On the boundary at infinity for branching random walk

We prove that supercritical branching random walk on a transient graph converges almost surely under rescaling to a random measure on the Martin boundary of the graph. Several open problems and conjectures about this limiting measure are presented.


Introduction
The asymptotic behavior of Markov chains on graphs is at the core of potential theory, which blends together tools from probability, algebra and analysis.Historically, much of this theory has focused on reversible Markov chains, with particular emphasis given to random walks on graphs and groups following the influential early work of Furstenberg [Fur63,Fur71a,Fur71b].A cornerstone of this theory is the Poisson boundary, a measure-theoretic object encoding all possible (positive probability) limiting behaviours of the random walk and which, by a foundational theorem of Blackwell [Bla55], may be defined equivalently either in terms of the invariant σ-algebra or the space of bounded harmonic functions on the Markov chain.The Martin boundary is a topological boundary that both recovers the measure-theoretic Poisson boundary and also includes information about possible singular (probability zero) behaviours of the walk at infinity.
While the Poisson and Martin boundaries of a Markov chain are rather abstract and difficult to study in general, they are known to coincide with concrete geometric compactifications in several classes of examples including hyperbolic groups [Anc87, Anc88, Kai00], planar graphs [Geo16, ABGGN16, HP17], lamplighter groups [KV83,Ers10,LP15], and Diestel-Leader graphs [Woe05].In general, however, the structure of the Martin boundary might be richer than what one would guess geometrically [PW94].For random walks on groups, various important and beautiful works have established characterizations and criteria for the non-triviality of the Poisson and Martin boundaries in terms of speed, entropy, heat kernel decay and volume growth [Ave74, KV83, PZ20, AK17, FHTF19].See [Woe00,LP16,Zhe21] for overviews of the literature.
In this work we aim to initiate a systematic investigation of the boundary theory of branching random walks.Let M = (S, P ) be a Markov chain with countable state space S and transition matrix P , and let µ be an offspring distribution, that is, a probability measure supported on the non-negative integers.Branching random walk (BRW) on M is a Markov process governed by P and µ in the following sense.Choose any state x ∈ S and at time 0 place one alive particle at x. Inductively, at each time step all alive particles reproduce independently of one another according to µ and then die, so that each particle is replaced by an independent random number of offspring with law µ.Subsequently, as part of the same time step, all (alive) particles take one step of the Markov chain, independently of one another, according to the transition kernel P .This defines a Markov process (B n ) n≥0 with state space Ω = {finitely supported functions S → N}, where B n (x) is the number of (alive) particles at the state x at time n.One can also equivalently consider the branching random walk as a tree-indexed Markov chain where the underlying genealogical tree of the particles is itself a random Galton-Watson tree [BP94a,BP94b].In particular, it follows from the classical theory of branching processes that the process dies almost surely if the mean μ = nµ(n) satisfies μ < 1 and survives forever with positive probability when μ > 1.A detailed account of what is known about branching random walk on Z, a rich subject that is largely disjoint from the kind of questions we consider here, can be found in [Shi15].
Previous works on the behaviour at infinity of branching random walk have focused either on the geometric properties of the trace (that is, the graph spanned by the edges crossed by the process) [BM12, CGM12, CR15, Hut20] or the set of points accumulated to by the process in some geometric compactification.The latter approach has been carried out primarily in the setting of trees and hyperbolic spaces, where a detailed and sophisticated theory has now been developed [LS97, KPS98, HL00, LS00, Lal06, CGM12, GL13, Gou14, SWX20].
Our main theorem states that branching random walks always converge under suitable rescaling to a random measure on the Martin boundary.We are optimistic that this random measure is both an interesting object of study and its own right and should lead to new perspectives and clarity on the limit-set theory in the hyperbolic case.We write P x and E x for probabilities and expectations taken with respect to the law of the branching random walk started with a single particle at x and write P x and E x for probabilities and expectations taken with respect to the underlying Markov chain started at x. Theorem 1.1.Let M = (S, P ) be a transient, irreducible Markov chain, let µ be an offspring distribution with mean μ > 1 satisfying the L log L condition ∞ n=1 µ(n)n log n < ∞, and let B = (B n ) n≥0 be a branching random walk on M started at some state o.Then (μ) −n B n almost surely converges weakly to a random measure W on the Martin compactification M of M that is supported on the harmonic Martin boundary of M and satisfies for every Borel set A ⊆ M.
A selection of open problems regarding this random measure are discussed in Section 4.
