Branching Brownian motion in an expanding ball and application to the mild obstacle problem

We first consider a d-dimensional branching Brownian motion (BBM) evolving in an expanding ball, where the particles are killed at the boundary of the ball, and the expansion is subdiffusive in time. We study the large-time asymptotic behavior of the mass inside the ball, and obtain a large-deviation (LD) result as time tends to infinity on the probability that the mass is aytpically small. Then, we consider the problem of BBM among mild Poissonian obstacles, where a random ‘trap field’ in Rd is created via a Poisson point process. The trap field consists of balls of fixed radius centered at the atoms of the Poisson point process. The mild obstacle rule is that when a particle is inside a trap, it branches at a lower rate, which is allowed to be zero, whereas when outside the trap field it branches at the normal rate. As an application of our LD result on the mass of BBM inside expanding balls, we prove upper bounds on the LD probabilities for the mass of BBM among mild obstacles, which we then use along with the Borel-Cantelli lemma to prove the corresponding strong law of large numbers. Our results are quenched, that is, they hold in almost every environment with respect to the Poisson point process.


Introduction
In this work, we first study a branching Brownian motion (BBM) in an expanding ball of fixed center, where the radius of the ball is increasing subdiffusively in time. We suppose that the boundary of the ball is killing in the sense that once a particle of the BBM hits the boundary, it is instantly killed. On this model, in Theorem 1, we obtain a large-deviation result, giving the large-time asymptotic behavior of the probability that the number of particles, i.e., the mass, of the BBM is atypically small in the expanding ball.
Then, we consider the model of BBM among mild Poissonian obstacles. We study the growth of mass of a BBM evolving in a random environment in R d , which is composed of randomly located spherical traps of fixed radius with centers given by a Poisson point process (PPP). The mild obstacle rule is that when a particle is inside the traps, it branches at a lower rate, which is allowed to be zero, than usual, that is, when it is not in a trap. The mild obstacle problem for BBM was proposed by Engländer in [6], and on a set of full measure with respect to the PPP, a kind of weak law of large numbers (WLLN) was obtained (see [6,Thm. 1]) for the mass of the process as well as a result on its spatial spread (see [6,Thm. 2]). In Theorem 2, by estimating successive large-deviation probabilities, we improve the WLLN in [6] to the strong law of large numbers (SLLN). We also include the possibility of no branching inside the traps, which was not covered in [6]. An essential ingredient in the proof of Theorem 2 turns out be Theorem 1, that is, the lower tail asymptotics for the mass of BBM in expanding balls. In the mild obstacle problem, a suitable time-dependent clearing (see Definiton 1) in the random environment in R d serves as the expanding ball of Theorem 1.

Formulation of the problems
We now describe the two sources of randomness in the models, and formulate the problems in a precise way.
1. Branching Brownian motion in an expanding ball: Let Z = (Z t ) t≥0 be a strictly dyadic d-dimensional BBM with branching rate β > 0, where t represents time. Strictly dyadic means that every time a particle branches it gives precisely two offspring. The process can be described as follows. It starts with a single particle, which performs a Brownian motion (BM) in R d for a random lifetime, at the end of which it dies and simultaneously gives birth to two offspring. Similarly, starting from the position where their parent dies, each offspring particle repeats the same procedure as their parent independently of others and the parent, and the process evolves through time in this way. All particle lifetimes are exponentially distributed with constant parameter β > 0. For each t ≥ 0, Z t can be viewed as a finite discrete measure on R d , which is supported at the positions of the particles at time t. For t ≥ 0, we use |Z t | to denote the total mass of Z at time t, and occasionally use N t := |Z t |. Also, for a Borel set A ⊆ R d and t ≥ 0, we write Z t (A) to denote the mass of Z that fall inside A at time t.
We also define a BBM with killing at a boundary. For a Borel set A ∈ R d , denote by ∂A the boundary of A. Consider a family of Borel sets B = (B t ) t≥0 , which we may view as a single timedependent Borel set. Let Z B = (Z Bt t ) t≥0 be the BBM with killing at ∂B, that is, a standard BBM with particles killed instantly upon hitting the boundary of B. The process Z B can be obtained from Z as follows. For each t ≥ 0, start with Z t , and delete from it any particle whose ancestral line up to t has exited B t to obtain Z Bt t . This means, Z Bt t consists of particles of Z t whose ancestral lines up to t have been confined to B t over the time period [0, t], and therefore it can be viewed as a finite discrete measure in B t .
We denote by Ω the sample space for the BBM, and use P x and E x , respectively, to denote the law and corresponding expectation of a BBM starting with a single particle at x ∈ R d . By an abuse of notation, we use P x and E x also for the BBM with killing at a boundary. For simplicity, we set P = P 0 . Also, we sometimes use n t := |Z Bt t | to denote the mass at time t of a BBM with killing at ∂B. Consider a radius function r : R + → R + with lim t→∞ r(t) = ∞, which is subdiffusively increasing, that is, r(t) = o( √ t) as t → ∞. For t > 0, let B t := B(0, r(t)), and p t be the probability of confinement to B t of a standard BM (starting from the origin) over [0, t]. In the first part of the current work, for a suitably decreasing function γ : R + → R + with lim t→∞ γ(t) = 0, we find the asymptotic behavior as t → ∞ of the LD probability where we have set γ t = γ(t) for convenience. It is easy to show that E[n t ] = p t e βt ; therefore, since lim t→∞ γ(t) = 0, for large t, γ t p t e βt is atypically small for the mass of a BBM with killing at ∂B.
