Rescaling the spatial lambda Fleming-Viot process and convergence to super-Brownian motion

We show that a space-time rescaling of the spatial Lamba-Fleming-Viot process of Barton and Etheridge converges to super-Brownian motion. This can be viewed as an extension of a result of Chetwynd-Diggle and Etheridge (2018). In that work the scaled impact factors (which govern the event based dynamics) vanish in the limit, here we drop that requirement. The analysis is particularly interesting in the biologically relevant two-dimensional case.


Introduction
Our purpose in this paper is to extend a result in [5] which shows that certain suitably rescaled spatial Lambda-Fleming-Viot (SLFV) processes converge weakly to super-Brownian motion (SBM). Our extension is analogous to that of allowing nearest neighbour interactions in interacting particle models, as opposed to taking long range limits, and is particularly delicate in the critical two-dimensional case. SBM is a well known measure-valued diffusion, introduced in [26] and [9], for which there is an extensive research literature (e.g., for reviews see [10], [13] and [22]). SLFV processes were introduced more recently, in [12], to serve as models for the evolution of allele frequencies in populations distributed across spatial continua. An analytic construction was given in [2], along with a discussion of the biological significance of the model. A more probabilistic construction was given in [25], one which gives a very useful connection between SLFV processes and their duals. Following [5], we consider here a neutral two-type version of the general SLFV model, taking "space" to be R d . Informally, our process (constructed below) is a Markov process (µ t ) t≥0 where for each x ∈ R d , µ t (x) is a probability distribution on the type space {0, 1}, with the interpretation that B µ t (x)({i})dx represents the proportion of the population of type i in a region B ⊂ R d at time t. We will consider an extension of the fixed radius case from [5] (Theorem 2.6 of that reference) and not the interesting variable radius case, also discussed there in Theorem 2.7, in which stable branching arises in the limit.
SBM arises as the limit under Brownian space-time rescaling of a range of critical spatially interacting models in mathematical physics and biology above the critical dimension including critical oriented percolation [17], critical lattice trees [18], the critical contact process [16], and the voter model [6]; it is believed to be the scaling of critical ordinary percolation in the same regime. The only scaling limit of the above which has been verified at the critical dimension is the voter model [6] where the critical dimension is two. In this case the simple nature of the dual process, a coalescing random walk, allows one to carry out the required explicit calculations. Now our challenge is to use the related but more complex dual of the Barton-Etheridge model to carry through the analysis. It is understood here that we are not taking "long-range" limits (e.g. as was done for the contact process in [11]) which will weaken the interaction and make the analysis considerably easier. In our setting this means not letting the impact factor (described below) approach zero in the rescaling.
We start a rigorous description of the model by recalling the definition of the fixed radius SLFV process given in [5]. Let r > 0 be the "interaction radius", let ρ ∈ [0, 1] be the "impact factor," and let Π be a Poisson point process on R d ⊗ (0, ∞) with intensity dx ⊗ dt.
We suppose the distribution of types in the population changes over time according to "reproduction events" determined by Π. Given µ t− , if (x, t) ∈ Π, choose an independent point z uniformly at random from the Euclidean ball B r (x) = {y : |y − x| ≤ r}, and (independently) a type α according to the distribution µ t− (z), and then set µ t (y) = (1 − ρ)µ t− (y) + ρδ α ∀y ∈ B r (x).
We keep µ t (y) = µ t− (y) for y / ∈ B r (x). Writing µ t (x) in the form w t (x)δ 1 + (1 − w t (x))δ 0 , we can reformulate the above dynamics more conveniently in terms of w t as follows. Starting from a Borel w 0 : R d → [0, 1] with compact support, for (x, t) ∈ Π, choose an independent parental location z uniformly at random from B r (x), independent of everything, and then: (i) with probability w t− (z) put w t (y) = (1 − ρ)w t− (y) + ρ for all y ∈ B r (x), (ii) with probability 1 − w t− (z) put w t (y) = (1 − ρ)w t− (y) for all y ∈ B r (x), (iii) for all y / ∈ B r (x) keep w t (y) = w t− (y). (1.1) As noted in Section 3 of [5], this description gives a well-defined w t : R d → [0, 1] which has compact support at all times. (See [25] for more details on the construction.) It will be useful to regard w t as the measure w t (x)dx, and for bounded Borel φ : Closely associated with the process w t is a dual process of coalescing "lineages". If we sample a finite number of spatial locations {x i } at time T , it is easy to see that the values w T (x i ) can be determined from w 0 by using Π to trace the lineages backward in time. Since Π run backwards is still a Poisson process, we may define a version of the lineages process starting at backwards time 0 from a finite number of locations {x i } as follows. If (x, t) ∈ Π, mark each lineage in B r (x) independently with probability ρ, and choose a point z uniformly at random from B r (x). If at least one of the lineages in B r (x) is marked, all marked lineages in B r (x) coalesce and the resulting lineage is moved to z. If no lineage is marked, no lineage moves. Lineages outside of B r (x) are not affected.
In this paper it will suffice to consider only the one and two-lineage systems, so we will ignore the higher lineage systems which are more complex to analyze.
We now give a more precise description of these Markov jump processes, using the language of "particles" instead of lineages. Let |Γ| be the Lebesgue measure of Γ ⊂ R d . Let U, U 1 , U 1 be independent random variables uniformly distributed on B r = B r (0), and letŪ have the law of U 1 + U 2 , i.e.,Ū has density P (Ū ∈ dz) = |B r (0) ∩ B r (z)| |B r (0)| 2 dz := hŪ (z)dz. (1.3) We letσ 2 1 d×d denote the covariance matrix ofŪ , so that if x = (x 1 , . . . , x d ), then (1.4) We will use this notation throughout, along with η t for the single particle dual and ξ t = (ξ 1 t , ξ 2 t ) for the two particle dual. (a) The single-particle dual η t . If we start with a single particle at x, it is easy to see that η t is the random walk on R d starting at x which makes jumps at rate ρ|B r | with jump distribution given in (1.3). We write P {x} for the underlying law of η.