The so-called 'L log L condition' ∞ n=1 µ(n)n log n < ∞ appearing here is needed for the limiting measure W to be non-zero.Indeed, if µ is supercritical and (|B n |) n≥0 is the underlying branching process of the branching random walk (B n ) n≥0 then the limit exists almost surely by the martingale convergence theorem.The Kesten-Stigum theorem [KS66,LPP95] states that the following are equivalent: 1. W > 0 with positive probability.

EW
Since any measure arising as the weak limit of (μ) −n B n must have total mass W , it follows that if µ does not satisfy the L log L condition then (μ) −n B n almost surely converges weakly to the zero measure.(In particular, the conclusions of Theorem 1.1 other than the formula (1.1) hold vacuously in this case.) Remark 1.2.The Martin boundary of branching processes (and hence of the Markov chain governing the number of particles in a branching random walk) has been investigated in [Ove94] and [Loo77], whose results imply in particular that the random variable W = lim n→∞ (μ) −n |B n | generates the entire tail σ-algebra of the branching process (|B n |) n≥0 .For more recent results we refer the reader to [AD19] and references therein.While it is natural to guess from this that the random measure W encodes all the tail information of the branching random walk (B n ) n≥0 , we observe in Section 4 that this is not the case even in very simple examples.
Remark 1.3.While we were in the final stages of preparing this paper, Kaimanovich and Woess [KW22] posted a preprint to the arXiv establishing, in independent work, similar results on the convergence of branching random walks to random measures; their results allow one to consider compactifications other than the Martin compactification and branching Markov chains where the offspring distribution depends on the location of the particle.

Background on the Martin boundary
We now briefly recall the definition and basic properties of the Martin boundary, referring the reader to [Woe00] and [Dyn69] for further details.Let P be the transition matrix of an irreducible, transient Markov chain M = (S, P ) on a countable state space S. Recall that a function h : S → R that is either non-negative or bounded1 is said to be harmonic if h(x) = y∈S P (x, y)h(y) for every x ∈ S and is said to be superharmonic if h(x) ≥ y∈S P (x, y)h(y) for every x ∈ S. The space of all non-negative superharmonic functions on S is denoted by S + .For each y ∈ S, the Martin kernel K( • , y) is the non-negative superharmonic function defined by where G(x, y) = n≥0 P n (x, y) is the Green's function.The Martin compactification M = M(M ) is defined to be the unique smallest compactification of the discrete set S for which all the Martin kernels K(x, • ) extend continuously.A detailed explanation of how this property uniquely determines M can be found in [Woe00, Section 24], where it is also shown that M is a Polish space.Thus, a sequence of states (y n ) n≥0 converges in the Martin compactification if and only if the Martin kernels K(x, y n ) converge for each x ∈ S. Note that this does indeed give a compact space since G(o, y) ≥ sup n P n (o, x)G(x, y) for every x, y ∈ S and hence 0 ≤ K(x, y) ≤ (sup n P n (o, x)) −1 < ∞ for every x, y ∈ S. The Martin boundary of M is the space ∂M = M\S.Note that if ξ ∈ ∂M M then we can define the Martin kernel K( • , ξ) := lim n→∞ K( • , y n ) where (y n ) n≥0 is any sequence of states converging to ξ; the choice of sequence does not affect the limit thus obtained.
The most important properties of the Martin boundary are as follows: 1. Convergence of the random walk [Woe00, Theorem 24.10]:If (X n ) n≥0 is a trajectory of the Markov chain M then X converges almost surely to a limit point X ∞ ∈ ∂M.Moreover, the Martin kernel K(x, • ) is the Radon-Nikodym derivative of the law of X ∞ under P x with respect to the law of X ∞ under P o in the sense that for every x ∈ S and every continuous function 2. Representation of positive harmonic functions [Woe00, Theorem 24.7]:If h : S → [0, ∞) is a non-negative harmonic function then there exists a finite Borel measure ν h on ∂M such that for every x ∈ S. Conversely, if ν is a finite Borel measure on ∂M then K(x, ξ)dν(ξ) is a non-negative harmonic function on S. The function h is bounded if and only if ν h may be taken to be absolutely continuous with respect to the law of X ∞ with essentially bounded Radon-Nikodym derivative; in particular, if h : S → R is a bounded harmonic function then there exists a bounded Borel function φ : ∂M → R such that for every x ∈ S [Woe00, Theorem 24.12].In both cases, the representation is not unique in general but can be made unique by restricting the boundary data to be supported on the minimal Martin boundary [Woe00, Theorem 24.9].