2. Trap field and mild obstacle problem for BBM: The setting of random obstacles in R d is formed as follows. Let Π be a Poisson point process (PPP) in R d with constant intensity ν > 0, and (Ω, P) be the corresponding probability space with expectation E. By a trap associated to a point x ∈ R d , we mean a closed ball of fixed radius a > 0 centered at x, and by a trap field, we mean the random set whereB(x, a) denotes the closed ball of radius a centered at x ∈ R d .
In the second part of the current work, the BBM is assumed to live in R d , to which K is attached. For ω ∈ Ω, we refer to R d with K(ω) attached simply as the random environment ω, and use P ω to denote the conditional law of the BBM in the random environment ω. The mild obstacle problem for BBM has the following rule: when a particle of BBM is in K c , it branches at rate β, whereas when in K, it branches at a lower rateβ with 0 ≤β < β. That is, under the law P ω , the BBM has a spatially dependent branching rate We emphasize that here we allow the possibility of complete suppression of branching in K, that is,β = 0, which was not considered in [6]. Our focus is on the total mass of BBM among mild obstacles. We first find upper bounds that are valid for large t on the LD probabilities that the mass is atypically small and atypically large. Then, via Borel-Cantelli arguments, we obtain the corresponding SLLN. The result is valid in almost every environment ω; hence, it is called a quenched SLLN.

History
The study of branching diffusions in restricted domains with absorbing boundaries goes back to Sevast'yanov [18], who studied the survival of such systems in bounded domains in R d . In [11], Kesten studied a BBM with negative drift in one dimension starting with a single particle at position x > 0 in the presence of absorption at the origin. He obtained a survival criterion depending on the drift of the process, and an asymptotic result on the number of particles in a given interval. Later, various further results were obtained on the one-dimensional model with absorption at a one-sided barrier. In [14], considering a BBM starting with a single particle at the origin and with a strong enough negative drift so as to make extinction almost sure, Neveu studied the process (Z x ) x≥0 formed by the total mass that is frozen upon exiting ((−∞, −x), x ≥ 0). Berestycki et al. followed up on Kesten's model of BBM with absorption at the origin, and in [1] and [2] studied, respectively, the survival probability of the BBM near the critical drift as a function of x > 0, and the genealogy of the process. In [8], on the same model, Harris et al. studied the one-sided FKPP travelling-wave equation, and obtained several results on the asymptotic speed of the rightmost particle, the almost sure exponential growth rate of particles having different speeds, and the asymptotic probability of presence of the BBM in the subcritical speed area. Then, in [13], Maillard improved on Neveu's work in the case where the process goes extinct almost surely, and obtained precise asymptotics on the number of absorbed particles at the linear one-sided barrier. More recently in [9], Harris et al. studied a BBM with drift in a fixed-size interval, that is, a two-sided barriered version of Kesten's model, and obtained a survival criterion involving a critical width for the interval, and also the asymptotics of the near-critical survival probability. The one-dimensional model involving a BBM with drift and a fixed barrier is equivalent to the model involving a BBM with no drift and a linearly moving barrier. The first part of the current work gives a large-deviation result in the downward direction on the population size of a BBM in a subdiffusively expanding (time-dependent) ball in d dimensions with killing boundary.
The second part of the current work is about a BBM among mild obstacles in R d . The study of branching diffusions among random obstacles in R d goes back to Engländer [3], who studied a BBM among hard Poissonian obstacles in the case of a uniform field, and obtained the asymptotic probability of survival for the system as t → ∞ in d ≥ 2. In the hard obstacle model, the process is killed instantly when a particle hits the traps. In search for an extension to d = 1, Engländer and den Hollander [4] then studied the more interesting case where the trap intensity was radially decaying in a particular way in d ≥ 2, and yielding uniform intensity in d = 1, so as to give rise to a phase transition in the survival probability and the optimal survival strategy of the system. In both [3] and [4], the branching rule was taken as strictly dyadic, and the main result was the exponential asymptotic decay rate of the annealed survival probability as t → ∞; in addition, in [4], several optimal survival strategies were proved. For a BBM with a generic branching law, denote by p 0 the probability that a particle gives no offspring at the end of its lifetime. In [15], the work in [3] was extended to a BBM with a generic branching law, including the case where p 0 > 0. Likewise in [16], the work in [4] on the radially decaying trap field was extended to a BBM with a generic branching law, with the possibility of p 0 > 0. Recently in [17], conditioning the BBM on the event of survival from hard Poissonian obstacles,Öz and Engländer proved several optimal survival strategies in the annealed environment, with particular emphasis on the population size.
