Torsional Rigidity for Regions with a Brownian Boundary

Let 𝕋m be the m-dimensional unit torus, m ∈ ℕ. The torsional rigidity of an open set Ω ⊂ 𝕋m is the integral with respect to Lebesgue measure over all starting points x ∈ Ω of the expected lifetime in Ω of a Brownian motion starting at x. In this paper we consider Ω = 𝕋m\β[0, t], the complement of the path ß[0, t] of an independent Brownian motion up to time t. We compute the leading order asymptotic behaviour of the expectation of the torsional rigidity in the limit as t → ∞. For m = 2 the main contribution comes from the components in 𝕋2\β0, t] whose inradius is comparable to the largest inradius, while for m = 3 most of 𝕋3\β[0, t] contributes. A similar result holds for m ≥ 4 after the Brownian path is replaced by a shrinking Wiener sausage Wr(t)[0, t] of radius r(t) = o(t-1/(m-2)), provided the shrinking is slow enough to ensure that the torsional rigidity tends to zero. Asymptotic properties of the capacity of ß[0, t] in ℝ3 and W1[0, t] in ℝm, m ≥ 4, play a central role throughout the paper. Our results contribute to a better understanding of the geometry of the complement of Brownian motion on 𝕋m, which has received a lot of attention in the literature in past years.


Background, Main Results and Discussion
Section 1.1 provides our motivation for looking at torsional rigidity, and points to the relevant literature. Section 1.2 introduces our main object of interest, the torsional rigidity of the complement of Brownian motion on the unit torus. Section 1.3 states our main theorems. Section 1.4 places these theorems in their proper context and makes a link with the principal Dirichlet eigenvalue of the complement. Section 1.5 gives a brief sketch of the main ingredients of the proofs and provides an outline of the rest of the paper.

Background on Torsional Rigidity
Let (M, g) be a geodesically complete, smooth m-dimensional Riemannian manifold without boundary, and let be the Laplace-Beltrami operator acting in L 2 (M). We will in addition assume that M is stochastically complete. That is, Brownian motion on M, denoted by (β(s), s ≥ 0;P x , x ∈ M), with generator exists for all positive time. The latter is guaranteed if for example the Ricci curvature on M is bounded from below. See [16]  It is straightforward to verify that v , the torsion function for , is the unique solution of The torsional rigidity of is the set function defined by The torsional rigidity of a cross section of a cylindrical beam found its origin in the computation of the angular change when a beam of a given length and a given modulus of rigidity is exposed to a twisting moment. See for example [28]. From a mathematical point of view both the torsion function v and the torsional rigidity T ( ) have been studied by analysts and probabilists. Below we just list a few key results. In analysis, the torsion function is an essential ingredient for the study of gamma-convergence of sequences of sets. See chapter 4 in [10]. Several isoperimetric inequalities have been obtained for the torsional rigidity when M = R m . If ⊂ R m has finite Lebesgue measure | |, and * is the ball with the same Lebesgue measure, centred at 0, then T ( ) ≤ T ( * ). The following stability result for torsional rigidity was obtained in [9]: Here, A( ) is the Fraenkel asymmetry of , and C m is an m-dependent constant. The Kohler-Jobin isoperimetric inequality [17,18] states that Stability results have also been obtained for the Kohler-Jobin inequality [9]. A classical isoperimetric inequality [27] states that v L ∞ ( ) ≤ v * (0).
In probability, the first exit time moments of Brownian motion have been studied in for example [4] and [20]. These moments are Riemannian invariants, and the L 1 -norm of the first moment is the torsional rigidity.
The heat content of at time t is defined as This quantity represents the amount of heat in at time t, if is at initial temperature 1, while the boundary of is at temperature 0 for all t > 0. By Eq. 1.2, 0 ≤ u ≤ 1, and so 0 ≤ H (t) ≤ | |.
Finally by Eqs. 1.4, 1.6 and 1.7 we have that (1.8) i.e., the torsional rigidity is the integral of the heat content.

