On the volume of the shrinking branching Brownian sausage

The branching Brownian sausage in $\mathbb{R}^d$ was defined by Engl\"ander in [Stoch. Proc. Appl. 88 (2000)] similarly to the classical Wiener sausage, as the random subset of $\mathbb{R}^d$ scooped out by moving balls of fixed radius with centers following the trajectories of the particles of a branching Brownian motion (BBM). We consider a $d$-dimensional dyadic BBM, and study the large-time asymptotic behavior of the volume of the associated exponentially shrinking branching Brownian sausage (BBM-sausage). Using a previous result on the density of the support of BBM, and some well-known results on the classical Wiener sausage and Brownian hitting probabilities, we obtain almost sure limit theorems as time tends to infinity on the volume of the shrinking BBM-sausage in all dimensions.


Formulation of the problem and background
Let X = (X(t)) t≥0 be a standard d-dimensional Brownian motion (BM) starting at the origin. The Wiener sausage of radius r associated to X is the set-valued process defined by X r t = 0≤s≤t B(X(s), r), where B(x, r) is the closed ball of radius r > 0 centered at x ∈ R d . For each t ≥ 0, X r t is then a random subset of R d , which looks like a 'sausage' scooped out over the period [0, t] by a moving ball of fixed radius centered at the Brownian trajectory. Note that the Wiener sausage is a non-Markovian functional of X. Now let Z = (Z(t)) t≥0 be a d-dimensional strictly dyadic branching Brownian motion (BBM). The process can be described as follows. It starts with a single particle at the origin, which performs a 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 each other and of the parent, and the process evolves through time in this way. All particle lifetimes are exponentially distributed with constant parameter β > 0, which is called the branching rate. For each t ≥ 0, Z(t) can be viewed as a finite discrete measure on R d , which is supported at the particle positions at that time. We use P and E, respectively, as the probability and corresponding expectation for Z. The range (accumulated support) of Z is the process defined by supp(Z(s)), (1.1) and the branching Brownian sausage (BBM-sausage) with radius r associated to Z is the process defined by B(x, r).
The Wiener sausage and various set functions of it, especially its volume, have been frequently studied going back to [13]. In [3], Donsker and Varadhan obtained an asymptotic result on the Laplace transform of the volume of the Wiener sausage, which is a large-deviation (LD) result giving information on the probability that the volume is aytpically small. In [1] and [14], the work in [3] was extended to the case of the so-called shrinking Wiener sausage. We refer the reader to [5,Sect. 1] and the references therein for a brief survey of limit theorems on the volume of the Wiener sausage.
The branching Brownian sausage was introduced by Engländer in [4] in analogy with the classical Wiener sausage, and an asymptotic result on the Laplace transform of its volume was obtained similar to the one in [3], by using an equivalence to a trapping problem of BBM among Poissonian traps. In more detail, consider a Poisson point process Π on R d with intensity measure ν, and for r > 0 define the random trap field as Define the first trapping time of the BBM as T := inf {t ≥ 0 : R(t) ∩ K = ∅}, and the event of survival of BBM from traps up to time t as S t := {T > t}. Then, denoting the annealed law of the traps and the BBM as P, the first trapping problem of BBM among a Poissonian field of traps in R d is related to the BBM-sausage by Fubini's theorem: For a Borel set A ⊆ R d , we say volume of A to refer to its Lebesgue measure, which we denote by vol(A). In [4], it was shown that when the trap intensity is uniform, that is, To the best of our knowledge, apart from [4], no further work was done on the BBMsausage. We note that (1.2) gives information on the probability that the volume of a BBM-sausage with a constant radius is atypically small, whereas in the current work we study the typical behavior of a BBM-sausage with an exponentially shrinking radius.

