Brownian Motion on Compact Manifolds: Cover Time and Late Points

: Let M be a smooth, compact, connected Riemannian manifold of dimension d ‚ 3 and without boundary. Denote by T ( x; † ) the hitting time of the ball of radius † centered at x by Brownian motion on M . Then, C † ( M ) = sup x 2 M T ( x; † ) is the time it takes Brownian motion to come within r of all points in M . We prove that C † ( M ) =† 2 ¡ d j log † j ! (cid:176) d V ( M ) almost surely as † ! 0, where V ( M ) is the Riemannian volume of M . We also obtain the \multi-fractal spectrum" f ( ﬁ ) for \late points", i.e., the dimension of the set of ﬁ -late points x in M for which lim sup † ! 0 T ( x; † ) = ( † 2 ¡ d j log † j ) = ﬁ > 0.


Introduction
Let M be a smooth, compact connected d-dimensional Riemannian manifold without boundary, and let {X t } t≥0 denote Brownian motion on M . {X t } t≥0 is a strongly symmetric Markov process with reference measure given by the Riemannian measure dV and infinitesimal generator 1/2 the Laplace-Beltrami operator ∆ M . We use d(x, y) to denote the Riemannian distance between In [7] we considered C (M ) for compact manifolds of dimension d = 2. By the use of isothermal coordinate systems this problem is reduced to the -covering time of the two-dimensional (flat) torus by a (standard) Brownian motion. In this paper we deal with manifolds of dimension d ≥ 3, for which in general there is no direct reduction to the Euclidean case. Consequently, we work directly on the manifold, taking advantage of the fact that the Brownian motion is "locally transient" on such manifolds, in sharp contrast with the situation for d = 2.
Here is the heuristic leading to (1.1): The ε-hitting time T (x, ε) grows with decreasing ε like 1 over the minimal eigenvalue of 1 2 ∆ M on M \ B(x, ε). The latter is known to be κ −1 M ε d−2 (1 + o(1)) (c.f. [3] and the references therein to earlier works by Ozawa and others). Since B(x, ε) and B(y, ε) have a substantial overlap whenever d(x, y) ε, the value of C ε (M ) is roughly the Furthermore, for any analytic set E ⊆ M we have The next theorem describes the multi-fractal structure of {T (x, ε)} for Brownian motion in M and those points x ∈ M for which T (x, ) is comparable with C (M ) as → 0.
Theorem 1.3. For Brownian motion in M and for any a ≤ d, We call a point x ∈ M a late point if x is in the set considered in (1.5) for some a > 0. This theorem may be compared with our results on the multi-fractal structure of thick points for Brownian motion, [4,6]. The first result of this type was the determination by Orey and Taylor [14] of the dimension of sets of fast points for Brownian motion.
Analytic tools provide in Section 2 simple uniform estimates on excursion times and exit probabilities for the annuli B(m, R) \ B(m, r). Using these estimates we obtain in Section 3 upper bounds, first on the tail probability of T (x, ε), then on the limits considered in Theorem 1.2 and the dimensions of the sets considered in Theorem 1.3. The complementary lower bounds are derived in Section 4 by an adaptation of the methods of [5,11] and of [12,13].
While outside the scope of this work, it is interesting to find the structure of consistently late points, where the lim sup in (1.4) and (1.5) is replaced by lim inf (or lim), changing, if needed, the scaling function.

Excursion time estimates
We start with a uniform estimate on the mean time to exit a small ball B(m, R), allowing us in the sequel to neglect the contribution of such times.
For R sufficiently small, and any m, Proof of Lemma 2.1: We use geodesic polar coordinates r, θ centered at m. When applied to radial functions f (r) (r = dist(m, x)), the Laplace-Beltrami operator takes in these coordinates the form which differs from the Euclidean Laplacian in the term denoted O(r) (see [2, page 106]). The left hand side of (2.1) satisfies Using (2.2) we see that for all R, hence r, sufficiently small, uniformly in m ∈ M and x ∈ B(m, R), Consequently, F − u ± are sub(super)-harmonic in B(m, R) with 0 boundary conditions on ∂B(m, R). Since 0 is the only function harmonic in B(m, R) with 0 boundary conditions on ∂B(m, R), we obtain (2.1).
The next lemma provides an estimate on the mean hitting time of a small ball B(m, r) starting at distance R > r from its center m. It is this estimate that give rise to the constant κ M .