Torsional Rigidity of the Complement of Brownian Motion
In the present paper we consider the flat unit torus T m . Let (β(s), s ≥ 0; P x , x ∈ T m ) be a second independent Brownian motion on T m . Our object of interest is the random set (see Fig. 1) In particular, we are interested in the expected torsional rigidity of B(t): Since |T m | = 1 and |β[0, t]| = 0, the torsional rigidity is the expected time needed by the first Brownian motionβ to hit β[0, t] averaged over all starting points in T m . As t → ∞, β[0, t] tends to fill T m . Hence we expect that lim t→∞ ♠(t) = 0. The results in this paper identify the speed of convergence. This speed provides information on the random geometry of B(t). In earlier work [6] we considered the inradius of B(t).
The case m = 1 is uninteresting. For m = 2, as t gets large the set B(t) decomposes into a large number of disjoint small components (see Fig. 1), while for m ≥ 3 it remains connected. As shown in [14], in the latter case B(t) consists of "lakes" connected by "narrow channels", so that we may think of it as a porous medium. Below we identify the asymptotic behaviour of ♠(t) as t → ∞ when m = 2, 3.
For m ≥ 4 we have ♠(t) = ∞ for all t ≥ 0 because Brownian motion is polar. To get a non-trivial scaling, the Brownian path must be thickened to a shrinking Wiener sausage This choice of shrinking is appropriate because for m ≥ 3 typical regions in B(t) have a size of order t −1/(m−2) (see [11] and [14]). The object of interest is the random set in particular, the expected torsional rigidity of B r(t) (t): Below we identify the asymptotic behaviour of ♠ r(t) (t) as t → ∞ for m ≥ 4 subject to a condition under which r(t) does not decay too fast.

Asymptotic Scaling of Expected Torsional Rigidity
Theorems 1.1-1.3 below are our main results for the scaling of ♠(t) and ♠ r(t) (t) as t → ∞.
In what follows we write f g when 0 < c ≤ f (t)/g(t) ≤ C < ∞ for t large enough.
where κ m is the Newtonian capacity of the ball with radius 1 in R m , We expect that similar results hold when T m is replaced by a smooth m-dimensional compact connected Riemannian manifold without boundary. We further expect that the torsional rigidity satisfies a strong law of large numbers for m ≥ 3 but not for m = 2.
A key ingredient in the proof of Theorem 1.3 is the following scaling behaviour of the capacity of the Wiener sausage for m ≥ 4. Let (1.16) Then there exist constants c m ∈ (0, ∞), m ≥ 4, such that In Section 7 we prove Eq. 1.17 for m ≥ 5 with the help of subadditivity. For m = 4, Eq. 1.17 is proven in [3].

Discussion
We refer the reader to [14] and [5] for an overview of what is known about the geometry of the complement of Brownian motion on the unit torus. The inradius of B(t) is the random variable ρ t defined by . A detailed analysis of ρ t and related quantities was given in [5,12] for m = 2 and in [11,14] for m ≥ 3. In [6] it was shown that for m = 2, while for m ≥ 3, A ball of radius r in T m with r sufficiently small has a torsional rigidity proportional to r m+2 . Theorem 1.1 and Eq. 1.19 show that log 4 for m = 2, while Theorem 1.2 and Eq. 1.20 show that ♠(t) t −2 E 0 (ρ t ) 5 for m = 3. Thus, for m = 2 the main contribution to the asymptotic behaviour of log ♠(t) comes from the components in B(t) that have a size of order ρ t (which are atypical; see [12] and [5]), while for m = 3 the main contribution to the asymptotic behaviour of ♠(t) comes from regions in B(t) that have a size of order t −1 (which are typical; see [11] and [14]), i.e., most of B(t) contributes.