Motivation
The current work can be regarded as a sequel to the recent works [10] and [11] under the common theme of spatial distribution of mass in BBM. In [10], the mass of BBM falling in linearly moving balls of fixed radius was studied, and an LD result on the large-time probability that this mass is atypically small on an exponential scale was obtained. The asymptotics of the probability of absence of BBM in linearly moving balls of fixed radius, emerged as a special case [10,Corollary 2]. It is well-known that the total mass of BBM typically grows exponentially in time. Also, it is known that typically for large t and any ε > 0, at time t there will be particles outside B(0, √ 2β(1 − ε)t) but no particles outside B(0, √ 2β(1 + ε)t). Definition 1.1 (Subcritical ball). We call B = (B(0, ρ t )) t≥0 a subcritical ball if there exists 0 < ε < 1 and time t 0 such that B(0, ρ t ) ⊆ B 0, √ 2β(1 − ε)t for all t ≥ t 0 .

Remark 1.2.
We use the term subcritical ball both in the sense of an expanding ball B = (B(0, ρ t )) t≥0 as in Definition 1.1, and also simply as a snapshot taken of an expanding ball at a fixed large time t as B(0, ρ t ).
In [11], the following was asked: how homogeneously are the exponentially many particles at time t spread out over a subcritical ball? This homogeneity question was formulated in terms of the degree of density of support of BBM at time t. Firstly, [10, Corollary 2] was extended to the case of the mass falling in linearly moving balls of exponentially shrinking radius r(t) = r 0 e −kt , and then via a covering by sufficiently many of such balls, an LD result concerning the r(t)-density of the support of BBM in subcritical balls was obtained. The concept of r(t)-density of the support of BBM naturally led to the following definition.
In [11], furthermore, the following results were obtained on the large-time behavior of r(t)-enlargement of BBM in R d . Theorem A below says that, with probability one, an r(t)-enlargement of BBM with r(t) decaying exponentially in t covers the subcritical ball B(0, θ √ 2βt) eventually provided that θ is smaller than a certain critical value. Theorem B is on the large-time behavior of the volume of r(t)-enlargement of BBM.
Theorem B (Almost sure volume of enlargement of BBM; [11]). Let 0 ≤ k ≤ 1/d, r 0 > 0 and r : R + → R + be defined by r(t) = r 0 e −βkt . Then, with probability one, Motivated by the results above, we ask the following question in the present work. For large t, by how much on the scale of t d , if at all, is the volume of the BBM-sausage with radius r(t) (i.e., the r(t)-shrinking BBM-sausage) larger than that of the r(t)-enlargement of BBM? The aim here is to answer this question in a precise way as t → ∞. Notation: We introduce further notation for the rest of the manuscript. For x ∈ R d , we use |x| to denote its Euclidean norm. 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 or c(p). We use R + to denote the set of nonnegative real numbers, and write o(f (t)) to refer to g(t), where g : R + → R + is a generic function satisfying g(t)/f (t) → 0 as t → ∞, unless otherwise stated. Also, for a function g : R + → R + , we use g t = g(t) for notational convenience. We denote by X = (X(t)) t≥0 a generic standard 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. In Section 3, we develop the preparation for the proofs of our main results, and then give the heuristic argument behind them. The proofs of the main results are given in Section 4.

Results
Theorem 2.1 and Theorem 2.2 are on the almost sure growth of exponentially shrinking BBM-sausages in d = 2 and d ≥ 3, respectively.
This can be explained as follows. In d = 2, the motion component of BBM plays a dominating role in the large-time behavior of the shrinking BBM sausage due to the almost sure neighborhood recurrence of BM. Note that the result does not depend on k.
For large t, a BBM-sausage with any exponentially shrinking radius (independent of how large the exponential rate of decay is for the radius) covers all subcritical balls, that is, for any 0 < ε < 1, the sausage Z rt t eventually covers B(0, Hence, Theorem 2.2 says that in d ≥ 3 for large t, provided that the decay of the sausage radius is slow enough, the accumulated support of BBM over [0, t) has a nontrivial contribution to the volume of the r t -shrinking sausage over [0, t] although the contribution is not significant enough to cover all subcritical balls; whereas, if the decay of r t is sharper than a certain threshold (i.e., if k > 1/(d − 2)), the accumulated support over [0, t) and the support at time t both have negligible contribution on the scale of t d .
which follows from the well-known result of Bramson [2] that the speed of strictly dyadic BBM converges to √ 2β as t → ∞ with probability one. On the other hand, Theorem B says that with probability one, Therefore, the large-time behavior of the volume of Z rt t is as different as it can be from that of Z rt t .

