ∈ {0, 1} for P z,α -a.e.(x ′ , γ).By our assumption about the points x and z, the measure P z,α is merely the joint law of exit position and time for x+z 2 + B α and Brownian motion X s starting at z and as such it has a continuous Radon-Nikodym density with respect to the product of the uniform surface measure ω d−1 on ∂( x+z 2 + B α ) and the Lebesgue measure on (0, ∞) (for example, see [Hs,Theorem 1 and 3]).Further, the latter density is strictly positive due to the continuity of (killed) Brownian transition kernel.Since the same applies to the corresponding Radon-Nikodym density between P x,α and dω d−1 × dt, we conclude that P z,α and P x,α are mutually It thus follows from (2.3) and the stochastic domination L 1 τ (a) starting at some position x ∈ ∂B a/2 that log m(− θa −2 , a/2) ≤ −C d θ/e for all a > 0 and θ ≤ 1, thereby establishing (2.13) with κ −1 − = C d /e positive, and completing the proof of the lemma.Proof of Theorem 1.15.This proof consists of six steps.First, for D t = B f (t) and f ∈ F * of (1.4), we prove in Step I the a.s.recurrence of the rbmg when J f = ∞, and in Step II its a.s.transience when J f < ∞.Relaxing these conditions, in Step III we prove part (a), and in Step IV get part (b) for K = B 1 .The a.s.sample-path recurrence when J f = ∞ is then established for K ∈ K of (1.5), when both ∂K and t → f (t) are C 3 -smooth (see Step V), and further extended to f (•) having isolated jump points (see Step VI).
and t → f (t) is positive, non-decreasing.(a).The sample path of the rbmg (W t , D t ) is a.s.transient whenever J f < ∞. (b).The sample path of the rbmg (W t , D t ) is a.s.recurrent whenever Assuming that J f = ∞, or equivalently that J 4g = ∞ (with g ∈ F * chosen as in Step III), we know from Step I that for any u fixed, {W ′′t | for the rbmg (W ′′ t , B 4g(t) ).