Tài liệu Đề tài " On the volume of the intersection of two Wiener sausages " ppt

Annals of Mathematics On the volume of the intersection of two Wiener sausages By M. van den Berg, E. Bolthausen, and F. den Hollander Annals of Mathematics, 159 (2004), 741–783 On the volume of the intersection of two Wiener sausages By M. van den Berg, E. Bolthausen, and F. den Hollander Abstract For a>0, let W a 1 (t) and W a 2 (t)bethea-neighbourhoods of two indepen- dent standard Brownian motions in R d starting at 0 and observed until time t. We prove that, for d ≥ 3 and c>0, lim t→∞ 1 t (d−2)/d log P  |W a 1 (ct) ∩ W a 2 (ct)|≥t  = −I κ a d (c) and derive a variational representation for the rate constant I κ a d (c). Here, κ a is the Newtonian capacity of the ball with radius a. We show that the optimal strategy to realise the above large deviation is for W a 1 (ct) and W a 2 (ct) to “form a Swiss cheese”: the two Wiener sausages cover part of the space, leaving random holes whose sizes are of order 1 and whose density varies on scale t 1/d according to a certain optimal profile. We study in detail the function c → I κ a d (c). It turns out that I κ a d (c)= Θ d (κ a c)/κ a , where Θ d has the following properties: (1) For d ≥ 3: Θ d (u) < ∞ if and only if u ∈ (u  , ∞), with u  a universal constant; (2) For d =3: Θ d is strictly decreasing on (u  , ∞) with a zero limit; (3) For d =4: Θ d is strictly decreasing on (u  , ∞) with a nonzero limit; (4) For d ≥ 5: Θ d is strictly decreasing on (u  ,u d ) and a nonzero constant on [u d , ∞), with u d a constant depending on d that comes from a variational problem exhibiting “leakage”. This leakage is interpreted as saying that the two Wiener sausages form their intersection until time c ∗ t, with c ∗ = u d /κ a , and then wander off to infinity in different directions. Thus, c ∗ plays the role of a critical time horizon in d ≥ 5. We also derive the analogous result for d = 2, namely, lim t→∞ 1 log t log P  |W a 1 (ct) ∩ W a 2 (ct)|≥t/ log t  = −I 2π 2 (c), ∗ Key words and phrases. Wiener sausages, intersection volume, large deviations, vari- ational problems, Sobolev inequalities. 742 M. VAN DEN BERG, E. BOLTHAUSEN, AND F. DEN HOLLANDER where the rate constant has the same variational representation as in d ≥ 3 after κ a is replaced by 2π. In this case I 2π 2 (c)=Θ 2 (2πc)/2π with Θ 2 (u) < ∞ if and only if u ∈ (u  , ∞) and Θ 2 is strictly decreasing on (u  , ∞) with a zero limit. Acknowledgment. Part of this research was supported by the Volkswagen- Stiftung through the RiP-program at the Mathematisches Forschungsinstitut Oberwolfach, Germany. MvdB was supported by the London Mathematical Society. EB was supported by the Swiss National Science Foundation, Contract No. 20-63798.00. 1. Introduction and main results: Theorems 1–6 1.1. Motivation. In a paper that appeared in “The 1994 Dynkin Festschrift”, Khanin, Mazel, Shlosman and Sinai [9] considered the following problem. Let S(n),n∈ N 0 , be the simple random walk on Z d and let R = {z ∈ Z d : S(n)=z for some n ∈ N 0 }(1.1) be its infinite-time range. Let R 1 and R 2 be two independent copies of R and let P denote their joint probability law. It is well known (see Erd¨os and Taylor [7]) that P (|R 1 ∩ R 2 | < ∞)=  0if1≤ d ≤ 4, 1ifd ≥ 5. (1.2) What is the tail of the distribution of |R 1 ∩R 2 | in the high-dimensional case? In [9] it is shown that for every d ≥ 5 and δ>0 there exists a t 0 = t 0 (d, δ) such that exp  − t d−2 d +δ  ≤ P  |R 1 ∩ R 2 |≥t  ≤ exp  − t d−2 d −δ  ∀t ≥ t 0 .