We refer the reader to [5] for a survey, and to [7] for a detailed treatment on the topic of BBM among random obstacles, and to [12] for a related problem where a critical BBM that is killed at a small rate inside the traps (such traps are called soft obstacles) is studied. We repeat that the mild obstacle model studied in the current work was proposed by Engländer in [6], and it is the partial aim of this work to improve the WLLN therein for the population size of the BBM to the SLLN.
Notation: We use c, c 0 , c 1 , . . . as generic positive constants, whose values may change from line to line. If we wish to emphasize the dependence of c on a parameter p, then we write c(p). We denote by f : A → B a function f from a set A to a set B. For two functions f, g : R + → R + , we write g(t) = o(f (t)) if g(t)/f (t) → 0 as t → ∞. Also, for a generic function g : R + → R + , we occasionally set g t = g(t) for notational convenience. We use N as the set of positive integers. For x ∈ R d , we use |x| to denote its Euclidean norm; also, for a generic finite set S, we use |S| to denote its cardinality. We use B(x, a) to denote the open ball of radius a > 0 centered at x ∈ R d . For an event A, we use A c to denote its complement, and ½ A its indicator function.
We denote by X = (X(t)) t≥0 a generic standard Brownian motion (BM) in d-dimensions, and use P x and E x , respectively, as the law of X started at position x ∈ R d , and the corresponding expectation.
Outline: The rest of the paper is organized as follows. In Section 2, we present our main results. Section 3 contains several introductory results, which serve as preparation for the proofs of Theorem 1 and Theorem 2. In Section 4 and Section 5, we present, respectively, the proofs of Theorem 1 and Theorem 2.

Main Results
The first main result gives the large-time asymptotic behavior of the probability that the mass of BBM inside a subdiffusively expanding ball B = (B t ) t≥0 with killing at the boundary of the ball, is atypically small. A subdiffusive expansion means that the ball is expanding slower than the rate at which a typical BM moves away from the origin, which means for large t it would be a 'rare event' for the BM to be confined in B t . For a generic standard Brownian motion X = (X(t)) t≥0 and a Borel set A ⊆ R d , define σ A = inf{s ≥ 0 : X(s) / ∈ A} to be the first exit time of X out of A.
Theorem 1 (LD for mass of BBM in an expanding ball). Let r : R + → R + be such that r(t) → ∞ as t → ∞ and r(t) = o( √ t). Also, let γ : where we use a ∧ b to denote the minimum of the numbers a and b.
Remark 1. The reason we call P (n t < γ t p t e βt ) with γ t = e −κr(t) a large-deviation (LD) probability is that with this choice of γ t , both P (n t < γ t p t e βt ) and P (n t = 0) decay as e −cr(t) to the leading order for large t, where the values of the constant c > 0 may differ. Indeed, start with P (n t < γ t p t e βt ) ≥ P (n t = 0).
One way to realize {n t = 0} is to completely suppress the branching and move the initial particle out of B t := B(0, r(t)) over [0, kr(t)], where k > 0 is a constant. The probability of realizing this joint strategy is where the second term in the exponent comes from Proposition A along with Brownian scaling.
To minimize the absolute value of the exponent in (4), set βkr(t) = r(t)/(2k), which yields k = 1/( √ 2β). With this choice of k, we arrive at Remark 2. In Theorem 1, we have only considered γ with γ t = e −κr(t) for the following reason. It can be shown that if γ t → 0 as t → ∞, then for all large t, P (n t < γ t p t e βt ) ≥ δγ t for some 0 < δ < 1 (see the proof of the lower bound of Theorem 1 in Section 4). Hence, when γ t is decaying sufficiently slowly so as to satisy (log γ t )/r(t) → 0 as t → ∞, due to (1) and (3), the event {n t < γ t p t e βt } would not be an LD event.
Our second main result is a quenched SLLN for the total mass of BBM among mild Poissonian obstacles in R d . Recall that (Ω, P) is the probability space for the PPP that creates the random environment, and N t := |Z t |. We now introduce further notation. Let λ d,r be the principal Dirichlet eigenvalue of − 1 2 ∆ on B(0, r) in d dimensions. Write λ d := λ d,1 , and let ω d be the volume of the d-dimensional unit ball. For d ≥ 1 and ν > 0, define the constant Theorem 2 (Quenched SLLN for BBM among mild obstacles). On a set of full P-measure, Remark 3. Note that the branching rateβ in the trap field K does not appear in the formula.

Remark 4.
It was shown in [6] that on a set of full P-measure, Theorem 2 is called a SLLN for BBM among mild obstacles, because it says that with P ω -probability one, the total mass of BBM among mild obstacles grows as its expectation as t → ∞. The reason why it is called a quenched SLLN is that it holds on a set of full P-measure.

Preparations
In this section, we present introductory results that serve as preparations for the proofs of the main theorems. The first two results are standard in the theory of Brownian motion. Proposition A is on the large-time asymptotic probability of atypically large Brownian displacements. For a proof, see for example [16,Lemma 5].
The following is a standard result on the large-time Brownian confinement in balls, and for instance can be deduced from [7, Prop. 1.6], along with the scaling λ d,r = λ d /r 2 . Recall that σ A = inf{s ≥ 0 : X(s) / ∈ A} denotes the first exit time of X out of A.