(b) The two-particle dual (ξ 1 t , ξ 2 t ). If we start with two particles, one at x 1 and the other at x 2 = x 1 , (ξ 1 t , ξ 2 t ) is the Markov jump process starting at (x 1 , x 2 ), and with law P {x1,x2} , which makes transitions at rate ρ|B r | if y 1 = y 2 = y (y 1 +Ū , y 2 ) at rate ρ(|B r | − ρ|B r (y 1 ) ∩ B r (y 2 )|) if y 1 = y 2 (y 1 , y 2 +Ū ) at rate ρ(|B r | − ρ|B r (y 1 ) ∩ B r (y 2 )|) if y 1 = y 2 (U + U y1,y2 , U + U y1,y2 ) at rate ρ 2 |B r (y 1 ) ∩ B r (y 2 )| if y 1 = y 2 , (1.5) where U y1,y2 is an independent random variable, uniformly distributed over B r (y 1 ) ∩ B r (y 2 ). For y 1 = y 2 , the total jump rate at (y 1 , y 2 ), y 1 = y 2 , is 2ρ|B r | − ρ 2 |B r (y 1 ) ∩ B r (y 2 )|. To see the above rates consider, for example, the second transition from (y 1 , y 2 ) to (y 1 +Ū , y 2 ) for y 1 = y 2 where (y 1 , y 2 ) is the current site of our two-particle dual. The next jump in the first coordinate can only occur at a point (x, t) ∈ Π with x ∈ B r (y 1 ) so let (x, t) be the next such point. At (x, t) such a jump (affecting the first coordinate but not the second) can occur in one of two ways: if x lands in B r (y 1 ) \ (B r (y 1 ) ∩ B r (y 2 )) and the particle ξ 1 at y 1 is marked, or if x lands in B r (y 1 ) ∩ B r (y 2 ) and the particle at y 1 is marked and the particle at y 2 is not. The total rate in t is obtained by integrating out x and so is ρ(|B r (y 1 )| − |B r (y 1 ) ∩ B r (y 2 )|) + ρ(1 − ρ)|B r (y 1 ) ∩ B r (y 2 )| = ρ|B r | − ρ 2 |B r (y 1 ) ∩ B r (y 2 )|.
In either of the above scenarios the particle at y 1 will jump to z, a uniformly selected site in B r (x). Given y 1 , x will be uniformly distributed on B r (y 1 ) and so x − y 1 will be uniform on B r . Clearly given (y 1 , x), z − x is uniformly distributed over B r and so (x − y 1 , z − x) is a pair of independent uniforms on B r . Therefore the jump in ξ 1 at time t is z − y 1 = (z − x) + (x − y 1 ) and so has lawŪ as claimed. The other transitions are similar to analyze. The coalescence time for the two-particle dual starting at (x 1 , x 2 ) is τ = inf{t ≥ 0 : ξ 1 t = ξ 2 t }. (1.6) Although (ξ 1 t , ξ 2 t ) is Markov, the individual coordinates ξ 1 t , ξ 2 t are not (i.e., ξ 1 is not Markov with respect to the filtration σ(ξ 1 s , 0 ≤ s ≤ t) t≥0 ). However, when B r (ξ 1 t ) ∩ B r (ξ 2 t ) = ∅, both coordinates move independently according to the single particle dynamics, while for t > τ , the coalesced coordinates move together according to the single particle dynamics. It is also clear from (1.5) that the two-particle dual is translation invariant, that is, P {x1+x,x2+x} ((ξ 1 , ξ 2 ) ∈ ·) = P {x1,x2} ((x + ξ 1 , x + ξ 2 ) ∈ ·) ∀x, x 1 , x 2 ∈ R d . (1.7) The two special cases of the general duality equation in Proposition 2.5 of [5] that we need are the following. For all t ≥ 0, By standard approximation arguments, these equations then hold for all Borel ψ 1 , ψ 2 which are either nonnegative or integrable (on one side or the other). In particular, letting 1 denote the constant function 1 on R d , we have (1.10) Before stating the main fixed radius result of [5], Theorem 2.6, we introduce super-Brownian motion using the martingale problem formulation. If (X t ) t≥0 is a stochastic process, (F X t ) t≥0 will denote the right-continuous filtration generated by X. Let M F (R d ) denote the space of finite Borel measures on R d endowed with the topology of weak convergence, and for µ ∈ M F (R d ) let µ(φ) = R d φdµ. The space of bounded continuous functions on R d is denoted by C b (R d ), and C 3 0 (R d ) is the space of continuous functions on R d which vanish at infinity and have bounded continuous partials of order 3 and less.
Then (see, e.g., Theorem A.1 of [6] for uniqueness, and Theorem II.5.1 and Remark II.5.5 of [22] for existence) Super-Brownian motion with diffusion coefficient σ 2 and branching rate b, denoted SBM(X 0 , σ 2 , b), is the unique M F (R d )-valued Markov process (X t ) t≥0 with continuous paths and initial state X 0 , such that such that for every φ ∈ C 3 0 (R d ), is a local (F X t )-martingale with predictable quadratic variation process with the Skorokhod (J1) topology. Theorem 1.0 (Theorem 2.6 in [5]). Suppose that for a compact set D 0 ⊂ R d , supp(w N 0 ) ⊂ D 0 for all N , and as elements of In addition, suppose there are constants C 1 , C 2 ∈ (0, ∞) such that, as N → ∞, (The constant C(d) in Definition 4.1 in [5] should be C(d) = B1 (x 1 ) 2 dx.) As noted in [5], this result is similar in spirit to Theorem 1.1 in [6], which proves convergence to SBM for certain sparse "long range" kernel voter models. Due to conditions (1) and (4) above, J N → ∞ and hence the impact factors ρ/J N → 0. It is this fact and the mass scaling condition (3) which make these SLFV processes analogous to the long range voter models in [6]. As for the duals, conditions (1) and (2) ensure that the single particle dual motion converges to Brownian motion, while the condition J N → ∞ ensures that the interactions between dual particles are weak.
If the sequence J N were bounded, so that the impact factors ρ = ρ/J N do not vanish in the limit, the resulting SLFV processes would correspond to the "fixed" kernel voter models in Theorem 1.2 in [6]. In biological terms this corresponds to keeping the "neighbourhood size" finite in the scaling limit, while letting J N → ∞ effectively allows this parameter to become infinite; see the discussion in Section 2 of [14] and especially Definition 2.2 there. In that work they showed in this fixed neighbourhood size setting (Theorem 2.7 of [14]) that, with an appropriate selection term, the dual particle process converges to a branching Brownian motion in the scaling limit. The purpose of this paper is to prove that in this setting, with no selection, there is also a forwards limit theorem giving convergence to SBM.
Throughout this work we will assume d ≥ 2, and N ≥ 3. (1.14) If we set J ≡ 1, and take C 1 = C 2 = 1 for simplicity, the conditions (1)-(3) in Theorem 1.0 suggest the choices for M and K above except for the logarithmic correction to K for d = 2. Without this correction, one can show that the limiting process in Theorem 1.2 would be nonrandom heat flow acting on X 0 , as is the case for the voter model [23]. We do not consider the case d = 1 in (1.14). For this case, the Wright-Fisher SPDE was obtained in [15] as an appropriate scaling limit of SLFV, but under that assumption that the scaled impact factors approach zero like N −1/3 (see [21] for the corresponding scaling limit for the voter model). If the impact factors were bounded away from zero, the strong recurrence of one-dimensional random walk would lead to heavy clustering, resulting in scaling limits with segregation of types; the corresponding scaling limit for the voter model is the Arratia flow [1], not super-Brownian motion.