Preliminary results
In this section, we develop preparatory results for the proofs of Theorem 2.1 and Theorem 2.2. The first result is about the large-time asymptotic probability of atypically large Brownian displacements. For a proof, see for example [9, Lemma 5]. As before, let X = (X(t)) t≥0 be a generic standard BM in d-dimensions, and P x the law of X started at position x ∈ R d , with corresponding expectation E x .

Heuristics
It is clear that since the largest particle distance from the origin is On the other hand, we know from Lemma 3.1 that in d = 2, the expected volume of the r t -shrinking Wiener sausage is asymptotically constant. Therefore, since there are typically e βt+o(t) particles at time t, of which at least e εt for some ε > 0 can be treated as independent particles over the second half of the interval [0, t], and since the volume of a subcritical ball grows only polynomially in t, we expect that in d = 2 for large t the r t -shrinking BBM-sausage covers B θ := B(0, θ √ 2βt) for each 0 < θ < 1 provided that the particles of BBM spread out sufficiently homogeneously over B θ (see Theorem A).
The situation is different in d ≥ 3.
This explains the upper bound in (2.1). For the lower bound, consider a ball of unit size, . Also, due to (3.2), the expected volume scooped out over [t − 1, t] by the r t -shrinking Wiener sausage is roughly . Then, provided that the particles are spread out sufficiently homogeneously over B, since θ < 1 − k(d − 2) and the volume of B is constant, we expect even the r t -shrinking Wiener sausages initiated by the particles present in B at time t − 1 to cover B over [t − 1, t]. Polynomially many balls of unit size suffice to cover B θ , and one can see with further analysis that a suitable union bound over these balls does not disturb the argument.
implies that with probability one, In the rest of the manuscript, let N t denote the set of particles of Z that are alive at time t, and set N t = |N 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 trajectory (X(s)) 0≤s≤t for each u ∈ N t . Recall the definition of R(t) from (1.1). For t > 0, let M t := inf{r ≥ 0 : R(t) ⊆ B(0, r)}. Then, using the union bound, for γ > 0, It is a standard result that E[N t ] = exp(βt) (see for example [7,Sect. 8.11]). Moreover, we know from Proposition A that P 0 sup 0≤s≤t |X(s)| > γt = exp[−γ 2 t/2 + o(t)]. Then, for fixed ε > 0, defining the events it follows from setting γ = 2β(1 + ε) in (4.2) that there exists a positive constant c(ε) such that P (A k ) ≤ e −βc(ε)k for all large k. Applying Borel-Cantelli lemma on the events (A k : k ≥ 1), and then choosing ε = 1/n, and finally letting n vary over N yields: with probability one,