2.
For m = 2 it is shown in [5] that   [0, t], and the growth of ♠ r(t) (t) depends on the global rather than the local properties of W r(t) [0, t]. 5. We saw in Section 1.1 that the torsional rigidity is closely related to the principal Dirichlet eigenvalue. In Section 2 we will exhibit a relation with the square-integrated distance function and the largest inradius. In Section 6 we will give a quick proof of the following inequality relating the torsional rigidity to (b) If m ≥ 4 and lim t→∞ ♠ r(t) (t) = 0, then for t large enough, Combining the result for m = 2 with what we found in Theorem 1.1, we obtain where f g means that f (t)/g(t) ≥ c > 0 for t large enough. In [6] we conjectured that log E 0 (λ 1 (B(t))) = [1 + o(1)] 2(π t) 1/2 , which fits the lower bound in Eq. 1.27. However, a better estimate than Eq. 1.27 is possible. Namely, in Section 2 we will see that λ 1 (B(t)) 1/ρ 2 t , and so Jensen's inequality gives the lower bound E 0 (λ 1 (B(t)) ≥ 1/E 0 (ρ t ) 2 . Assuming that the scaling in Eq. 1.21 also holds in mean (which is expected but has not been proved), we get which is better than Eq. 1.27 by a factor t 3/8+o(1) . Presumably Eq. 1.28 captures the correct scaling behaviour.

Brief Sketch and Outline
For m = 2, B(t) consists of countably many connected component and the expected lifetime is sensitive to the starting point. We make use of the Hardy inequality to relate the time-integrated heat content to the space integral T 2 dist(x, β[0, t]) 2 dx. Because of the symmetry of T 2 , the problem boils down to studying the distribution of dist(x, β[0, t]) 2 with x ∈ T 2 chosen uniformly at random. This can be done by using a domain perturbation formula for the Dirichlet Laplacian eigenvalues. For m ≥ 3, B(t) has only one connected component and the proof is probabilistic. The starting point is the representation It is easy to see thatβ hits β[0, t] within time o((log t) −1 ) with a very high probability. For s ≤ (log t) −1 , the above integrand is the probability that β avoids the small setβ[0, s] for a long time t. We appeal to a recursive argument to evaluate this probability. Roughly speaking, in each unit of time β hitsβ[0, s] with probability ≈ cap (β[0, s]).
Outline The remainder of this paper is organised as follows. In Section 2 we recall some analytical facts about the torsional rigidity. In Sections 3-5 we prove Theorems 1.1-1.3, respectively. The proof of Theorem 1.4 is given in Section 6, while the proof of the scaling in Eqs. 1.16-1.17 for m ≥ 5 is given in Section 7.

Analytical Facts for the Torsional Rigidity
Let M be an m-dimensional Riemannian manifold without boundary that is both geodesically and stochastically complete. In most of this paper we focus on the case where M is the m-dimensional unit torus T m . However, the results mentioned below hold in greater generality. We derive certain a priori estimates on the torsional rigidity that will be needed later on.
For an open set ⊂ M with boundary ∂ , and with finite Lebesgue measure | |, we denote the Dirichlet heat kernel by p (x, y; t), x, y ∈ , t > 0. Recall that the Dirichlet heat kernel is non-negative, monotone in , symmetric. Thus, we have that Since | | < ∞, there exists an L 2 ( ) eigenfunction expansion for the Dirichlet heat kernel in terms of the Dirichlet eigenvalues λ 1 ( ) ≤ λ 2 ( ) ≤ · · · , and a corresponding orthonormal set of eigenfunctions ϕ 1 , ϕ 2 , · · · in L 2 ( ): and Lemma 2.1 below provides an upper bound on the Dirichlet eigenfunctions in terms of the Dirichlet eigenvalues. This bound will show that the eigenfunctions are in L ∞ (T m ), which by Hölder's inequality implies that they are in L p (T m ) for all 1 ≤ p ≤ ∞. Lemma 2.2 below states upper and lower bounds on the torsional rigidity that will be needed later on.
Proof By Eq. 2.1 and the domain monotonicity of the Dirichlet heat kernel ( [16]), we have that Taking first the supremum over x ∈ M in the right-hand side of Eq. 2.4 and subsequently in the left-hand side of Eq. 2.4, we get Eq. 2.3. Let