In order to state our limit theorem for scaled SLFV processes assuming (1.14), we must first identify certain constants γ (d) e that appear in the limiting SBM branching rate. These constants are determined by the asymptotic tail behavior of the coalescence times τ for the unscaled two-particle dual process defined in (1.5). Introduce (1.16) Recall that when outside B 2r , ξ 1 t − ξ 2 t behaves like a rate 2ρ|B r | random walk with jump distribution given in (1.3), and τ = inf{t ≥ 0 : the difference will escape to infinity with positive probability by transience, and so the limit in (1.15), which exists by monotonicity, will have a non-zero limit. For d = 2 the situation is more delicate. One can predict the 1/ log t behaviour of γ e (t) from the corresponding non-return probabilities for irreducible symmetric random walk on Z 2 with diagonal covariance matrix (see, e.g., Lemma A.3(ii) of [6]), but the slowing rates when the difference ξ 1 − ξ 2 is in B 2r complicates things. The limit (1.16) can be derived from Lemma 4.10 in [14]. The analysis there is based on a construction using successive "inner" and "outer" excursions of ξ 1 − ξ 2 from certain balls before coalescence occurs. Our argument represents the difference process as a time change of a rate 2ρ|B r | random walk with step distribution hŪ , and makes use of a reflection coupling. We feel the proof is of independent interest and so have included it in an Appendix. One advantage of the excursion approach in [14] is that it should also allow inclusion of a random "interaction radius", that is However, as is discussed below, our time-change representation of the dual difference process in the fixed radius case will also play an important role in the analysis of the martingale square function which is the key ingredient in the proof of our main convergence result, Theorem 1.2 below. With the choice of renormalization constants in (1.14) we now give a different description of the rescaled SLFV processes X N , which will clarify the comparison with Theorem 1.2 below of the fixed kernel voter model result in [6]. Assume X 0 ∈ M F (R d ) and the compactly supported initial conditionsw N . (1.18) For each N , letw N be the (original, unscaled) SLFV process defined in (1.1) with fixed interaction radius r, fixed impact factor ρ and initial condition w N 0 =w N 0 , and define the rescaled SLFV process by This process has the same law as w N defined using Π N right before Theorem 1.0, with J and M given in (1.14).
. Thus the interaction radius for w N is r/ √ N .) Finally our approximating empirical measures are given by (1.20) so that (1.18) just asserts that X N 0 → X 0 . A simple change of variables shows that in terms of the unscaled SLFV processes,w N , we have for any bounded Borel φ on R d , Here is our main result for the scaled SLFV process. For a measure or function H, we let supp(H) denote its closed support. Recall the definition ofσ 2 from (1.4).
It is important to note that in our scaling regime with J = 1, the originalw N we are working with is an ordinary SLVF process with fixed interaction range r and impact factor ρ, but with an initial condition in which type 1's are scarce.
Equation (1.21) should be compared to the corresponding rescaled empirical measures in [6] associated with a sequence of voter models ξ In that reference it is shown thatX N converges weakly in D([0, ∞), M F (R d )) to an appropriate SBM, whose branching rate is determined by the asymptotics of the escape probability (from 0) for a continuous time random walk starting at a uniformly chosen neighbour of 0 in the integer lattice through a two-particle dual calculation. This suggests the same should hold (as it does) for the SLFV but now with the asymptotics of the non-coalescing probability of our two particle dual playing the role of the random walk escape probability.
The proof follows a familiar outline, based in part on methods in [6]. For appropriate test functions φ the semimartingale decomposition from [5], recalled in Section 2, states that is a local martingale, and D N (φ) is a drift term of bounded variation.
In Section 2 we provide some elementary simplifications for the explicit expressions for both D N (φ) and the predictable quadratic variation process M N (φ) t from [5]. In Section 3 we use the above and the one-and two-particle duals to calculate the first moments of X N t and give uniform L 2 bounds on the total mass X N t (1) (Corollary 3.2) which will be used throughout.
Assuming the key Proposition 4.1 which is proved in Section 7, tightness of {X N } is then established in Section 4, where Theorem 1.2 is also proved by showing that any weak limit satisfies the martingale problem for SBM(X 0 , σ 2 , b). The term D N (φ) is easy to handle (Lemma 2.3); it is the asymptotic behavior of the quadratic variation process M N (φ) which requires some work. The key result here is the aforementioned Proposition 4.1 which we present here for the discussion below.
After establishing preliminary random walk results in Section 5 and facts about twoparticle duals in Section 6, it is proved in Section 7. Its proof uses Proposition 1.1 but the issues go well beyond this result. The behavior of the quadratic variation process is the main difference in the proofs of Theorem 1.2 and its counterpart in [5], Theorem 1.0. Lemma 4.3 in [5] shows that a key term in the variation process is negligible in the limit N → ∞. This fact is a consequence of the assumption J → ∞. In our case, with J ≡ 1, this term is nonnegligible, and in fact determines the limiting SBM branching rate. Its analysis is the main objective of Section 7. The analysis for d ≥ 3 is straightforward; it is the 2-dimensional case (the most relevant from a biological perspective) that is the most interesting. In this setting the proof requires an extension of the arguments in [6] and [8] used to analyze the voter model and stochastic Lotka-Volterra models, respectively. The analogues of Proposition 4.1 in [6] ((I1) in that reference) and [8] (Proposition 4.7 in this work) involved L 2 and L p (p > 1 is used) norms, respectively, instead of the L 1 norm in Proposition 4.1, but also had no supremum over time in the expectation. When the L 2 norm is expanded in the voter model paper this leads to a four-particle dual calculation, while for the more general stochastic Lotka-Volterra models considered in [8], a trick using the Markov property reduced this to a three-particle dual calculation. Here, because of the non-Markovian property of individual coordinates in the dual, similar calculations seem out of reach and we are led to the L 1 convergence in Proposition 4.1 which must be established using only one-and two-particle duals. The first issue here is that squares are easier to handle than absolute values (the p > 1 in [8] is bounded eventually by a square using a stopping argument), and here the innocuous looking Lemma 7.9 below allows one to handle the square (even with a supremum over time) by using a martingale argument. This then enables us to take absolute values inside the time integral where two-particle duals (albeit more complicated ones than those in [8]) can handle the calculation. Here a second issue arises as even in handling a second moment calculation in Proposition 7.2 of [8], the use of stochastic calculus there leads to a three-particle calculation. We follow a more efficient path in its analogue, Lemma 7.8, in Section 7 which only involves the two-particle dual. A third issue is the fact that the weaker L 1 convergence in Proposition 4.1 will require some additional technical work to establish the local uniform integrability of the {M N (φ) 2 t : N }, and hence identify the limiting square function. This is what occupies most of the proof of Theorem 1.2 in Section 4. As a small bonus, the fact that Proposition 4.1 controls the square functions uniformly in time means it also allows one to establish tightness without any higher moments. The required properties of the two-particle dual are established in Section 6. Lemma 6.1 represents the difference of the coordinates of the dual as the time change of a continuous time random walk and this result is then used to obtain several probability estimates on the two-particle dual. These results (notably Lemmas 6.3 to 6.7) then play a central role in Section 7. The time-change is particularly useful when controlling the two-particle dual when the particles are close together and the dual motions slow down.