(1.3) Noteworthy about this result is the subexponential decay. The following prob- lems remained open: (1) Close the δ-gap and compute the rate constant. (2) Identify the “optimal strategy” behind the large deviation. (3) Explain where the exponent (d−2)/d comes from (which seems to suggest that d = 2, rather than d = 4, is a critical dimension). In the present paper we solve these problems for the continuous space-time setting in which the simple random walks are replaced by Brownian motions and the ranges by Wiener sausages, but only after restricting the time horizon to a multiple of t. Under this restriction we are able to fully describe the large deviations for d ≥ 2. The large deviations beyond this time horizon will ON THE VOLUME OF THE INTERSECTION OF TWO WIENER SAUSAGES 743 remain open, although we will formulate a conjecture for d ≥ 5 (which we plan to address elsewhere). Our results will draw heavily on some ideas and techniques that were developed in van den Berg, Bolthausen and den Hollander [3] to handle the large deviations for the volume of a single Wiener sausage. The present paper can be read independently. Self-intersections of random walks and Brownian motions have been stud- ied intensively over the past fifteen years (Lawler [10]). They play a key role e.g. in the description of polymer chains (Madras and Slade [13]) and in renor- malisation group methods for quantum field theory (Fern´andez, Fr¨ohlich and Sokal [8]). 1.2. Wiener sausages. Let β(t), t ≥ 0, be the standard Brownian motion in R d – the Markov process with generator ∆/2 – starting at 0. The Wiener sausage with radius a>0 is the random process defined by W a (t)=  0≤s≤t B a (β(s)),t≥ 0,(1.4) where B a (x) is the open ball with radius a around x ∈ R d . Let W a 1 (t), t ≥ 0, and W a 2 (t), t ≥ 0, be two independent copies of (1.4), let P denote their joint probability law, let V a (t)=W a 1 (t) ∩ W a 2 (t),t≥ 0,(1.5) be their intersection up to time t, and let V a = lim t→∞ V a (t)(1.6) be their infinite-time intersection. It is well known (see e.g. Le Gall [11]) that P (|V a | < ∞)=  0if1≤ d ≤ 4, 1ifd ≥ 5, (1.7) in complete analogy with (1.2). The aim of the present paper is to study the tail of the distribution of |V a (ct)| for c>0 arbitrary. This is done in Sections 1.3 and 1.4 and applies for d ≥ 2. We describe in detail the large deviation behaviour of |V a (ct)|, including a precise analysis of the rate constant as a function of c. In Section 1.5 we formulate a conjecture about the large deviation behaviour of |V a | for d ≥ 5. In Section 1.6 we briefly look at the intersection volume of three or more Wiener sausages. In Section 1.7 we discuss the discrete space-time setting considered in [9]. In Section 1.8 we give the outline of the rest of the paper. 1.3. Large deviations for finite-time intersection volume.Ford ≥ 3, let κ a = a d−2 2π d/2 /Γ( d−2 2 ) denote the Newtonian capacity of B a (0) associated with the Green’s function of (−∆/2) −1 . Our main results for the intersection volume of two Wiener sausages over a finite time horizon read as follows: 744 M. VAN DEN BERG, E. BOLTHAUSEN, AND F. DEN HOLLANDER Theorem 1. Let d ≥ 3 and a>0. Then, for every c>0, lim t→∞ 1 t (d−2)/d log P  |V a (ct)|≥t  = −I κ a d (c),(1.8) where I κ a d (c)=c inf φ∈Φ κ a d (c)   R d |∇φ| 2 (x)dx  (1.9) with Φ κ a d (c)=  φ ∈ H 1 (R d ):  R d φ 2 (x)dx =1,  R d  1 − e −κ a cφ 2 (x)  2 dx ≥ 1  . (1.10) Theorem 2. Let d =2and a>0. Then, for every c>0, lim t→∞ 1 log t log P  |V a (ct)|≥t/ log t  = −I 2π 2 (c),(1.11) where I 2π 2 (c) is given by (1.9) and (1.10) with (d, κ a ) replaced by (2, 2π). Note that we are picking a time horizon of length ct and are letting t →∞ for fixed c>0. The sizes of the large deviation, t respectively t/ log t, come from the expected volume of a single Wiener sausage as t →∞, namely, E|W a (t)|∼  κ a t if d ≥ 3, 2πt/ log t if d =2, (1.12) as shown in Spitzer [14]. So the two Wiener sausages in Theorems 1 and 2 are doing a large deviation on the scale of their mean. The idea behind Theorem 1 is that the optimal strategy for the two Brow- nian motions to realise the large deviation event {|V a (ct)|≥t} is to behave like Brownian motions in a drift field xt 1/d → (∇φ/φ)(x) for some smooth φ: R d → [0, ∞) during the given time window [0,ct]. Conditioned on adopting this drift: – Each Brownian motion spends time cφ 2 (x) per unit volume in the neigh- bourhood of xt 1/d , thus using up a total time t  R d cφ 2 (x)dx. This time must equal ct, hence the first constraint in (1.10). – Each corresponding Wiener sausage covers a fraction 1 − e −κ a cφ 2 (x) of the space in the neighbourhood of xt 1/d , thus making a total intersection volume t  R d (1 − e −κ a cφ 2 (x) ) 2 dx. This volume must exceed t, hence the second constraint in (1.10). The cost for adopting the drift during time ct is t (d−2)/d  R d c|∇φ| 2 (x)dx. The best choice of the drift field is therefore given by minimisers of the variational problem in (1.9) and (1.10), or by minimising sequences. ON THE VOLUME OF THE INTERSECTION OF TWO WIENER SAUSAGES 745 Note that the optimal strategy for the two Wiener sausages is to “form a Swiss cheese”: they cover only part of the space, leaving random holes whose sizes are of order 1 and whose density varies on space scale t 1/d (see [3]). The local structure of this Swiss cheese depends on a. Also note that the two Wiener sausages follow the optimal strategy independently. Apparently, under the joint optimal strategy the two Brownian motions are independent on space scales smaller than t 1/d . 1 A similar optimal strategy applies for Theorem 2, except that the space scale is  t/ log t. This is only slightly below the diffusive scale, which explains why the large deviation event has a polynomial rather than an exponential cost. Clearly, the case d =2iscritical for a finite time horizon. Incidentally, note that I 2π 2 (c) does not depend on a. This can be traced back to the recurrence of Brownian motion in d = 2. Apparently, the Swiss cheese has random holes that grow with time, washing out the dependence on a (see [3]). There is no result analogous to Theorems 1 and 2 for d = 1: the variational problem in (1.9) and (1.10) certainly continues to make sense for d = 1, but it does not describe the Wiener sausages: holes are impossible in d =1. 1.4. Analysis of the variational problem. We proceed with a closer analysis of (1.9) and (1.10). First we scale out the dependence on a and c. Recall from Theorem 2 that κ a =2π for d =2. Theorem 3. Let d ≥ 2 and a>0. (i) For every c>0, I κ a d (c)= 1 κ a Θ d (κ a c),(1.13) where Θ d :(0, ∞) → [0, ∞] is given by Θ d (u) = inf  ∇ψ 2 2 : ψ ∈ H 1 (R d ), ψ 2 2 = u,  (1 − e −ψ 2 ) 2 ≥ 1  .(1.14) (ii) Define u  = min ζ>0 ζ(1 − e −ζ ) −2 =2.45541 Then Θ d = ∞ on (0,u  ] and 0 < Θ d < ∞ on (u  , ∞). (iii) Θ d is nonincreasing on (u  , ∞). (iv) Θ d is continuous on (u  , ∞). (v) Θ d (u)  (u −u  ) −1 as u ↓ u  . Next we exhibit the main quantitative properties of Θ d . 1 To prove that the Brownian motions conditioned on the large deviation event {|V a (ct)| ≥ t} actually follow the “Swiss cheese strategy” requires substantial extra work. We will not address this issue here. 746 M. VAN DEN BERG, E. BOLTHAUSEN, AND F. DEN HOLLANDER Theorem 4. Let 2 ≤ d ≤ 4. Then u → u (4−d)/d Θ d (u) is strictly decreas- ing on (u  , ∞) and lim u→∞ u (4−d)/d Θ d (u)=µ d ,(1.15) where µ d =        inf  ∇ψ 2 2 : ψ ∈ H 1 (R d ), ψ 2 =1, ψ 4 =1  if d =2, 3, inf  ∇ψ 2 2 : ψ ∈ D 1 (R 4 ), ψ 4 =1  if d =4, (1.16) satisfying 0 <µ d < ∞. 2 Theorem 5. Let d ≥ 5 and define η d = inf{∇ψ 2 2 : ψ ∈ D 1 (R d ),  (1 − e −ψ 2 ) 2 =1}.(1.17) (i) There exists a radially symmetric, nonincreasing, strictly positive min- imiser ψ d of the variational problem in (1.17), which is unique up to transla- tions. Moreover, ψ d  2 2 < ∞. (ii) Define u d = ψ d  2 2 . Then u → θ d (u) is strictly decreasing on (u  ,u d ) and Θ d (u)=η d on [u d , ∞).