Proposition B (Brownian confinement in small balls). For t > 0, let B t = B(0, r(t)), where r : Then, as t → ∞, The following result is well-known in the theory of branching processes. For a proof, see for example [10,Sect. 8.11].
Proposition C (Distribution of mass in branching systems). For a strictly dyadic continuous-time branching process N = (N t ) t≥0 with constant branching rate β > 0, the probability distribution at time t is given by We now focus on the model of Poissonian traps in R d . Recall that a random environment in R d is created via a PPP, called Π, with being the trap field attached to R d .
By a clearing of radius r, we mean a ball of radius r which is a clearing.
The following result is Lemma 4.5.2 in [19].
Then, on a set of full P-measure, there exists ℓ 0 > 0 such that for each ℓ ≥ ℓ 0 the cube (−ℓ, ℓ) d contains a clearing of radius We now prove a somewhat stronger version of Proposition D, which will be needed in the proof of the lower bound of Theorem 2 (see Section 5). For a Borel set B and x ∈ R d , we define their sum in the sense of sum of sets as x + B := {x + y : y ∈ B}.
Proof. Let x 1 , x 2 , . . . be a sequence of points in R d , and C j,ℓ := x j + (−ℓ, ℓ) d for j = 1, 2, . . . For k ≥ 0, let A ℓ,k be the event that there is a clearing of radius R ℓ + k in each C 1,ℓ , C 2,ℓ , . . . , C ⌈(2ℓ) n ⌉,ℓ . Also, for k ≥ 0, define Due to the homogeneity of the PPP, it is clear that for all x ∈ R d and k > 0, Then, the union bound gives We now estimate P(E c ℓ,k ). Partition (−ℓ, ℓ) d into smaller cubes of side length 2(R ℓ + k). Then, a ball of radius R ℓ + k can be inscribed in each smaller cube, and we can bound P(E c ℓ,k ) from above as where we have used the estimate 1 + x ≤ e x . Let Then, using (11), and that log⌊ℓ/(R ℓ + k)⌋ ≥ log ℓ 2(R ℓ +k) , we obtain for all large ℓ, where we have used in the first inequality that (13) and (14) that for a given k > 0, for all large ℓ, , which along with (12) and (14) implies where c(m 0 ) is a constant that depends on m 0 . Applying Borel-Cantelli lemma to the cubes (−2 m , 2 m ) d , we conclude that with P-probability one, only finitely many A c 2 m ,k occur. That is, P(Ω 0 ) = 1, where Ω 0 = {ω : ∃m 0 ∀m ≥ m 0 , each C 1,2 m , . . . , C ⌈(2 m+1 ) n ⌉,2 m has a clearing of radius R (2 m ) + k}. (15) Let ω 0 ∈ Ω 0 , and m 0 be as in (15). If we choose k ≥ a, then to complete the proof, it suffices to show that in the environment ω 0 for each m ≥ m 0 and 2 m ≤ ℓ ≤ 2 m+1 , each C 1,ℓ , . . . , C ⌈ℓ n ⌉,ℓ contains a clearing of radius R ℓ + a. Let ℓ ≥ 2 m 0 so that 2 m ≤ ℓ ≤ 2 m+1 for some m ≥ m 0 . Fix this integer m. Observe that Choose k = R 0 log 2 + a (so far the choice of k > 0 was arbitrary). Then, since R ℓ is increasing in ℓ for large ℓ, we have Furthermore, Then, setting ℓ 0 = 2 m 0 , (15), (16) and (17) imply that for ℓ ≥ ℓ 0 , each of C 1,ℓ , . . . , C ⌈ℓ n ⌉,ℓ contains a clearing of radius R ℓ + a. This completes the proof since the choice of ω 0 ∈ Ω 0 was arbitrary and P(Ω 0 ) = 1.

Proof of Theorem 1
Recall that by assumption r : R + → R + satisfies r(t) → ∞ as t → ∞ and r(t) = o( √ t). Also, we set B t = B(0, r(t)) for t ≥ 0. Throughout this section, we will use that the law of |Z Bt t | is the same as the law of number of particles of Z which are present at t and whose ancestral lines over [0, t] have been confined to B t .
Consider the joint strategy of suppressing the branching over [0, f (t)], and then letting the BBM evolve 'normally' in the remaining interval [f (t), t]. To be precise, recall that n t := |Z Bt t |, σ A denotes the first exit time out of A, and p t := P 0 (σ Bt ≥ t); and let f : We will show that P (E c t | A t ) tends to a constant smaller than one as t → ∞ for suitable f . Let (Y 1 (s)) 0≤s≤τ 1 be the path of the initial particle, where τ 1 denotes the particle's lifetime. Conditional Then, where the second term on the right-hand side vanishes. Write Define Applying the Markov property of a standard BM at time s with 0 < s < t gives Furthermore, it follows from (19) and (20) that Now apply the Markov property of BBM at time f (t), and use the many-to-one lemma (see for instance [7,Lemma 1.6]) to obtain Using (22) with s therein replaced by f (t), it then follows from (22), (23) and (24) that Then, by the Markov inequality, Choose Then, noting that P (A t ) = e −βf (t) , it follows from (18) that (1)) .