Remark 2.1.Let us warn the reader that the Martin boundary can be quite ill-behaved in general, and in particular that the harmonic extension of a continuous function need not be continuous on the Martin compactification.

Proof of the convergence theorem
For the remainder of this section we fix a countable-state Markov chain M = (S, P ) and a vertex o such that sup n P n (o, x) > 0 for every x ∈ S, fix an offspring measure µ satisfying the L log L condition, and let B n be a branching random walk on M with offspring measure µ, started with a single particle at o.We write Bn = (μ) −n B n for each n ≥ 0 and, given a function f : S → R, write f, Bn := x∈S f (x) Bn (x).
Lemma 3.1.If h : S → R is harmonic and is either bounded or non-negative then ( h, Bn ) n≥0 is a martingale with respect to the filtration generated by (B n ) n≥0 .Similarly, if h : S → R is superharmonic and is either bounded or non-negative then ( h, Bn ) n≥0 is a supermartingale with respect to the filtration generated by (B n ) n≥0 .
Proof of Lemma 3.1.Let F n be the σ-algebra generated by (B i ) n i=0 .If h : S → R is either bounded or non-negative then we have by linearity of expectation that where in the second equality we have used that E [B n+1 (x) | F n ] = μ y∈S B n (y)P (y, x) by linearity of expectation and the definition of branching random walk.The claim follows since P h = h when h is harmonic and P h ≤ h when h is superharmonic.
Corollary 3.2.If h : S → R is superharmonic and is either bounded or non-negative then the sequence h, Bn converges almost surely as n → ∞.
Lemma 3.3.Let V ⊆ ∂M be closed and let h(x) := P x (X ∞ ∈ V ) be the harmonic extension of ½ V to S. Then sup{W(V ) : W is a subsequential weak limit of Bn } ≤ lim n→∞ h, Bn almost surely.
Proof of Lemma 3.3.We have by continuity of measure that for each ε > 0 there exists an open neighbourhood U of V in M such that (Here we are implicitly using that, since ∂M is metrisable, each closed set can be written as the intersection of countably many open sets, namely the 1/k neighbourhoods of the set in some compatible metric.)Fix one such choice of ε and U and let f : S → [0, 1] be the function f (x) := P x (X n ever visits U ).
Note that f is harmonic on S \ U , superharmonic on S ∩ U , and satisfies f ≥ h and f ≥ ½ U everywhere.Since f, Bn is a non-negative supermartingale, it converges a.s., and since U is open we have by the portmanteau theorem that sup{W(V ) : W is a subsequential weak limit of Bn } ≤ sup{W(U ) : W is a subsequential weak limit of Bn } On the other hand, since f − h is non-negative, superharmonic, and satisfies and hence by Markov's inequality that P sup{W(V ) : W is a subsequential weak limit of Bn } ≥ lim n→∞ h, Bn + δ for every δ > 0. Since ε, δ > 0 were arbitrary it follows that sup{W(V ) : W is a subsequential weak limit of Bn } ≤ lim n→∞ h, Bn almost surely as claimed.
We next apply this one-sided estimate for closed sets to deduce an equality holding for each open set, closed set, and continuous function.1.Let A ⊆ ∂M be either open or closed and let h(x) := P x (X ∞ ∈ U ) be the harmonic extension of ½ U to S. The event "every subsequential limit W of Bn satisfies W(A) = lim n→∞ h, Bn " has probability one.
2. Let φ : ∂M → R be continuous and let h be the harmonic extension of φ to S. The event "every subsequential limit W of Bn satisfies φ(ξ)dW(ξ) = lim n→∞ h, Bn " has probability one.
Proof of Lemma 3.4.We begin with claim 1.We first prove that if A = U ⊆ ∂M is open then sup{W(U ) : W is a subsequential weak limit of Bn } ≤ lim of ½ U c to S is given by 1 − h.Thus, it follows from Lemma 3.3 that sup{W(U c ) : W is a subsequential weak limit of Bn } ≤ lim almost surely, and since 1dW(ξ) = lim n→∞ 1, Bn = W < ∞ almost surely for every subsequential weak limit W of Bn , we can rearrange (3.2) to deduce that inf{W(U ) : W is a subsequential weak limit of Bn } ≥ lim We now turn to the second claim.We may assume without loss of generality that φ takes values in [0, 1].For each t ∈ [0, 1] let V t = {ξ ∈ ∂M : φ(ξ) ≥ t} and let h t (x) := P x (X ∞ ∈ V t ) be the harmonic extension of ½ Vt to S, so that φ = 1 0 ½ Vt dt and h = 1 0 h t dt.We have by Lemma 3.3 that sup φ(ξ)dW(ξ) : W is a subsequential weak limit of Bn The open-set case of Lemma 3.4 has the following immediate corollary.Recall that the harmonic Martin boundary ∂M harm is defined to be the support of the law of X ∞ under P o , that is, the smallest closed subset of ∂M for which P o (X ∞ ∈ ∂M harm ) = 1.