It would be interesting to see if it is possible to extend Theorem 1.2 to the variable but bounded radius case discussed above.

Constants.
In proofs, C will denote a positive constant whose value may change from line to line. We will use C T and C φ for constants depending on T > 0 or functions φ in a similar way. In some cases constants will be numbered and dependence on various quantities indicated explicitly. Finally, most constants will have an implicit dependence on the impact radius r, this dependence will be pointed out in some cases for clarity.

Semimartingale characterization of the SLFV
for all s ≥ 0.
Proof. By the dynamics (1.1), for (x, s) ∈ Π N (we may assume there is at most one such x), w N s (y) = w N s− (y) for all y / ∈ B N r (x), and for y ∈ B N r (x), Thus, |w N s (y) − w N s− (y)| ≤ ρ1 B N r (x) (y), and so, The martingale characterization below is provided by Lemma 3.1 of [5]. The filtration below is implicit in their argument. Although φ = 1 is not included in that result it is easy to handle it by a localization argument using the stopping times T n = inf{t ≥ 0 : has the semimartingale decomposition: (2.6) Implicit in the above is the fact that the local martingale M N t (φ) is locally square integrable, but this is already clear from the fact that it has bounded jumps. The latter follows from Lemma 2.1 and (2.4) which imply for all s ≥ 0. (2.7) For the drift term D N s (φ) we will need only the following facts.
Part (a) follows easily from (2.1), and (b) is the special case of Lemma 4.2 (and its proof) in [5] for our choices of J, M, K in (1.14). (We note that the constant C(d) in Definition 4.1 in [5] is B1 (x 1 ) 2 dx.) Turning next to the martingale square function, for (2.10) Proof. Define Then replacing X N s with Kw N s in (2.2), after expanding and rearranging, we find that On account of 0 ≤ w N s ≤ 1, I is nonnegative, hence (a) follows for m N from the above expression, and is immediate form n from (2.9) (integrate out z 3 in the first line on the right-hand side).
Consider the integrals over B N r (x) in (2.11). By a change of variables and order of integration, Plugging this into (2.11), and using the definition ofm N Using the fact that |I( This proves (c).

Total mass bounds
We start with the dual particle systems for the rescaled SLFV process w N t in (1.19).
If η and (ξ 1 , ξ 2 ) are as in (1.8), (1.9), introduce the rescaled duals, (3.1) Then (1.8) and (1.9) imply for Borel ψ 1 on R d , and Borel ψ 2 on (R d ) 2 , and t ≥ 0 (recall As before, either ψ i ≥ 0, or one side is integrable for the above to hold. A simple change of variables shows that (3.2) implies (for ψ 1 as above) (a) There exists C 3.5 > 0 such that for s ≥ 0, and so conclude that This completes the proof of (3.5).
(b) For d = 2, using the two particle duality equation If we plug this bound and (3.8) into (3.7), we get Plugging these bounds into (3.9) we obtain (3.6).
Proof. By Proposition 2.2 and Lemma 2.3(a), , and so is a non-negative martingale. By Doob's L 2 submartingale inequality, As we don't know the square integrability yet, the first inequality holds by considering a sequence of localizing stopping times and applying monotone convergence. By Combining the above bounds we obtain (3.10) and hence the next to last statement as well.
It is easy to repeat the above reasoning using Lemma 2.4(a) and see that This in turn shows that the local martingale M N (φ) is in fact a square integrable martingale.

Proof of main result
The proof of Theorem 1.2 proceeds by taking limits as N → ∞ in Proposition 2.2 to derive the martingale problem for the limiting super-Brownian motion. The main issue is the identification of the square function of the limiting martingale part and the key here is the following result: This will be proved in Section 7. In this section we will establish Theorem 1.2, assuming this result. If S is a metric space, recall that a sequence of laws on D(R + , S) is C-tight iff it is tight and all limit laws are continuous. C-tightness on D(R + , S)×C(R + , S) is then defined in the obvious manner. The first step is to prove: {A N : N ≥ 3} is tight, and hence relatively compact, in C(R + , R) by Prohorov's theorem. It then follows from Proposition 4.1 that the sequence of continuous (recall (2.6)) increasing processes { M N (φ) · : N ≥ 3} is relatively compact in C(R + , R), and so also tight by Prohorov's theorem again.
Therefore if 0 ≤ s < t ≤ T , then by the above and Corollary 3.2, Using (4.2) and the C- Proof. By the Kurtz-Jakubowski theorem (e.g. see Proposition 3.1 in [6]) it suffices to show: The last ( We are ready to turn to the main result. Proof of Theorem 1.2. By Proposition 4.3 it suffices to show that every weak subsequential limit is the super-Brownian motion described in the Theorem. Fix φ ∈ C 3 0 (R d ). By Lemma 4.2 and Skorokhod's theorem, and then taking a further subsequence, we may assume that we are on a probability space where Since the limit is continuous a.s. one has in fact a.s. uniform convergence on compact time intervals. It also follows from the above and Corollary 3.2 that → 0 a.s. and in L 1 as k → ∞ for all T > 0. This and Proposition 4.1 show that It follows from (4.6), (4.7), and Proposition 4.1 that Define an a.s. continuous process by Then the above, the convergence of the initial conditions in (1.18), and the semimartin- )-martingale by Corollary 3.2, it follows from the above that M (φ) is a continuous martingale and a standard argument (e.g. see the proof of Theorem 3.5 in [6]) shows it is in fact an (F X t )-martingale. Recalling (1.12), (4.8), and the value of b in Theorem 1.2, it remains to identify the square function of M (φ) as A φ by showing For d ≥ 3 this is fairly easy, but we give a stopping argument to include the more delicate The convergence in (4.11) readily shows that We claim that (4.14) The reason there is an issue here is that we do not know whether or not lim k T N k J = T J a.s. It follows from (4.13) that for t ≤ T J we have lim k T N k J ∧ t = t = T J ∧ t (the convergence is uniform for t ≤ T J ∧ T for any fixed T ) and therefore by (4.11) and (4.9), A simple calculation using (4.13) shows that (sup ∅ := 0) with probability one for any In view of the above and (4.15), to prove (4.14) it suffices to show that for T > 0 fixed, By (4.9) and (4.11) this would follow from lim sup For this we will use the following lemma, whose proof is deferred to the end of this section.