Proof of the upper bound
For the proof of the upper bound, we follow a method that is based on Chebyshev's inequality, similar to the proof of [6, Thm. 1]. Let g : R + → R + satisfy g(t) → 0 as t → ∞. Later, we will choose g t := g(t) in a precise way. For t ≥ 0, let N t = |Z t | as before, and estimate We first bound the second term on the right-hand side of (26) from above. It follows from (9) that P (N t ≤ k) = 1 − (1 − e −βt ) k ≤ ke −βt for k ≥ 1. Set k = ⌊e βt g t ⌋ to obtain P N t ≤ e βt g t = P N t ≤ ⌊e βt g t ⌋ ≤ ⌊e βt g t ⌋e −βt ≤ g t .
Next, for t > 0 define P t ( · ) = P ( · | N t > e βt g t ), and let E t be the corresponding expectation. We now bound the first term on the right-hand side of (26) from above. Let N t denote the set of particles of Z that are alive at time t. For u ∈ N t , let (Y u (s)) 0≤s≤t denote the ancestral line up to t of particle u. By the ancestral line up to t of a particle present at time t, we mean the continuous trajectory traversed up to t by the particle, concatenated with the trajectories of all its ancestors including the one traversed by the initial particle. Note that (Y u (s)) 0≤s≤t is identically distributed as a Brownian path (X(s)) 0≤s≤t for each u ∈ N t . Let us pick randomly, independent of their genealogy and position, ⌊e βt g t ⌋ particles from N t . Note that this is possible under P t ( · ). Denote this collection of particles by M t , set M t := |M t |, and definê Observe thatn t counts, out of M t , the particles whose ancestral lines are confined to B t over [0, t]. Since the collection M t is chosen independently of the motion process, each particle u in M t has an ancestral line (Y u (s)) 0≤s≤t that is Brownian. Then, since the branching and motion mechanisms are independent of each other, the many-to-one lemma implies that for t > 0, where p t is as before the probability of confinement of a standard BM to B t over [0, t]. It is clear thatn t ≤ n t . At this point, choose g such that g t ≥ γ t for all t > 0. Then, using Chebyshev's inequality, it follows from (26), (27), and (28) that P (n t < γ t p t e βt ) ≤ P t n t < γ t p t e βt + g t where Var t denotes the variance associated to P t . In the rest of the proof, we estimate Var t (n t ).
Let P be the probability under which the pair (i, j) is chosen uniformly at random among the M t (M t − 1) possible pairs in M t , and let E be the corresponding expectation. Also, for a generic Brownian motion X, let Var denote its variance, and let A = {X(s) ∈ B t ∀ 0 ≤ s ≤ t}. Then, Let Q (t) be the distribution of the splitting time of the most recent common ancestor of ith and jth particles under E ⊗ P t . Applying the Markov property at this splitting time, we obtain where p (t) (x, s, dy) and p t s,x are as defined in (21). Set p t s = p t s,0 . Then, it follows from (22) and (31) that For t > 0 define Then, by (30) and (32), we have Var t (n t ) ≤ g t p t e βt + g 2 t p 2 t e 2βt (J t − 1).
It is clear that J t − 1 ≥ 0. Next, we bound J t − 1 from above. Recall that r(t) is a distance scale. For k > 0, we will use kr(t) as a time scale. Note that for large t it is atypical for a BM starting at the origin to escape B t = B(0, r(t)) over [0, kr(t)]. For large t, split J t up as We first bound J (1) t − 1 from above. Observe that p t s is nonincreasing in s, and estimate . (34) It then follows from (34) and (35) that To bound J (2) t from above, we will use the following fact on the distribution Q (t) from [6, Prop. 5]: Q (t) is absolutely continuous with respect to the Lebesgue measure, which we denote by ds, and its density function, which we denote by g (t) , satisfies ∃ C > 0, s 0 > 0 such that ∀ s ≥ s 0 , g (t) (s) ≤ Cse −βs .
Since r(t) = o( √ t) by assumption, this implies that for all large t we have r(t) ≤ t, which implies 1/p t s ≤ 1/p r(t) s . Here, p r(t) s = P 0 (σ B r(t) ≥ s) with B r(t) = B(0, r(r(t))) in accordance with previous notation. Then, since r(t) → ∞ as t → ∞, for all large t and for kr(t) ≤ s ≤ t, where we have used Proposition B. Then, we continue with where we have used integration by parts. From (36) and (37), we have To optimize the smallest absolute exponent on the right-hand side of (38), choose k so that This yields k = 1 √ 2β . With this choice of k, we have (1)) .
It then follows from (29) and (33) that By assumption, γ t = e −κr(t) with κ > 0. Choose g t = 2γ t . Then, we can continue (39) with Using Proposition B, and the assumptions that r(t) → ∞ and Then, using that g t = 2γ t = 2e −κr(t) , it follows from (40) that This completes the proof of (3) and the upper bound of (1).