Corollary 3.5.The event "every subsequential limit of Bn is supported on ∂M harm " has probability one.

Proof. The complement ∂M c
harm is open and has harmonic extension h(x) = P x (X ∞ ∈ ∂M c harm ) ≡ 0 by definition.The claim therefore follows immediately from Lemma 3.4.
We next use the separability of ∂M harm to exchange the order of the quantifiers in the second item of Lemma 3.4.
Proof.The Martin compactification ∂M is compact and metrisable, and since S is dense in ∂M it is separable also.Since ∂M harm is a closed subset of ∂M, it is also a compact, separable, metrisable space.As such, the space C(∂M harm ) of continuous functions on ∂M harm is separable also.(Indeed, if d is a metric compatible with the topology of ∂M harm and Q is a countable dense subset of ∂M harm then the Q-algebra generated by the continuous functions of the form {d( • , q) : q ∈ Q} is a countable dense subset of C(∂M harm ).) Let A be a countable dense subset of C(∂M harm ).We have by Corollary 3.2 and Lemma 3.4 that the event Ω = {W = lim n→∞ 1, Bn exists and is finite, and for each function φ ∈ A with harmonic extension h the sequence ( h, Bn ) n≥0 converges and every subsequential weak limit W of Bn satisfies φ(ξ)dW(ξ) = lim n→∞ h, Bn } has probability one.Now, if ψ is an arbitrary continuous function on ∂M, then for each ε > 0 there exists φ ∈ A with |φ − ψ| ≤ ε, so that if h and h ′ denote the harmonic extensions of h and h ′ to S then |h − h ′ | ≤ ε.Thus, on the event Ω we have that sup ψ(ξ)dW(ξ) : W a subsequential limit of Bn where the final inequality follows since | h − h ′ , Bn | ≤ ε 1, Bn for every n ≥ 0. We also have symmetrically that inf ψ(ξ)dW(ξ) : W a subsequential limit of Bn holds pointwise on the event Ω.The claim follows since ψ and ε > 0 were arbitrary.
We may now conclude the proof of the main theorem.Note that we have not yet used the L log L condition.This will be needed only to verify the identity (1.1); if the L log L condition does not hold then Bn converges to the zero measure!Proof of Theorem 1.1.We have by Corollary 3.5 and Lemma 3.6 that the event " every subsequential weak limit of Bn is supported on ∂M harm , and if φ : ∂M harm → R is continuous then φ(ξ)dW 1 (ξ) = φ(ξ)dW 2 (ξ) for any two subsequential weak limits W 1 , W 2 of Bn " holds almost surely.(Again, it is very important that the quantifier over all continuous functions and subsequential limits is inside the event being defined!)The Riesz-Markov-Kakutani representation theorem implies that measures on a compact metric space are determined by their integrals against continuous functions, so that all subsequential weak limits of Bn are identical almost surely.Since the space of measures on M is a compact metrisable space under the weak topology, it follows that Bn converges almost surely to a (random) limit measure W as claimed.Now, if the offspring measure µ satisfies the L log L condition then we have by the Kesten-Stigum theorem that the martingale ( 1, Bn ) n≥0 is uniformly integrable.As such, if the L log L condition holds then ( h, B n ) n≥0 is a uniformly integrable martingale for each bounded harmonic function h and we have by Lemma 3.4 that if φ : ∂M → R is a continuous with harmonic extension h then This implies that the expectation of W and the law of X ∞ determine the same measure on ∂M, which is equivalent to the claimed equality (1.1).