It follows from our jump bounds in (2.7) that Recalling that M N (φ) is a square integrable martingale (from Corollary 3.2), we have by optional stopping, Next use (4.13), and the convergence in (4.9) and (4.11), together with Fatou's lemma, to see that where the last is by (4.20). Let J → ∞ and then n → ∞ to prove the result for s < t fixed, as required.
hŪ (z) given in (1.3). We will need basic information about this random walk, as well as a way to compareξ t to it.
Throughout the paper, Y t = Y x t will denote a rate 2ρ|B r | random walk with jump distribution that ofŪ starting at x under P x . That is, Y x t will be the pure-jump Markov process on R d with generator defined for suitable f . We will often make use of the Poisson process construction . . which have the same law asŪ , and S n =Ū 1 + · · ·Ū n , n ≥ 1 In particular, for all x ∈ R d , t > 0, and nonnegative Borel f , Proof. (a) According to Theorem 19.1 of [3], there is a uniform bound on the densities of S n / √ n, n = 1, 2, . . . , so that By a standard large deviations estimate, for 0 < α < 1, where we have used the large deviation bound with α = 1/2. This proves (a) for Y starting at x. The result for YŪ t follows from the observation that YŪ t has the same law as S N (t)+1 and a slight alteration in the above calculation.
To see this we switch to component notation, and writeŪ j = (Ū n is a sum of bounded, mean zero independent random variables, so a martingale square function argument (e.g. see Theorem 21.1 of [4]) shows that for The first sum is bounded by (k − 1) k . The second sum is bounded by This proves (5.4) for t ≥ 1.
(c) This is immediate from (b) and Markov's inequality.
Proof. Let A > |x|. By radial symmetry and (5.1), f (x) = |x| 2−d is a harmonic function for Y . If we let σ = t a ∧ T A , then |Y s∧σ | 2−d is a bounded martingale (recall a > 2r), and so proving (5.8).  Proof. By radial symmetry and (5.1), log |x| is a harmonic function for Y . If σ = t a ∧ T A as before then log |Y s∧σ | is a bounded martingale, and (5.13) Using |Y T A | ≥ A and |Y ta | > a − 2r in the above gives Rearranging gives (5.10). A similar argument yields (5.11). For (5.12), rearranging (5.13) gives (5.14) Proof. By (5.5) with k = 2, for all x, A as in the Lemma, To handle P x (T A ≥ A 2 log A) we must first estimate E x (T 2 A ). Let σ 2 = E(|Ū | 2 ), σ 4 = E(|Ū | 4 ) and λ = 2ρ|B r |, and define the functions It is a straightforward calculation to check that both u = u 2 and u = u 4 satisfy This and the fact that for p = 2, 4, |y| p , and A Y (|y| p ) are bounded on {|y| ≤ A+2r}, so that where we have used (5.16). Let t → ∞ on the left-hand side of the above to conclude On account of this bound and Markov's inequality, we have for |x| < A/2 and A > 2, Together with (5.15) this proves (5.14).
The following technical result will play a key role in the proof of Lemma 6.5 below.
Proof. We may suppose |w| > 3r/ √ N , because otherwise C 5.18 can be chosen large enough so that the right side of (5.18) is at least one. Now for any A > |w| To handle the first term, we apply Lemma 5.3 with x = w √ N and a = 3r, Using s ≤ t and |w| ≤ (log N ) −α , and taking N ≥ N 0 (t), we see that for some C(t) > 0, Plug the above bounds in (5.20) to see that for N ≥ N 0 (t) ∨ N 1 (α, β), For the second term in (5.19), take k ≥ 1/α and use (5.5) to get  1.13)).

The two particle dual
In this section we collect some properties of the two-particle dual which will be needed in our analysis of the martingale square functions. Our main focus will be on the difference of the two particles. Define and observe that ψ r (a) is decreasing in |a| and 0 ≤ ψ r (a)/ρ|B r | ≤ 1. Consider the two-particle dual ( , and the coalescence time τ defined in (1.6). By the dynamics defining the two-particle dual (recall (1.5)), the fact that |B r (a) ∩ B r (b)| = |B r (0) ∩ B r (a − b)| shows that for y = 0,ξ makes transitions y → y +Ū at rate 2ρ|B r | − 2ψ r (y) 0 at rate ψ r (y), A is given bỹ Recall from Section 5 that Y x t is the rate 2ρ|B r | random walk starting at x ∈ R d under P x , and with jump distribution that ofŪ and generator A Y given in (5.1) for f ∈ B(R d ).
For a random variable V we let Y V denote the same random walk with initial law that of V , and will use this notation with other Markov processes below.
We will construct a version ofξ x t by absorbing a random time change of Y x at 0. Define β(y) = 1 − ψr(y) ρ|Br| and ds. (6.4) Note that for x = 0, and thus inf s≤t β(Y x s ) > 0 a.s. This implies that I(t) is finite and strictly increasing a.s. for all t. Evidently I(t) = ∞ for all t > 0 if x = 0. We will allow x = 0 later, but until otherwise indicated we will take our initial point x = 0. From the definition of I we see that for 0 < s < t, t − s ≤ I(t) − I(s).
(6.5) Therefore I −1 (t) exists for all t a.s., and , then it follows from (6.6) that for all but countably many t, and therefore that Clearly, I −1 (t) ≤ t. For x = 0 it is natural to define I −1 (t) = 0 for all t ≥ 0, which means thatỸ 0 t := Y 0 I −1 (t) = 0 for all t ≥ 0. Thus (6.7) holds for all x and We may apply Theorems 1.1 and 1.3 of Sec. 6.1 of [19] to see thatỸ x is the unique solution of the martingale problem for Here we note that the continuity of f is not needed for Theorem 1.3 of [19] in our jump process setting as the proof there shows. Uniqueness of the martingale problem is classical for such bounded jump generators, e.g., see Theorem 4.1 in Chapter 4 of [19]), and soỸ x is the unique Feller process with generator AỸ , and in particular is strong Markov. Finally we sendỸ x to its absorbing state, 0 according to the continuous additive For an independent mean one exponential random variable, e, define the absorbing time κ = κ x = inf{t ≥ 0 : C x t > e}, (6.9) and the absorbed processξ Thenξ x is a Feller jump process and an elementary calculation shows that it solves the From (6.1) we see that the two-particle dual difference,ξ, is the Feller jump process satisfying the same well-posed martingale problem, and so, as the notation suggests,ξ x has the same law asξ x . We have proved: where κ = κ x is as in (6.9), then  We often denote the starting point x ofξ in the underlying probability as P x . The tail behaviour of the coalescing time κ x will be important for us. Introduce , a ∈ R d . (6.11) Proof. By definition of κ, The following result shows that I −1 (t) is close to t, and so Y x t is a good approximation toỸ x t .

Lemma 6.3.