(1.18) 0 s u  u d η d (iii) 0 u  µ 4 (ii) 0 u  (i) Figure 1 Qualitative picture of Θ d for: (i) d =2, 3; (ii) d = 4; (iii) d ≥ 5. Theorem 6. (i) Let 2 ≤ d ≤ 4 and u ∈ (u  , ∞) or d ≥ 5 and u ∈ (u  ,u d ]. Then the variational problem in (1.14) has a minimiser that is strictly positive, radially symmetric (modulo translations) and strictly decreasing in the radial component. Any other minimiser is of the same type. (ii) Let d ≥ 5 and u ∈ (u d , ∞). Then the variational problem in (1.14) does not have a minimiser. 2 We will see in Section 5 that µ 4 = S 4 , the Sobolev constant in (4.3) and (4.4). ON THE VOLUME OF THE INTERSECTION OF TWO WIENER SAUSAGES 747 We expect that in case (i) the minimiser is unique (modulo translations). In case (ii) the critical point u d is associated with “leakage” in (1.14); namely, L 2 -mass u − u d leaks away to infinity. 1.5. Large deviations for infinite-time intersection volume. Intuitively, by letting c →∞in (1.8) we might expect to be able to get the rate constant for an infinite time horizon. However, it is not at all obvious that the limits t →∞and c →∞can be interchanged. Indeed, the intersection volume might prefer to exceed the value t on a time scale of order larger than t, which is not seen by Theorems 1 and 2. The large deviations on this larger time scale are a whole new issue, which we will not address in the present paper. Nevertheless, Figure 1(iii) clearly suggests that for d ≥ 5 the limits can be interchanged: Conjecture. Let d ≥ 5 and a>0. Then lim t→∞ 1 t (d−2)/d log P  |V a |≥t  = −I κ a d ,(1.19) where I κ a d = inf c>0 I κ a d (c)=I κ a d (c ∗ )= η d κ a (1.20) with c ∗ = u d /κ a . The idea behind this conjecture is that the optimal strategy for the two Wiener sausages is time-inhomogeneous: they follow the Swiss cheese strategy until time c ∗ t and then wander off to infinity in different directions. The critical time horizon c ∗ comes from (1.13) and (1.18) as the value above which c → I κ a d (c) is constant (see Fig. 1(iii)). During the time window [0,c ∗ t] the Wiener sausages make a Swiss cheese parametrised by the ψ d in Theorem 5; namely, (1.9) and (1.10) have a minimising sequence (φ j ) converging to φ =(c ∗ κ a ) −1/2 ψ d in L 2 (R d ). We see from Figure 1(ii) that d =4iscritical for an infinite time horizon. In this case the limits t →∞and c →∞apparently cannot be interchanged. Theorem 4 shows that for 2 ≤ d ≤ 4 the time horizon in the optimal strategy is c = ∞, because c → I κ a d (c) is strictly decreasing as soon as it is finite (see Fig. 1(i–ii)). Apparently, even though lim t→∞ |V a (t)| = ∞ P - almost surely (recall (1.7)), the rate of divergence is so small that a time of order larger than t is needed for the intersection volume to exceed the value t with a probability exp[−o(t (d−2)/d )] respectively exp[−o(log t)]. So an even larger time is needed to exceed the value t with a probability of order 1. 1.6. Three or more Wiener sausages. Consider k ≥ 3 independent Wiener sausages, let V a k (t) denote their intersection up to time t, and let 748 M. VAN DEN BERG, E. BOLTHAUSEN, AND F. DEN HOLLANDER V a k = lim t→∞ V a k (t). Then the analogue of (1.7) reads (see e.g. Le Gall [11]) P (|V a k | < ∞)=  0if1≤ d ≤ 2k k−1 , 1ifd> 2k k−1 . (1.21) The critical dimension 2k/(k −1) comes from the following calculation: E|V a k | =  R d P  σ B a (x) < ∞  k dx =  R d  1 ∧  a |x|  d−2  k dx,(1.22) where σ B a (x) = inf{t ≥ 0: β(t) ∈ B a (x)}. The integral converges if and only if (d − 2)k>d. It is possible to extend the analysis in Sections 1.3 and 1.4 in a straight- forward manner, leading to the following modifications (not proved in this paper): (1) Theorems 1 and 2 carry over with: – V a replaced by V a k ; – c replaced by kc/2 in (1.9); –  R d (1 − e −κ a cφ 2 (x) ) 2 dx replaced by  R d (1 − e −κ a cφ 2 (x) ) k dx in (1.10). (2) Theorems 3, 4 and 5 carry over with: –  (1 − e −ψ 2 ) 2 replaced by  (1 − e −ψ 2 ) k in (1.14) and (1.17); – u  = min ζ>0 ζ(1 −e −ζ ) −k ; – ψ 4 replaced by ψ 2k in (1.16). For k = 3, the critical dimension is d = 3, and a behaviour similar to that in Figure 1 shows up for: (i) d = 2; (ii) d = 3; (iii) d ≥ 4, respectively. For k ≥ 4 the critical dimension lies strictly between 2 and 3, so that Figure 1(ii) drops out. 1.7. Back to simple random walks. We expect the results in Theorems 1 and 2 to carry over to the discrete space-time setting as introduced in Section 1.1. (A similar relation is proved in Donsker and Varadhan [6] for a single random walk, respectively, Brownian motion.) The only change should be that for d ≥ 3 the constant κ a needs to be replaced by its analogue in discrete space and time: κ = P(S(n) =0∀n ∈ N ),(1.23) the escape probability of the simple random walk. The global structure of the Swiss cheese should be the same as before; the local structure should depend on the underlying lattice via the number κ. ON THE VOLUME OF THE INTERSECTION OF TWO WIENER SAUSAGES 749 1.8. Outline. Theorem 1 is proved in Section 2. The idea is to wrap the Wiener sausages around a torus of size Nt 1/d , to show that the error com- mitted by doing so is negligible in the limit as t →∞followed by N →∞, and to use the results in [3] to compute the large deviations of the intersection volume on the torus as t →∞for fixed N. The wrapping is rather delicate because typically the intersection volume neither increases nor decreases under the wrapping. Therefore we have to go through an elaborate clumping and re- flection argument. In contrast, the volume of a single Wiener sausage decreases under the wrapping, a fact that is very important to the analysis in [3]. Theorem 2 is proved in Section 3. The necessary modifications of the argument in Section 2 are minor and involve a change in scaling only. Theorems 3–6 are proved in Sections 4–7. The tools used here are scaling and Sobolev inequalities. Here we also analyse the minimers of the variational problems in (1.14) and (1.17). 2. Proof of Theorem 1 By Brownian scaling, V a (ct) has the same distribution as tV at −1/d (ct (d−2)/d ). Hence, putting τ = t (d−2)/d ,(2.1) we have P  |V a (ct)|≥t  = P  |V aτ −1/(d−2) (cτ)|≥1  .(2.2) The right-hand side of (2.2) involves the Wiener sausages with a radius that shrinks with τ. The claim in Theorem 1 is therefore equivalent to lim τ→∞ 1 τ log P  |V aτ −1/(d−2) (cτ)|≥1  = −I κ a d (c).(2.3) We will prove (2.3) by deriving a lower bound (§2.2) and an upper bound (§2.3). To do so, we first deal with the problem on a finite torus (§2.1) and afterwards let the torus size tend to infinity. This is the standard compactifi- cation approach. On the torus we can use some results obtained in [3]. 2.1. Brownian motion wrapped around a torus. Let Λ N be the torus of size N>0, i.e., [− N 2 , N 2 ) d with periodic boundary conditions. Let β N (s), s ≥ 0, be the Brownian motion wrapped around Λ N , and let W aτ −1/(d−2) N (s), s ≥ 0, denote its Wiener sausage with radius aτ −1/(d−2) . Proposition 1. (|W aτ −1/(d−2) N (cτ)|) τ>0 satisfies the large deviation prin- ciple on R + with rate τ and with rate function J κ a d,N (b, c)= 1 2 c inf ψ∈Ψ κ a d,N (b,c)   Λ N |∇ψ| 2 (x)dx  ,(2.4) [...]