Open problems
As mentioned in Remark 1.2, since W generates the tail σ-algebra of the branching process (|B n |) n≥0 [Loo77,Ove94] and X ∞ generates the invariant σ-algebra of the underlying random walk, it is natural to wonder whether the random measure W generates the tail σ-algebra of the branching random walk.Unfortunately this is not the case even in some very simple examples.Indeed, consider branching random walk on Z 3 with deterministic branching governed by the degenerate offspring measure µ(16) = 1.Since the Martin boundary of Z 3 consists of a single point {1} and W = 16, W is deterministically equal to a point mass of mass 16 at 1. On the other hand, if we start with a single particle at the origin, then the number of particles at (n, 0, 0) at time n is itself a branching process with Binomial(16, 1/8) offspring distribution.Since this offspring distribution is supercritical but has positive probability to die after a single step, it survives forever with probability strictly between 0 and 1.As such, {B n (n, 0, 0) > 0 for infinitely many n} is a non-trivial tail event that is not in the σ-algebra generated by W. Similarly, {max{x : B n (x, 0, 0) > 0} > − max{x : B n (−x, 0, 0) > 0} for all sufficiently large n} is a non-trivial invariant event for the branching random walk that does not belong to the σ-algebra generated by W. This examples suggests that the following problem is non-trivial.
Problem 4.1.Characterise the σ-algebra generated by the limiting measure W.
As discussed in the introduction, much of the existing literature on the boundary behaviour of branching random walk focuses on the set of accumulation points of the walk [LS97, KPS98, SWX20], and it would be interesting to reinterpret these analyses through the lens of Theorem 1.1.To do this, an important first step would be to compare the set of limit points with the support of W. Again, simple examples show that these are not always the same: If one performs a supercritical branching random walk on the lamplighter group Z 2 ≀ Z then every vertex is visited infinitely often, so that the set of limit points is the entire Martin boundary, which has more than one point by, say, the results of [AK17].Meanwhile, the measure W is supported on the harmonic Martin boundary, which in this case is the single point {1} since Z 2 ≀ Z is Liouville.Still, we expect that in 'sufficiently nice' examples, such as hyperbolic groups, the two sets will coincide.
Problem 4.2.Give conditions under which the support of W coincides almost surely with the set of accumulation points of B.
For hyperbolic groups there are many further questions one could ask about the limiting measure W, mirroring what is known about the limit set.For example one can try to compute the Hausdorff dimension of the measure, or study the tail probability that an unusually large mass is placed on a small ball.
Problem 4.3.Study the fine properties of the limiting measure W for branching random walk on a hyperbolic group.
Much of the interest in branching random walk on nonamenable groups stems from the fact that there are two phase transitions: The transition between certain death and possible survival when μ = 1 and the transition between transience and recurrence when μ = 1/ρ, where ρ is the spectral radius of the random walk: when μ ≤ 1/ρ each point is visited at most finitely often almost surely while when μ > 1/ρ every point is visited infinitely often almost surely on the event that the walk survives forever [BP94b,BZ08,GM06].It is likely that the answers to each of the questions above are different in the two cases μ ≤ 1/ρ and μ > 1/ρ, with the possibility of particularly interesting behaviour at or near the boundary case μ = 1/ρ.
It is also natural to wonder how this phase transition manifests itself in the behaviour of the limiting measure W. One plausible scenario is as follows: For random walks on nonamenable groups, the random measure W is always singular with respect to harmonic measure (i.e. the law of X ∞ ), but is, in some qualitative sense, 'more singular' in the transient regime μ ≤ 1/ρ than in the recurrent regime μ > 1/ρ.While we are not confident enough to make a formal conjecture, it is also plausible that W is almost surely supported on a strict subset of ∂M harm whenever μ ≤ 1/ρ.This is true for hyperbolic groups by the results of [SWX20].The results of [Hut20] also suggest that the support of W is always totally disconnected in the transient regime.
where h is the harmonic extension of ½ U to S. Since ∂M is metrisable, the open set U may be written as a countable union of closed sets U = m≥1 V m .(For example, if d is a compatible metric on ∂M then U can be written as the union of the sequence of closed sets V m = {ξ ∈ ∂M : d(ξ, U c ) ≥ 1/m}.)Fix one such representation U = m≥1 V m and for each m ≥ 1 let h m (x) := P x (X ∞ ∈ V m ) be the harmonic extension of ½ Vm to S. Since h m ≤ h for each m ≥ 1, we have by continuity of measure and Lemma 3.3 that sup{W(U ) : W a subsequential weak limit of Bn } = sup m sup{W(V m ) : W a subsequential weak limit of Bn } ≤ sup m lim n→∞ h m , Bn ≤ lim n→∞ h, Bn almost surely as claimed.Now, if U is open then U c = ∂M\U is closed and the harmonic extension Together (3.1) and (3.3) imply the open case of the claim, and the closed case follows by taking complements as in the proof of (3.2).