) Suppose that M and satisfy the hypotheses in (a). Then
Proof (a) Since the eigenfunctions are in all L p ( ), we have by Eqs. 2.1, 2.2 and Parseval's identity that (2.11) (Inequality Eq. 2.6 goes back to [22]. For a recent discussion and further improvements we refer the reader to [8]). (b) By Eq. 1.8 and the first identity in Eq. 2.11, we have that (2.12) By Lemma 2.1, we have that ϕ 1 L ∞ ( ) < ∞, and so Choose y = x, integrate over x ∈ and use Eq. 1.6, to get the claim. (d) It was shown in [2] that the Dirichlet Laplacian on a simply connected proper subset of R 2 satisfies a strong Hardy inequality: Theorem 1.5 in [7] implies Eq. 2.9. (e) Recall that the metric on T m is given by We therefore conclude that I = ∅. Finally, Eq. 2.10 follows from Eq. 2.8 for m = 2 and Eq. 2.9.

Torsional Rigidity for m = 2
In Section 3.1 we show that the inverse of the principal Dirichlet eigenvalue of B(1) = T 2 \β[0, 1] has a finite exponential moment. In Section 3.2 we use this result to prove Theorem 1.1.

Exponential Moment of the Inverse Principal Dirichlet Eigenvalue
Proof Let cap (A) denote the logarithmic capacity of a measurable set A ⊂ R 2 . It is well known (see [19]) that if cap (A) > 0 and A is a homothety of A by a factor , then In particular, if L ε is a straight line segment of length ε, then there exists a c ∈ (0, ∞) such that which is finite when c/c < 2.

Proof of Theorem 1.1
Proof The proof comes in 6 Steps, and is based on Lemmas 3.2-3.5 below. We use the following abbreviations (recall Eqs. 1.18 and 1.26): is open and its components are open and countable. Let { 1 (t), 2 (t), · · · } enumerate these components. Let and abbreviate Since the torsional rigidity is additive on disjoint sets we have that 2. The first term in the right-hand side of Eq. 3.2 is estimated from above by Lemma 2.2(d). This gives (recall Eq. 2.5) The second term in the right-hand side of Eq. 3.2 is estimated from above by Lemma 2.2(a). This gives By Cauchy-Schwarz, this term contributes to ♠(t) at most To bound the probability in the right-hand side of Eq. 3.3 from above, we let  (1), we find that β [1,2] starting at β(1) has a positive probability p of making a closed loop around each of these translated squares and staying insideQ N, + β (1). Continuing this way, by induction we find that the probability of β[0, t] not making any of these closed translated loops is at most (1 − p ) t , where · denotes the integer part. Hence P 0 (sup i∈N φ i (t) > 1 2 ) ≤ (1 − p ) t , and so for some p > 0. We conclude that Since t → λ t is non-decreasing, Lemma 3.1 implies that the second term decays exponentially fast in t, and therefore is harmless for the upper bound in Eq. 1.11. 3. To derive a lower bound for ♠(t), we note that by Lemma 2.2(e) we have where in the last inequality we use that δ i (t) (x) ≤ diam(T 2 ) = 1 2 √ 2 and |T 2 | = 1. We conclude by Eq. 3.4 that The second term is again harmless for the lower bound in Eq. 1.11. 4. The estimates in Eqs. 3.5 and 3.6 show that ♠(t) E 0 (D 2 t ) up to exponentially small error terms. In order to obtain the leading order asymptotic behaviour of E 0 (D 2 t ), we make a dyadic partition of T 2 into squares as follows. Partition T 2 into four 1-squares of area 1 4 each. Proceed by induction to partition each k-square into four (k+1)-squares, etc. In this way, for each k ∈ N, T 2 is partitioned into 2 2k k-squares. We define a k-square to be good when the path β[0, t] does not hit this square, but does hit the unique (k − 1)-square to which it belongs. Clearly, if x belongs to a good k-square, then dist(x, β[0, t]) ≤ (2 √ 2)2 −k . Hence, as the area of each k-square is 2 −2k , we get where we write E = T 2 dx E x , which is the same as E 0 for the quantity under consideration, by translation invariance. To estimate the right-hand side of Eq. 3.7 we need three lemmas. Proof Let p T 2 \S k (x, y; t) be the Dirichlet heat kernel for T 2 \S k . By the eigenfunction expansion in Eq. 2.1, we have that where we use Parseval's identity in the last equality.