... The proof of Proposition 4 will require quite a bit of work The hard part is to show that the intersection volume of the Wiener sausages on Rd is close to the intersection volume of the Wiener sausages wrapped around ΛN when N is large Note that the intersection volume may either increase or decrease when the Wiener sausages are wrapped around ΛN , so there is no simple comparison available Proof The. .. T )/T , which is the factor in the second term on the right-hand side of the analogue of (2.86) The integral on the right-hand side of (3.9) is of order 1/2 log(1/ ) Hence we get C 5/6 log(1/ ) for the second term on the right-hand side of the analogue of (2.93) 4 Proof of Theorem 3 In Sections 4–6 we prove Theorems 3–5 The proof follows the same line of reasoning as in [3, §5], but there are some subtle... the rate function (compare (2.4) with (2.7)) comes from the fact that both Brownian motions have to follow the drift field ∇φ/φ The proof is a straightforward adaptation and generalization of the proof of Proposition 3 in [3] We outline the main steps, while skipping the details Step 1 One of the basic ingredients in the proof in [3] is to approximate the volume of the Wiener sausage by its conditional... √ These two bounds will play a crucial role in the sequel We will pick η = N and M = log N , do our reflections with respect to the central hyperplanes in ON THE VOLUME OF THE INTERSECTION OF TWO WIENER SAUSAGES 759 J S√N ,N , and use the fact that for large N both the number of crossings and the J intersection volume in S√N ,N are small because of (2.56) This fact will allow us to control both the. .. N The rest of the proof of Proposition 4 will be based on Propositions 5 and 6 below Proposition 5 states that the intersection volume has a tendency to clump: the blocks where the intersection volume is below a certain threshold have a negligible total contribution as this threshold tends to zero Proposition 6 states that, at a negligible cost as N → ∞, the Brownian motions can be J reflected in the. .. 2 ≥b ΛN The last equality, showing that the variational problem reduces to the diagonal φ1 = φ2 , holds because if φ2 = 1 (φ2 + φ2 ), then 2 2 1 (2.34) 2|∇φ|2 ≤ |∇φ1 |2 + |∇φ2 |2 , 2 2 (1 − e−cκa φ1 )(1 − e−cκa φ2 ) ≤ (1 − e−cκa φ )2 This completes the proof of Proposition 2 2 2 2 ON THE VOLUME OF THE INTERSECTION OF TWO WIENER SAUSAGES 755 2.2 The lower bound in Theorem 1 In this section we prove:... we proceed with the proof of Proposition 6(ii) Proof Each reflection of an excursion beginning with an exit-point and ending with an entrance-point costs a factor 2 in probability On the event Ccτ,log N,√N , the total number of excursions of the two Brownian motions is √N bounded above by d logN cτ Moreover, on the event Ocτ the number of central hyperplanes available for the reflection is bounded above... Lemma 6 This completes the proof of Proposition 4 and hence of Theorem 1 3 Proof of Theorem 2 In this section we indicate how the arguments given in Section 2 for the Wiener sausages in d ≥ 3 can be carried over to d = 2 The necessary modifications are minor and only involve a change in the choice of the scaling parameters By Brownian scaling, V a (ct) has the same distribution as × (c log t), t > 1... expectation given a discrete skeleton We do the same here Abbreviate (2.9) −1/(d−2) aτ Wi (cτ ) = Wi,N (cτ ) , V (cτ ) = W1 (cτ ) ∩ W2 (cτ ) i = 1, 2, ON THE VOLUME OF THE INTERSECTION OF TWO WIENER SAUSAGES 751 Set X i,cτ,ε = {βi (jε)}1≤j≤cτ /ε , (2.10) i = 1, 2, where βi (s), s ≥ 0, is the Brownian motion on the torus ΛN that generates the Wiener sausage Wi (cτ ) Write E cτ,ε for the conditional expectation... that on the complement of the event on the left-hand side of (2.62) we have 0 ≤ |V aτ (2.63) −1/(d−2) J |Vcτ,√N ,N,in (z)| ≤ 2δ (cτ )| − z∈Z J,N The sum on the right-hand side is invariant under the reflections (because the J |Vcτ,√N ,N,in (z)| with z ∈ Z J,N end up in disjoint N -boxes), and therefore the estimate in (2.63) implies that most of the intersection volume is unaffected by the reflections . bit of work. The hard part is to show that the intersection volume of the Wiener sausages on R d is close to the intersection volume of the Wiener sausages. completes the proof of Proposition 2. ON THE VOLUME OF THE INTERSECTION OF TWO WIENER SAUSAGES 755 2.2. The lower bound in Theorem 1. In this section we prove: Proposition