Proof of the upper bound
The following upper bound was proved in [6, Section 6.1] via a first moment argument, using (8) and the Markov inequality. On a set of full P-measure, say Ω 0 , for any ε > 0, To pass from (41) to the upper bound of the corresponding SLLN, we use a standard Borel-Cantelli argument. Recall that Ω is the sample space for the BBM. For t > 0, define and let Let ω ∈ Ω 0 . We will show that P ω ( Ω 0 ) = 1. For n ∈ N, define By (41), there exists n 0 ∈ N such that for all n ≥ n 0 , P ω (A n ) ≤ e −c(ε)n(log n) −2/d . Then, By the Borel-Cantelli lemma, it follows that P ω (A n occurs i.o.) = 0, where i.o. stands for infinitely often. Choosing ε = 1/k, this implies that for each k ≥ 1, we have P ω ( Ω k ) = 1, Ω k := {̟ ∈ Ω : ∃ n 0 = n 0 (̟) such that ∀ n ≥ n 0 , Y n ≤ −c(d, v) + 1/k}.

Proof of the lower bound
Let ε > 0. We will find an upper bound for that is valid for large t on a set of full P-measure, and then use this upper bound along with the Borel-Cantelli lemma to pass to the corresponding SLLN. The proof is split into four parts for better readability. The first three parts are based on a bootstrap argument, where in part one, we find an upper bound on P (N t < e δt ) for 0 < δ < β, and then use this upper bound in parts two and three to find a similar upper bound on (42). We mainly follow the proof strategy given in [6]. We significantly improve the first and third parts of the corresponding proof in [6] in order to extend the WLLN therein to SLLN, where the extra work is due to finding rates of decay to zero for the relevant probabilities as t → ∞ as opposed to merely showing that they tend to zero.
In the first part of the proof, we use probabilistic arguments alone, including Theorem 1, in contrast to the partial differential equations (PDE) approach used in [6]. The main challenge is due to the possibility ofβ = 0 (no branching inside the traps), which makes it difficult to show that even in the presence of mild obstacles the system produces exponentially many particles with 'high' probability. We emphasize that the caseβ = 0 was not covered in [6], and the PDE approach used therein exploits the conditionβ > 0. The second part of the proof is similar to that in [6]; here, with minor further work, we find the rate of convergence to zero of the probability of the relevant unlikely event. The third part of the proof is an application of Theorem 1, where we argue that with 'high' probability sufficiently many particles are produced in a certain expanding clearing in R d , which exists in almost every environment. The fourth part of the proof uses a Borel-Cantelli argument along with the upper bound on (42) from part three to obtain the lower bound of the SLLN in (7).

Part 1: Upper bound on exponentially few total mass
In the first part of the proof, we will find an upper bound for P ω (N t < e δt ) with 0 < δ < β, that is valid for large t on a set of full P-measure. The argument will be based on the following lemma of independent interest, which is on the hitting probability of a standard BM to clearings of a certain size. Recall that X = (X(t)) t≥0 denotes a standard BM in d dimensions, and P x is the law of X started at position x ∈ R d . Also, recall the definition of R 0 from (10).
Lemma 2 (Hitting probability of BM to large clearings). Let r : R + → R + be such that For ω ∈ Ω and t > 0, define Let P ω x be the conditional law of X started at position x ∈ R d in the random environment ω. Then, there exists Ω 1 ⊆ Ω with P(Ω 1 ) = 1 such that for every ω ∈ Ω 1 , there exists t 0 = t 0 (ω) such that for all t ≥ t 0 , Proof. Introduce a time scale h(t), and two different space scales ρ(t) and r(t), as follows. Let h, ρ, r : R + → R + satisfy: where t 0 is large enough. Later, we will choose h(t) and ρ(t), and hence r(t), in a precise way so that r(t) will be as in (43). For notational convenience 1 , suppose that t/h(t) is an integer. Split Let x 1 , . . . , x t/h(t) be any set of t/h(t) points in R d . For j = 1, 2, . . . , t/h(t), define the intervals I j,t and the balls B j,t , respectively, as The balls (B j,t : 1 ≤ j ≤ t/h(t)) need not be disjoint. First, we establish a suitable almost-sure environment in R d with sufficient concentration of 'large' clearings. Let Ω 1 ⊆ Ω consist of environments ω with the property that there exists t 0 = t 0 (ω) such that for all t ≥ t 0 there exists a clearing of radius 2r(t) inside each B j,t , 1 ≤ j ≤ t/h(t). By assumption, t/h(t) = ρ n (t) for some n ≥ 2. Also, it follows from (11) and (c) that 2r(t) ≤ R ρ(t) for all large t. Then, since ρ(t) → ∞ as t → ∞, Lemma 1 implies that P(Ω 1 ) = 1.