Lemma 3.3
There exists C < ∞ such that, for all k ∈ N, Proof By [21, Theorem 1] we have that, for any disc D ⊂ T 2 with radius , This implies, by monotonicity and continuity of → λ 1 (T 2 \D ), the existence of C < ∞ such that For S k ⊂ T 2 there exist two discs D 1 and D 2 , with the same centre and radii 2 −k−1 and , and Eq. 3.8 follows by applying Eq. 3.9 with = 2 −k−1 and = 2 −k−1 √ 2, respectively.
Proof Let E k be the event that S k is not hit. Since S k is a good k-square if and only if the event E k ∩ E c k−1 occurs, the lemma follows because E k−1 ⊂ E k .

5.
We are now ready to estimate E(D 2 t ). By Eq. 3.7) and Lemma 3.4, where p 0 (t) = 0. In order to bound this sum from above we consider the contributions coming from k = 1, . . . K and k = K + 1, . . . , 1 4 t 1/2 and k > 1 4 t 1/2 , respectively, where · denotes the integer part, and we choose K = (C log 2)/π (3.11) with C the constant in Eq. 3.8. Since the first contribution is exponentially small in t. For k = K + 1, . . . , 1 4 t 1/2 we have C/k 2 ≤ π/k log 2, and hence by Lemmas 3.2-3.3, (3.13) and so the second contribution is o(t 1/4 e −4(πt) 1/2 ). Finally, for k > 1 4 t 1/2 we have e Ct/k 2 ≤ e 16C , and hence (3.14) The summand is increasing for 1 ≤ k ≤ (π t) 1/2 / log 2 and decreasing for k ≥ (π t) 1/2 / log 2. Moreover, it is bounded from above by e −4(πt) 1/2 . We conclude that for t → ∞, This is the desired upper bound in Eq. 1.11. 6. To obtain a lower bound for E(D 2 t ), we consider a good k-square. This square contains a square with the same centre, parallel sides and area 2 −2k−2 . The distance from this square to β[0, t] is bounded from below by 2 −k−2 . Hence since p 0 (t) = 0. The following lemma provides a lower bound for the right-hand side of Eq. 3.16.

Lemma 3.5
There exists k 0 ∈ N such that for all k ≥ k 0 , Proof By the eigenfunction expansion in Eq. 2.1 we have that By the results of [21], Combining Eqs. 3.8, 3.10, 3.16 and Lemma 3.5, we have that Now let t be such that πt/ log 2 > k 0 . Then Because the summand is strictly decreasing in k, we can replace the sum over k by an integral with a minor correction. This gives = (π t) 1/2 log 2 K 1 4(π t) 1/2 = π 3/4 2 3/2 log 2 t 1/4 e −4(πt) 1 This is the desired lower bound in Eq. 1.11.

Torsional Rigidity for m = 3
It is well known that β[0, 1] has a strictly positive Newton capacity when m = 3. In Section 4.1 we show that the inverse of the capacity of β[0, 1] on R 3 has a finite exponential moment. In Section 4.2 we show that for every closed set K ⊂ T 3 that has a small enough diameter the principal Dirichlet eigenvalue of T 3 \K is bounded from below by a constant times the capacity of K. (The same is true for m ≥ 4, a fact that will be needed in Section 5.) In Section 4.3 we use these results to prove Theorem 1.2.

Exponential Moment of the Inverse Capacity
Proof We use the fact that, for any compact set A ⊂ R 3 , As test probability measure we choose the sojourn measure of β[0, t], that is It therefore suffices to prove that for small enough c > 0. A proof of this fact is hidden in [13]. For the convenience of the reader we write it out here. By Cauchy-Schwarz and Jensen, we have that It therefore suffices to prove that the right-hand side is finite for small enough c > 0.
Expanding the exponent, we get The integrand equals where F t is the sigma-algebra of β up to time t. However, where in the second inequality we use that |x + β(1)| is stochastically larger than |β(1)| for any x = 0. Iterating Eqs. 4.2, 4.3, we get where t 0 = 0. Hence which is finite for c < 1/4γ . Integrating both sides of this identity over x ∈ T m \K, we get