There is a constant C 6.3 > 0 such that for all 0 < α < 1 and t > 1, and Proof. Let Y 0 = x, |x| > 2r. By (6.4) and Y s = 0 for all s, ds. (6.14) By an elementary argument, there is a constant C 6.15 = C 6.15 (d, r) > 0 such that We are assuming |x| > 2r, so using the density bound (5.2), we see that On account of (6.15), plugging this bound into (6.14) gives This proves (6.13), because by (6.5), P x t − I −1 (t) ≥ t α ≤ P x I(t) − t ≥ t α , and we also have I −1 (t) ≤ t by I(t) ≥ t. The proof for YŪ is essentially the same.
(6.21) By (6.13) with α = 1/2 and N ≥ N 0 (q) (recall |x| > 2r), Next, using the Markov property at time u N , we have for N ≥ N 0 (q), In the next to last line we have used the d = 2 bound; if d ≥ 3, We will also need a bound on the two-particle dual ξ t = (ξ 1 , ξ 2 t ) after the coalescing time κ for any d ≥ 2. In this setting assume W 1,x1 , W 2,x2 and W 3,0 are independent rate ρ|B r | random walks in R d with step (6.23) distributionŪ (now in R d ) and starting at points x 1 , x 2 , 0 ∈ R d , respectively.
Proof. The jump rate of W to the diagonal becomes unbounded as it approaches the diagonal (for ρ = 1), so we proceed more carefully than in the proof of Lemma 6.1, making use of optional stopping. Let Here U y is uniformly distributed on B r (y 1 ) ∩ B r (y 2 ) and is independent of the uniform (on B r ) r.v. U . It is easy to check that W n solves the martingale problem forĀ n on the LetT n = inf{t ≥ 0 :ξ t ∈ R n } ≤ ∞. Using the properties ofĪ −1 and (6.28), it is easy to check that It follows that If we defineξT n t =ξ(t ∧T n ), the above impliesξT n  Here we recall again that the continuity of f assumed in Ch. 6 Theorem 1.3 of [19] is not needed in our jump process setting. A bit of arithmetic shows G n f (y) = (ρ|B r | − ψ r (y 1 − y 2 ))[E(f (y 1 +Ū , y 2 ) + f (y 1 , y 2 +Ū ) − 2f (y))]1(|y 1 − y 2 ≥ 1/n) If ξ = (ξ 1 , ξ 2 ) is the two-particle dual process, as described in (1.5), the above is the generator of the Feller pure jump process ξ(t ∧ T n ), where T n = inf{t ≥ 0 : ξ t ∈ R n } and so ξ Tn (t) = ξ(t ∧ T n ) also solves the martingale problem for G n (f ∈ B(R d × R d )).
By well-posedness of this martingale problem ((Section 2 and Thm. 4.1 of Chapter 4 of [19]) we concluded that ξ Tn andξT n are identical in law for all n ∈ N. Since R n ↓ ∅ and ξ(T n ),ξ(T n ) ∈ R n when these times are finite, it follows that T n ,T n ↑ ∞ a.s. as n → ∞ (in fact for large n they will be infinite a.s.), and therefore ξ andξ are identical in law.
The following result is now an easy consequence of (6.24), Lemma 6.6 and the bound I −1 (t) ≤ t for all t ≥ 0. Lemma 6.7. Assume ξ x is the two-particle dual in R d × R d , starting at x = (x 1 , x 2 ). Then we may assume there are random walks W i,xi (i = 0, 1, 2, x 0 = 0) as in (6.23) such that Proof. By a change of variables, Returning to the definition ofm N,1 by the martingale property of X N s (1) (Corollary 3.2). Next, we bound the difference |E(X N s (φ 2 )) − X N 0 (φ 2 )|. By the single particle duality equation (3.2) and a change of variables, Using the smoothness of φ and scaling, we see that Lemma 5.1(b) (it applies to the rate ρ|B r | walk η as well) implies Combining this bound with (7.5) and (7.6) gives (7.4).
To handlem N,2 s (φ) we apply the two-particle duality equation (3.3) and then split the resulting expression into two pieces, obtaining 1{τ N > s} dz 1 dz 2 dx. (7.9) Convergence of SLFV to SBM Lemma 7.2. There is a constant C 7.10 = C 7.10 (φ) > 0 such that for s ≥ 0, Proof. By translation invariance, changing of variables and order of integration, we see Changing variables again with x = x + ξ N,1 s and adding and subtracting φ 2 (x ), the right-side above equals For fixed z 1 , z 2 ∈ B r , letting z 3 = 0, Lemma 6.7 implies that using Lemma 5.1(b) for the last inequality. Plugging this bound into (7.11), we obtain which proves (7.10).
Using Lemmas 7.1 and 7.2 in (7.1) and (7.7), we arrive at the following: We turn now to the analysis of J N,2 (1) which is (7.13).
Proof. Let s ∈ [s N , 2s N ], let t N = N s/4, and define

τ > N s dy
To prove (7.15), by Lemma 7.4 it suffices to show that each E i is uniformly bounded in s ∈ [s N , 2s N ] by terms in the right side of (7.15).
where we have used (5.5). With this bound and X N 0 (·) ≤ K we obtain from the definition of E 1 that The above bound is then extended to all N ≥ 3 by increasing C. Using X N 0 (·) ≤ K again, we have It follows from Lemma 6.4, taking β = 1/3, that for N ≥ N 0 (q), As before, the above bound is then valid for all N ≥ 3 by increasing C. We split E 3 into two parts, letting G(a, b) = |YŪ I −1 (u) | > 2r for all u ∈ [a, b] . If we let G N = G(N s/2, N s) then we can write Using the above and Lemma 6.1, we see that on the above event and for u as above, where we recall that F Y t is the right-continuous filtration generated by the random walk Y . Then I −1 (s) is an (F Y t )-stopping time. By the strong Markov property of Y ,Ŷ 0 is a copy of Y starting at 0 and is independent ofF t N . SinceỸ t N is F t N -measurable, we may conclude from (7.17) and (7.16) that uniformly in x, Use this and the fact that P (τ (Ū ) > t N ) ≤ C K (from Proposition 1.1 if d = 2) in (7.18), to see that where the last line is very crude if d ≥ 3, and is an equality if d = 2. Combining the bounds for E 1 , E 2 , E 3 , E 3 , we establish (7.15). Corollary 7.3, Lemma 7.5, and q > 4 imply the following: Remark 7.7. To identify the square function of M N (φ) we will need to use the above and the Markov property to bound This means we will need to bound the expected value of the last term in (7.19) with X N s−s N replacing X N 0 . For d ≥ 3 we only need to bound the resulting double integral on the right-hand side of (7.19) by X N s−s N (1) 2 , but for d = 2 we require the following additional result.