We call x ∈ R d a good point for ω ∈ Ω at time t if B(x, r(t)) is a clearing (see Definition 1) in the random environment ω. That is, is the set of good points associated to the pair (ω, t). Now choose x j = X((j − 1)h(t)), 1 ≤ j ≤ t/h(t), 1 We would like to avoid the floor function in notation. and define Ω 1 as above. We know that P(Ω 1 ) = 1. In what follows, we will use Ω 1 as the almostsure, i.e., quenched, environment for the BBM. We now estimate the conditional probability that X does not hit Φ ω t up to a large time t given that ω ∈ Ω 1 . Let ω ∈ Ω 1 , and choose t large enough so that t ≥ t 0 (ω), where t 0 (ω) is as above. Then, for each 1 ≤ j ≤ t/h(t), B j,t = B(x j , ρ(t)) contains a clearing of radius 2r(t), hence a ball of radius r(t), say B j,t , that is entirely contained in Φ ω t . That is, Then, for t > 1, define the events In words, E t is the event that X does not hit a good point associated to (ω, t) over [0, t], i.e., Next, for t > 1, define the sets Σ t and the events F t as Equivalently, F t is the event that the number of intervals I j,t where X is confined to B j,t over I j,t is greater than or equal to t/(2h(t)) out of a total of t/h(t) intervals. Recall that ρ(t) is the radius of the ball B j,t and h(t) is the length of the time interval I j,t . Then, since a BM typically moves a distance of order √ s over a time period of length s and since ρ(t) = o( h(t)), F t is an unlikely event for large t. Set P ω = P ω 0 . We estimate P ω (E t ) as Let q 0 (t) be the probability that X stays inside B j,t over the period I j,t , and q 1 (t) be the probability that it doesn't hit B j,t conditional on exiting B j,t over I j,t . Recall that x j = X((j − 1)h(t)) by choice, and B j,t ⊇ B j,t is a ball of radius r(t). If X is conditioned to exit B j,t = B(x j , ρ(t)) over I j,t = [(j − 1)h(t), jh(t)], over this same period it must also exit B(x j , r(t)), where r(t) is the distance between x j and the center of B j,t . Therefore, since the Brownian exit distribution out of a ball centered at the starting point has rotational invariance (even under the conditioning), by comparing the surface area of the r(t)-ball that intersects B j,t to the total surface area of the r(t)-ball, and since r(t) ≤ ρ(t) for each t > 1, we obtain where κ d is a constant that only depends on the dimension d. Then, apply the Markov property of the Brownian path (X(s)) 0≤s≤t at times h(t), 2h(t), . . . , t − h(t) to continue the estimate in (44) as and where we have used the estimate 1 + x ≤ e x in passing to the last inequality. Then, from (44), (45), and (46), From Proposition B, we have We now choose h(t), ρ(t), and r(t) in a suitable way so as to keep P ω (E t ) sufficiently small in view of (47), while respecting the previously stated requirements (a)-(c): With these choices, since q 0 (t) → 0 as t → ∞, it follows from (47) that for all large t, where the last inequality follows since 1 3 < 1 2 − d−1 6d . Hence, we reach the following conclusion. There exists Ω 1 ⊆ Ω with P( Next, we use Lemma 2 to complete the first part of the proof of the upper bound of Theorem 2. Recall that 0 ≤ δ < β. Choose α such that 0 < α < 1 − δ/β. Split the interval [0, t] into two pieces as [0, αt] and [αt, t]. We argue that with 'high' probability, the BBM hits a good point, say z 0 ∈ R d , associated to (ω, αt) over [0, αt], and then the sub-BBM emanating from the particle that hits z 0 produces at least e δt particles over [αt, t] inside B(z 0 , r(αt)).
Let Y 1 = (Y 1 (s)) s≥0 be a randomly 2 chosen ancestral line of the BBM in the random environment ω. Note that even under P ω , since branching and motion mechanisms are independent of each other, (Y 1 (s)) s≥0 is identically distributed as a standard Brownian motion. The range (accumulated support) of Z is the process defined by Since Y 1 is an ancestral line of Z, we have ∪ 0≤s≤t Y 1 (s) ⊆ R(t) for each t ≥ 0. Then, since Y 1 is Brownian, Lemma 2 implies that for 0 < α < 1, on a set of full P-measure, say Ω 2 , for all large t, Observe that {R(αt) ∩ Φ ω αt = ∅} is the event that Z doesn't hit a good point associated to (ω, αt) over [0, αt]. Now let τ = τ (ω) = inf{s > 0 : R(s) ∩ Φ ω αt = ∅} be the first hitting time of Z to Φ ω αt . Let Y 2 be the ancestral line of Z that first hits Φ ω αt , and let z 0 = Y 2 (τ ). Conditional on τ < αt, apply the strong Markov property at time τ , and then apply Theorem 1 to the growth inside B(z 0 , r(αt)) of the sub-BBM initiated by Y 2 at time τ . Note that B t := B(z 0 , r(αt)) is a clearing in the random environment ω by definition of τ , z 0 and Φ ω αt . In detail, for u ≥ 0, let Z Bt [τ,τ +u] denote the mass at time τ + u of the sub-BBM initiated at position z 0 and time τ by Y 2 with killing at ∂B t . Let s := (1 − α)t,r : R + → R + be such that for large s, and B s := B(0,r(s)). Observe the equality of events {τ ≤ αt} = {R(αt) ∩ Φ ω αt = ∅}, and that t − τ ≥ (1 − α)t conditional on τ ≤ αt, andr(s) = r(αt). Then, on Ω 2 with P(Ω 2 ) = 1, applying the strong Markov property at τ = τ (ω), and taking γ s = exp[− β/2r(s)] for instance, Theorem 1 implies that for all large t, where, in the first inequality, we have used that δt < (1 − α)βt = βs due to the choice α < 1 − δ/β, and in the first equality we have used that B t is a clearing in ω followed by translation invariance. Then, in view ofr(s) = r(αt) and the definition of r(t) from (43), we reach the following conclusion via (49) and (50): on Ω 2 ⊆ Ω with P(Ω 2 ) = 1, where c = c(d, ν, β, δ) > 0. (The dependence of c on ν is through R 0 , which appears in the definition of r(t); see (10) and (43).) This gives a quenched upper bound on the probability that N t = |Z t | is exponentially few, and completes the first part of the proof of the lower bound of Theorem 2.