Proof of Theorem 2.1
Note that ω 2 = π. The upper bound comes from (4.3). We will show that for every ε > 0 there exists a positive constant c 1 such that for all large t, Then, the lower bound for Theorem 2.1 will follow from (4.4) via a standard Borel-Cantelli argument.
Let ε > 0, and for t > 0 let ρ t := 2β(1 − ε)t and B t := B(0, ρ t ). To prove (4.4), we choose a well-spaced net of points in B t , and argue that for large t with overwhelming probability, each ball of radius one centered at a net point has sufficiently many particles at time t − 1 so that even simple BMs initiated (rather than sub-BBMs) from the positions of these particles at time t − 1 are enough to ensure that there is no ball of radius r t with center lying in B t that remains not hit over the period [t − 1, t]. In other words, the following occurs with overwhelming probability: over [0, t − 1], the system produces sufficiently many particles which are sufficiently well-spaced over B t at time t − 1, and then (neglecting the branching over [t − 1, t]), Wiener sausages initiated from the positions of the particles at this time are enough to cover B t .
In this subsection, d = 2. However, in some of the notation and arguments that follow, we prefer to keep d general as they will be used in the next subsection as well, where d ≥ 3.
For t > 0 define Then, n t = c 2 t/r t d for some c 2 = c 2 (ε, β, d). For t > 0, define the events First, we prepare the setting at time t − 1. Let C(0, ρ t ) be the cube centered at the origin with side length 2ρ t so that B(0, ρ t ) is inscribed in C(0, ρ t ). Consider the simple cubic packing of C(0, ρ t ) with balls of radius 1/(2 √ d). Then, at most m t balls are needed to completely pack C(0, ρ t ), say with centers (x j : 1 ≤ j ≤ m t ). For each j, let B j = B(x j , 1/(2 √ d)). (We suppress the t-dependence in x j and B j for ease of notation.) Consider a simple cubic packing of R d by balls (B j : j ∈ Z + ) of radius r > 0, and let x ∈ R d be any point. Then, it is easy to see that min j max z∈Bj |x − z| < ( √ d/2)4r, where √ d/2 is the distance between the center and any vertex of the d-dimensional unit cube, i.e., C(0, 1/2). Then, since the packing ball radius is 1/(2  Typically, the mass of BBM that fall inside a linearly moving ball of fixed radius a > 0, say B t := B(θ √ 2βte, a) for some unit vector e and 0 < θ < 1, is exp[β(1 − θ 2 )t + o(t)]. Quoting [10, Thm. 1], in any dimension d ≥ 1, for 0 ≤ a < 1 − θ 2 , lim t→∞ 1 t log P Z t (B t ) < e βat = −β × I for some positive rate function I = I(θ, a). Then, since x j ∈ B(0, 2β(1 − ε)t) for each j, βε/2 is an atypically small exponent (typical exponent is at least β[1 − ( √ 1 − ε) 2 ] = βε) for the mass of BBM in each B j at time t − 1. It follows from (4.8) that there exists a positive constant c(ε) such that for all large t, where we have used the union bound and that m t is only a polynomial factor in t. It follows from (4.6), (4.7) and (4.9) that at time t − 1, with overwhelming probability, there are at least e β(ε/2)t particles in the 1-neighborhood of each point in B t . That is, there exists c = c(ε) > 0 such that for all large t, P (G t ) ≤ e −ct , G t := inf x∈Bt Z t−1 (B(x, 1)) < e β(ε/2)t . √ d). Then, at most n t balls are needed to completely pack C(0, ρ t ), say with centers (y j : 1 ≤ j ≤ n t ). For each j, let B j = B(y j , r t /(2 √ d)). By an argument similar to the one leading to (4.6), it follows that ∀ x ∈ B t , min 1≤j≤nt max z∈ Bj |x − z| < r t . (4.11) For j ∈ {1, 2, . . . , n t }, define the events F j : It then follows from (4.11) that H t ⊆ 1≤j≤nt F j , and therefore P (H t | G c t ) ≤ P (∪ 1≤j≤nt F j | G c t ). Now, the union bound gives P (H t | G c t ) ≤ n t max 1≤j≤nt P (F j | G c t ). (4.12) In view of (4.10), (4.12), and the estimate and since n t is only an exponential factor in t, to complete the proof of (4.4), it suffices to show that max 1≤j≤nt P (F j | G c t ) is super-exponentially small in t for large t. Observe that conditional on the event G c t , the event F j for any j can be realized only if the sub-BBMs initiated by each of the at least exp[β(ε/2)t] many particles present in B(y j , 1) at time t − 1 does not hit B(y j , r t /(2 √ d)) in the remaining time interval [t − 1, t]. Apply the Markov property at time t − 1, and neglect possible branching of particles over [t − 1, t] for an upper bound on P (F j | G c t ). Then, by (3.7) in Lemma 3.2, and the independence of particles present at time t − 1, there exists c > 0 such that for all large t, P (F j | G c t ) ≤ 1 − P e min 0≤s≤1 |X(s)| < r t 2 √ d e β(ε/2)t ≤ 1 − c βkt e β(ε/2)t , (4.13) where e is a unit vector. It is clear that the right-hand side of (4.13) is super-exponentially small in t for large t. This completes the proof of Theorem 2.1.

Proof of Theorem 2.2
We will show that for every ε > 0 there exist positive constants c 1 and c 2 such that for all large t,