Proof. By the duality equation (3.3) and then a change of variables, Here E 1 = (log N ) First, integrating y 2 out in E 1 yields By Lemma 6.1, switching toξ y √ N , and using I −1 (u) ≤ u, By Lemma 5.5, which is applicable because T ≥ s ≥ δ N and we consider only |y| ≤ √ δ N , the last probability above is bounded by C log 1/|y| log N (C = C(T )). It follows that By definition, We use the representation of ξ = (ξ 1 , ξ 2 ) in Lemma 6.6 and the fact that on G N ∩ {τ > N s}, ξ sN = W sN . Then, dropping the indicator of G N ∩ {τ > N s}, we see that With this bound, integrating out y 2 first and then y, it follows that Using X N 0 (·) ≤ K = log N and integrating out y 2 gives By the representation of ξ = (ξ 1 , ξ 2 ) given in Lemma 6.6, for any y, Using Lemma 5.5 again, for δ N ≤ s ≤ T , we have Combining the bounds on E 1 , E 2 , E 3 gives (7.20).
We are almost ready for the proof of Proposition 4.1. The proof is lengthy, so we separate out one of its key steps in the following lemma. For s ≥ s N define Now if j T = 2k − 1, then by (7.22), which also holds if j T is even. DefineX N T (1) = sup t≤T X N t (1). By Lemma 2.4(a), (3.5) and (3.10) we have where we have used Lemma 2.4(a), (3.5), and the martingale property of X N s (1) (Corollary 3.2). Thus from the above, (7.26), and Corollary 3.2, the left side of (7.21) is bounded above by and so to prove (4.1) it suffices to show sup and therefore So by our choice of δ, q (recall (7.14)), we need only integrate over [3δ N , T ] in (7.30).

Appendix: Proof of Proposition 1.1
From the discussion following the statement of Proposition 1.1 we may assume d = 2 throughout. Recall the definitions and time change construction from Section 6 (especially Lemmas 6.1 and 6.2), using the rate 2ρ|B r | random walk Y t , the difference processξ x t , and absorption timeτ = κ. By Lemma 6.
As the exact form of the killing rate k(x) will not be important in our arguments, we will replace it with a general radial function φ : where ↓ in |x| means non-increasing in |x|, and similarly for ↑. We assume throughout that φ has these properties. Recall the stopping times t A and T A from (5.7).
To prove (8.3) we first establish a number of properties of It is elementary that 0 ≤ Φ(x, A) ≤ 1 and that by recurrence, Φ(x, A) → 0 as A → ∞. The next two results will show that Φ(x, A) is increasing in |x|. Lemma 8.2. Let N ∈ N. If 0 = s 0 < s 1 < · · · < s N and f : R N +1 → R is bounded and ↑ in each coordinate then Proof. Let U, U 1 , U 2 , . . . be iid rv's uniform on B r , and let S m = U 1 + · · · + U m . The first step is to prove that if N = 1 then for m = 1, 2 . . . , E[f (|y|, |y + S m |)] is increasing in |y|. (8.10) Let u ≥ 0, and define h u (y) = P (|y + U | ≤ u) = |B r ∩ B u (−y)| |B r | = |B r ∩ B u (y)| |B r | . (8.11) It is easy to see that h u (y) is decreasing in |y|, so that |y + S 1 | is stochastically increasing in |y|, which proves (8.10) for m = 1. Now suppose m = 2, and consider P (|y + S 2 | ≤ u) = P (|y Clearly h u (y) depends only on |y|, and having established it is decreasing in |y|, the m = 1 case of (8.10) implies that E[h u (y + S 2 )] is decreasing in |y|, which shows |y + S 2 | is stochastically increasing in |y|, proving (8.10) for m = 2. The general inductive step for (8.10) is similar.
Proof. By monotone convergence, we may assume A, t are finite and g is bounded. Let g(0) = lim s↓0 g(s). For N ∈ N let M N ∈ N and 0 = s N 0 < s N 1 < · · · < s N M N = t satisfy s N i+1 − s N i < 2 −N for 0 ≤ i < M N , and define τ N = min{s N i : |Y s N i | > A} ∧ t. By right-continuity of |Y s |, τ N ↓ T A a.s. as N → ∞. By continuity of g on [0, ∞) and dominated convergence, It is easy to check that G N i is increasing in each of its variables, and hence applying A consequence of the strong Markov property we will use repeatedly is (8.14) Lemma 8.4. There exists C 8.15 = C 8.15 (r) > 1 such that for all k ≥ 2 and 0 < |x| ≤ k < A, . (8.15) Proof. By the monotonicity in Lemma 8.3, it suffices to prove (8.15) for x = x k = (k, 0). Assume additionally k > 6r ∨ r −1 and A > r 2 . By (8.14), |Y t3r | ≤ 3r, and monotonicity, we where we have set α(r) = Φ(x 3r , 4r) < 1. Insert this into (8.16) and rearrange to conclude where the second inequality uses k > 1/r and A > r 2 . In view of (8.17), letting C = 8/(1 − α(r)), we now have for all k > 6r ∨ r −1 ∨ 2 and A > k ∨ r 2 . It is easy to see that C can be increased so that (8.19) will hold for all k ≥ 2 and A > k, completing the proof of (8.15).
We will construct a coupling of the random walks Y t started at x = x in order to obtain good bounds on the difference Φ(x , A) − Φ(x, A). We start in discrete time. Let {U i } be iid r.v.'s which are uniformly distributed over B r , and for x ∈ H r = {(x 1 , x 2 ) : Let π denote the reflection mapping π(x 1 , x 2 ) = (−x 1 , x 2 ) and set x = π(x ). We will use a reflection coupling to define (S x n : n ≥ 0). Let H = {(x 1 , x 2 ) : x 1 ≤ 0}, , and define N c = N x,x c = min{n ≥ 1 : S x n ∈ B r (π(S x n−1 ))}.
Lemma 8.5. N c ≤ N := min{n ≥ 0 : S x n ∈ H } a.s., and so S x n ∈ H r for all 0 ≤ n < N c a.s.
The result follows.
We now define (S x n ) n≥0 by Then S x 0 = x, and it follows from Lemma 8.5 that for n < N c , S x n is in H r and so S x n = S x n , which implies that N c = min{n ≥ 0 : S x n = S x n }.
That is, N c is the coupling time of (S x n ) and (S x n ). If we let F S x n = σ(S x m , m ≤ n), then N c is an (F S x n )-stopping time, and S x is (F S x n )-adapted. We next show that S x n is an (F S x n )-random walk starting at x with step distribution U 1 , as the notation suggests.