Part 2: Time scales within [0, t] and moving a particle into a large clearing This part of the proof is not new; it is essentially taken from [6] with minor improvements, where we also estimate the rate of decay to zero as t → ∞ of the probabilities of the relevant unlikely events as opposed to merely showing that they tend to zero. Introduce two different time scales, ℓ(t) and m(t), where ℓ(t) = o(m(t)) and m(t) = o(t), and split the interval [0, t] into [0, ℓ(t)], [ℓ(t), m(t)] and [m(t), t]. More precisely, let ℓ, m : R + → R + be two functions satisfying ℓ(t) < m(t) < t for all t > 0, and For concreteness, we fix the following choices of ℓ and m that satisfy (i)-(v): let ℓ(t) and m(t) be arbitrarily defined with ℓ(t) < m(t) < t for t ∈ (0, e], and ℓ(t) = t 1−1/(log log t) , m(t) = t 1−1/(2 log log t) , for t > e.
Firstly, using Part 1 of the proof, we prepare the setting at time ℓ(t). Fix δ ∈ (0, β), and define Recall the definition of R(t) from (48), and for t > 0, let Next, for t > 0, define the families of events Recall that we write Z t (B) to denote the mass of Z that fall inside B at time t, and define further the family of events F t := Z ℓ(t) B(0, 2β + ε ℓ(t)) ≥ I(t) .
Since lim t→∞ ℓ(t) = ∞, by (51), on a set of full P-measure, which we had called Ω 2 , with c = c(d, ν, β, δ) > 0. Next, we establish some control on the spatial spread of the BBM at time ℓ(t). Observe that M t /t is a measure of the spread of Z over [0, t]. As before, let N t denote the set of particles of Z that are alive at time t, and for u ∈ N t , let (Y u (s)) 0≤s≤t denote the ancestral line up to t of particle u. Note that N t = |N t |. Then, using the union bound, for γ > 0, Here, as before, we use P for the law of free 3 BBM with branching rate β everywhere. It is a standard result that E[N t ] = exp(βt) (one can deduce this, for example, from Proposition C), and we know from Proposition A that P 0 sup 0≤s≤t |X(s)| > γt = exp[−γ 2 t/2(1 + o(1))]. Moreover, the following stochastic domination is clear: for all B ⊆ R d Borel, all k ∈ N, and t ≥ 0, Then, taking B = B(0, √ 2β + ε ℓ(t)) c and k = 1 in (54), and γ = √ 2β + ε in (53), and replacing t by ℓ(t) in both (53) and (54), it follows that on Ω, for any ε > 0, (1)) . (55) , which, in view of (52) and (55) implies that on Ω 2 , This means, on a set of full P-measure, with 'high' P ω -probability, there are at least I(t) particles in B(0, √ 2β + ε ℓ(t)) at time ℓ(t) for large t. Next, we prepare the setting at time m(t). Recall (11) and define Since lim t→∞ ℓ(t) = ∞, Lemma 1 implies that on a set of full P-measure, say Ω 3 , there is a clearing B(x 0 , R(t) + 1) such that |x 0 | ≤ ℓ(t) for all large t. Let ω ∈ Ω 2 ∩ Ω 3 . Conditional on the event F t , the distance between x 0 and each of the at least I(t) many particles in B(0, √ 2β + ε ℓ(t)) at time ℓ(t) is at most (1 + 2β + ε)ℓ(t).
This means, on a set of full P-measure, with 'high' P ω -probability, there is at least one particle of Z inside B(x 0 , 1) at time m(t) for large t, where |x 0 | ≤ ℓ(t). Let us generically call this particle v, and denote by y 0 := X v (m(t)) its position at time m(t).
Finally, using assumption (ii), (57) along with R(s) = R(t), and (61), we reach the following conclusion. On Ω 0 , which is a set of full P-measure, for any ε > 0, where c = c(d, ν, β) > 0. In the next part of the proof, we will exploit the fact that the right-hand side of (66) does not depend on ε.

Part 4: Borel-Cantelli argument
We will show that on a set of full P-measure, for any ε > 0, Recall the definition of A t,ε from (62). It follows from (66) that there exists c = c(d, ν, β)/2, independent of ε, such that on Ω 0 , for all large t, P ω A t,ε/2 ≤ e −c(log t) 1/d .