Principal Dirichlet Eigenvalue and Capacity
where T K is the first hitting time of K by Brownian motion on T m . It follows that for any t > 0, where we use the inequality − log(1 − z) ≥ z, z ∈ [0, 1). Letβ be Brownian motion on R m , and letTK be the first hitting time ofK byβ. Then whereL K is the last exit time from K byβ. Let μ K denote the equilibrium measure on K in R m . Then (see [23]) By Eqs. 4.6-4.7, We note that if m = 3 and K = B ⊂ T 3 is a closed ball with radius , then λ 1 (T 3 \B ) = cap (B ) [1 + o(1)] as ↓ 0 (see [19]). In that case, since k 3 = 0.0101 . . . , we see that the constant in Eq. 4.4 is off by a large factor.

Torsional Rigidity for m ≥ 4
The same estimates as in the proof of Theorem 1.

Proof of Theorem 1.3 for m ≥ 5
Proof In the proof we assume that The estimate in Step 2 shows that, because of the first half of Eq. 5.2, P 0 (E c t ) with E t defined in Eq. 4.16 decays faster than any negative power of t, so that we can remove the intersection with B 1/4 (0) at the expense of a negligible error term. Since Via an estimate similar as in Eq. 4.15 with c replaced by cη(t)/r(t) 2 , we obtain, with the help of Lemma 7.1 below (which is the analogue of Lemma 4.1 and is proved in Section 7.1), Hence the right-hand side of Eq. 5.3 is O(K(t) −1 ) when we pick The second half of Eq. 5.2 ensures that K(t) grows faster than any positive power of t, and so Eq. 5.3 is negligible. The contribution can again be estimated in a similar way by reversing the roles of β andβ. This leads to a term that is even much smaller.
Step 5 is unaltered. 6-7. In Step 6 we use that cap where With the change of variable u = t 1/(m−2) √ s, the integral becomes where with (recall Eq. 1.10) and where ε(t) = t 1/(m−2) r(t). Now, Eq. 1.23 tells us that in P 0 -probability as t → ∞ for every u ∈ (0, ∞) and m ≥ 5, where we use that ε(t) = o(1) by the first half of Eq. 5.1. Therefore with the help of Eq. 5.2 and dominated convergence, we find that , t → ∞, Step 7 the first line in Eq. 4.25 is replaced by the statement that lim δ↓0 δ m−2 e δ = 1/κ m . Combining Eqs. 5.4, 5.5 and 5.8, and letting δ ↓ 0, we get the scaling in Eq. 1.14.

Proof of Theorem 1.3 for m = 4
Proof In the proof we assume that where we note that by the second half of Eq. 5.10 there exists a choice of η(t) satisfying Eq. 5.11. The estimate uses Eq. 4.15 with c replaced by c(η(t)/r(t) 2 )/ log(η(t)/r(t) 2 ), and also Lemma 7.1 below (which is the analogue of Lemma 4.1 and is proved in Section 7.1). 3-5. These steps are unaltered.

Capacity of Wiener Sausage for m ≥ 4
In Section 7.1 we derive the analogue of Lemma 4.1, showing that the inverse of C(t) for m ≥ 4 defined in Eq. 1.16 has a finite exponential moment uniformly in t ≥ 2. In Section 7.2 we prove Eqs. 1.16-1.17 for m ≥ 5. where ω m = |B 1 (0)|. Since μ has support in W 1 [0, t], we have

Exponential Moment of the Inverse Capacity
Moreover, there exists C = C(m) > 0 such that for all u and v, We first prove the claim for m ≥ 5. Letc = c C/κ m ω 2 m . We have that Taking the expectation and using the translation invariance of Brownian motion, we obtain the t-independent bound and so it remains to show that the right-hand side is finite for c small enough. Arguing in the same way as in the proof of Lemma 4.1, we obtain Therefore it remains to prove the finiteness of the integral. That that end, we estimate To prove thatc m > 0 it suffices to show that the right-hand side has a finite expectation. To that end, we estimate and note that, as shown in Eq. 7.6, the integral converges as t → ∞ when m ≥ 5.