Lemma 8.6. For any Borel
Proof. This is obvious on {N c ≤ n} (in F S x n ) since then S x n and S x n+1 equal S x n and S x n+1 , respectively. Suppose now that N c > n, and defineB =B(ω) = B r (S x n ) ∩ B r (π(S x n )), so that π(B) =B, (8.22) andB ⊂ B r (S x n ). (8.23) This last inclusion holds because S x n = S x n or π(S x n ) for all n. For simplicity we will write F n for F S x n in the rest of this proof. By the definition of S x n , P (S x n+1 ∈ A|F n )1(N c > n) = P (S x n+1 ∈ B r (π(S x n )), N c > n, S x n + U n+1 ∈ A|F n ) + P (S x n+1 / ∈ B r (π(S x n )), N c > n, π(S x n+1 ) ∈ A|F n ) = P (S x n+1 ∈B ∩ A, N c > n|F n ) + P (S x n+1 / ∈B, N c > n, π(S x n+1 ) ∈ A|F n ) = P (π(S x n+1 ) ∈B ∩ π(A), N c > n|F n ) + P (π(S x n+1 ) ∈B c ∩ A, N c > n)|F n ) (by (8.22)) = [P (S x n + π(U n+1 ) ∈B ∩ π(A)|F n ) + P (S x n + π(U n+1 ) ∈B c ∩ A|F n )]1(N c > n).
Next introduce the dependence on ω in the above, and use the fact that, conditionally on F n , S x n (ω) + π(U n+1 ) is uniformly distributed over B r (S x n (ω)) to see that if |C| is the Lebesgue measure of C, then the above evaluated at ω is a.s. equal to The result follows.
n ) denote a copy of the random walk starting at x under P x . Lemma 8.7. There is a constant C 8.7 so that for all x in the positive x 1 -axis and all n ∈ N, Proof. Use Lemma 8.5 and then the reflection principle to see that The step distribution of (S n ) has density f (u) = 2 √ r 2 − u 2 /|B r | ≤ 1/r on [−r, r]. It follows from the d = 1 version of (5.6) applied to random variables with this distribution that for a constant C = C(r), so we are done.
We now use translation invariance to extend the above to points x, x ∈ {(x 1 , 0) : = N c = min{n ≥ 1 : S x n ∈ B r (π m (S x n−1 ))} ≤ N x = min{n ≥ 0 : S x n ∈ H m }, (8.24) where the inequality is by Lemma 8.5, and The above results imply that both S x and S x are (F S x n )-random walks with step distribution U 1 , N x,x c = min{n ≥ 0 : S x n = S x n } (8.26) is their coupling time, and Next define coupled copies of the discrete time random walk with step distribution , and also setF x n = F S x 2n . We will writeF n for F x n if there is no ambiguity. Then it follows from Lemma 8.6 that bothŶ x n andŶ x n are (F n )-random walks with step distribution U 1 + U 2 , that is, they are (F n )-adapted and P (Ŷ x n+1 ∈ A|F n )(ω) = P (Ŷ x n (ω) + U 1 + U 2 ∈ A) a.s., (8.28) and similarly forŶ x . It follows easily from (8.26 Proof. By (8.29) The result follows from (8.27).
We move now to the continuous time random walks. Let N (t) be an independent Poisson process with rate λ = 2ρ|B r | and jump time sequence (s n ) n∈Z+ , i.e., s n = inf{t ≥ 0 : N t = n}. For K > 0 put x = (K + 2r, 0), and let x ∈ [K, K + 2r) × {0}. Define coupled continuous time rate λ random walks with step distribution U 1 + U 2 , starting at x and x, respectively, by The coupling time of these random walks is , and so by setting n = N t in (8.30), we have Let F t be the right-continuous filtration generated by (Y x , Y x , N ), and let Y t (respec-tivelyŶ n ) denote a generic rate λ continuous time (respectively, discrete time) random walk with step distribution U 1 + U 2 , starting at 0 under P 0 . Lemma 8.10. (a) Both Y x and Y x are rate λ continuous time (F t )-random walks (and (F t )-strong Markov processes) with jump distribution U 1 + U 2 . That is for y = x or x , t > 0, and any a.s. finite (F t )-stopping time S, P (Y y S+t ∈ A|F S )(ω) = P 0 (Y y S (ω) + Y t ∈ A) a.s. for any Borel A ⊂ R 2 . For y = x or x and 2r ≤ δ < A we let t y δ = inf{t ≥ 0 : |Y y s | ≤ δ}, T y A = inf{t ≥ 0 : |Y y t | ≥ A}, and also set We definet y δ ,T y A ,t x,x δ , andT x,x A in a similar way, using the discrete time random walkŝ Y x ,Ŷ x , for example,t y δ = min{n ≥ 0 : |Ŷ y n | ≤ δ}. Lemma 8.11. Let K > 3r and x = (K + 2r, 0).  (c) LetŶ (1) be the first coordinate ofŶ , and let x, x , m be as above, with K > δ ≥ 3r, so that δ < |x| ≤ |x | < 2K. Let n = K 2−2ε . Then, using Lemma 8.9 for the second inequality and symmetry for the second to last inequality, we have k | ≥ K/2 , (8.36) provided K is larger than some K 0 (δ) > 0. We recall Theorem 21.1 in [4], which in the present context implies If we take p = p 0 (ε) large enough so that K pε > K 1−ε , substituting into (8.36) we obtain for a constant C > 0 depending on ε, . Multiplication of C by a large enough constant depending on δ allows us to remove the restriction K > K 0 (δ). That is, for some C 8.34 (δ, ε) > 0, So (c) is now immediate from (8.38).
As a consequence, so that ∆ = ∆ 1 + ∆ 2 , and bound ∆ 1 , ∆ 2 separately. For ∆ 1 , using (8.40) and T x A = t x 3r , ). (8.41) It suffices to consider the first term, as the second follows in the same way. By the strong Markov property (T A < t 3r )]. Now taking a = 3r in Lemma 5.3, and noting that 3r a.s.
Choose K 0 large enough so that K > K 0 implies K 2 > 2K r + 2. If, in addition we have K > K 0 and A > r 2 , then By replacing 8 with a sufficiently large constant C we may drop the additional conditions K > K 0 and A > r 2 , and so obtain for all K > 5r ∧ 2 and A > 2K + 2r, Plug this bound into (8.42) and use the coupling bound Lemma 8.11(c) with δ = 3r, ε = 1/2 to obtain .
The above and (8.41) imply .
Now consider ∆ 2 . Recalling from (8.40) that t x 3r ≤ t x 3r , ∆ 2 is bounded by the sum of In ∆ 2a , the event in the indicator function belongs to F t x . (8.44) Finally, consider ∆ 2b . Dropping the exponential and applying the strong Markov property to Y x at time t x 3r , we have (T A < t 3r ) . 3r | ≤ 2K + 10r. Let K 0 be large enough so that K > K 0 implies K 2 > (2K/r) + 10, and assume additionally that K > K 0 and A > (2K + 10r) ∨ r 2 . By the hitting probability bound (5.11), with a = 3r we see that if |Y x t x 3r | > 3r, then The same bound holds if |Y x T x 3r | ≤ 3r because then the left-hand side is zero. Now the additional restrictions on K, A can be dropped by replacing 8 with a larger constant C, so we may conclude that for A, K as in the Lemma and on {t x 3r < S c }, (T A < t 3r ) ≤ C log K log(A 2 ) .