Thus by the maximum principle for all
Our lemma follows immediately (using (2.1) to provide the bound (2.4) also for x ∈ B(m, r) and r sufficiently small).
The next lemma shows that the probability of hitting a small ball B(m, ) upon exit of a small annuli B(m, R) \ B(m, ) is (uniformly) comparable to that for M = R d .
Proof of Lemma 2.3: We follow an argument similar to that used in proving Lemma 2.1.
It is easy to check that for R small enough and R ≥ 2r ≥ 4ε, Note that g − u ± is 0 at the boundary of the annulus B(m, R) \ B(m, ε) and by (2.2), for all R small enough, uniformly in m ∈ M and x in this annulus, are sub(super)-harmonic in this annulus with 0 boundary conditions. Since 0 is the only function harmonic in this annulus with 0 boundary conditions, we obtain (2.5).
is the length of the j'th excursion E j from ∂B(x, R) to itself via ∂B(x, r), and σ (j) is the amount of time it takes to hit ∂B(x, r) during the j'th excursion E j .
Proof of Lemma 2.4: Applying Kac's moment formula for the first hitting time τ r of the strong Markov process X t (see [9,Equation (6)]), it follows by (2.4) that for any integer k, all m ∈ M and r ≤ r 0 ≤ 1, sup Hence, for some λ > 0, sup Reducing r 0 as needed, by the same argument, Lemma 2.1 implies that By the strong Markov property of X t at τ (0) and at τ (0) + σ (1) we then deduce that (see (2.1)). Fixing x ∈ M and 0 < 2r ≤ R ≤ r 0 , let τ = τ (1) and v = κ M (r 2−d −R 2−d ). It follows from (2.3) and (2.12) that there exists a universal constant c 4 < ∞ such that for ρ = c 4 r 2(2−d) and all θ ≥ 0, Since τ (0) ≥ 0, using Chebycheff's inequality we bound the left hand side of (2.10) by To prove (2.11) we first note that for λ > 0 as in (2.13), it follows that where c 5 < ∞ is a universal constant and c 6 = c 6 (R, r) > 0 does not depend upon N , δ, x 0 or x, and is bounded below by some c 7 (δ 0 ) > 0 when r 1−δ 0 < R. Thus, the proof of (2.11), in analogy to that of (2.10), comes down to bounding the proof of (2.11) now follows as in the proof of (2.10).

Hitting time estimates and upper bounds
The first step in getting upper bounds is to control the tail probabilities of T (x, ), uniformly in x and the initial position x 0 .
Lemma 3.1. For any δ > 0 we can find c < ∞ and ε 0 > 0 so that for all ε ≤ ε 0 and y ≥ 0 for all x, x 0 ∈ M .
Proof of Lemma 3.1: We use the notation of the last lemma and its proof, with 2r < R < R 0 (δ). Let n ε : . It is easy to see that It follows from Lemma 2.4 that for some C = C (δ) > 0. On the other hand, the first probability in the right hand side of (3.2) is bounded above by the probability of not hitting the ball B(x, ) during n ε excursions, each starting at ∂B(x, r) and ending at ∂B(x, R), so that by Lemma 2.3, for all ε > 0, small enough and (3.1) follows.
We next provide the required upper bound in Theorem 1.3. Namely, with the notation we will show that for any a ∈ (0, d], Fix δ > 0,˜ 0 , and define set˜ n inductively so that Since, for˜ n+1 ≤ ≤˜ n we have it is easy to see that for any a > 0, Fix x 0 ∈ M and let {x j : j = 1, . . . ,K n }, denote a maximal collection of points in M , such that inf =j d(x , x j ) ≥ δ˜ n . Let A n be the set of 1 ≤ j ≤K n , such that It follows by Lemma 3.1 that for some c = c(δ) < ∞, all sufficiently large n and any x ∈ M . Thus, for all sufficiently large n, any j and a > 0, implying that (3.7) Let V n,j = B(x j , δ˜ n ). For any x ∈ M there exists j ∈ {1, . . . ,K n } such that x ∈ V n,j , hence B(x,˜ n ) ⊇ B(x j , (1 − δ)˜ n ). Consequently, ∪ n≥m ∪ j∈An V n,j forms a cover of D a by sets of maximal diameter 2δ˜ m . Fix a ∈ (0, 2]. Let d(B) denote the diameter of a set B ∈ M . Since d(V n,j ) = 2δ˜ n , it follows from (3.6) that for γ = d − (1 − 11δ)a > 0, Thus, ∞ n=m j∈An d(V n,j ) γ is finite a.s. implying that dim(Late ≥a ) ≤ dim(D a ) ≤ γ a.s. Taking δ ↓ 0 completes the proof of the upper bound (3.4).
We conclude this section with the derivation of the upper bound for (1.3), that is, for any E ⊆ M lim sup Fix x 0 ∈ M and let {x j : j = 1, . . . , k n }, denote a maximal collection of points in E, such that inf =j d(x , x j ) ≥ δ˜ n . Let A n (E) be the set of 1 ≤ j ≤ k n , such that If dim m (E) = γ, we have that for any δ > 0 Thus, as in (3.7) we have Therefore with a = (γ + 2δ) By Borel-Cantelli, it follows that A n (E) is empty a.s. for all n > n 0 (ω) and some n 0 (ω) < ∞. For any x ∈ E there exists j ∈ {1, . . . , k n } such that x ∈ B(x j , δ˜ n ), hence B(x,˜ n ) ⊇ B(x j , (1−δ)˜ n ). We then see from (3.5) that for some C = C(γ, d) < ∞, all δ > 0 small enough and n > n 1 (δ, ω) and (3.8) follows by taking δ ↓ 0.

Lower bounds
For any E ⊆ M we define The following is a restatement of (1.3). Hence it suffices to show that when dim m (E) = γ > 0. For any 1 > δ > 0 we can find a sequence n ↓ 0 and a collection of points {x n,j : Recall that h d ( ) = 2−d log 1 , so by Theorem 2.6 of [12] this implies that for all n ≥ N Combining (4.4) and (4.5) we see that for some c 1 < ∞ and all n, By the Markov property of X t at kξ n , for any x ∈ M and k = 0, 1, . . ., Hence, C n (E)/ξ n is stochastically dominated by a Geometric(1/2) random variable (for any n). Considering (4.6), the sequence {C n (E)/E x 0 (C n (E))} n is then uniformly integrable, implying by an extension of Fatou's lemma (c.f. [1,Theorem 7.5.2]), that, Thus, by (4.4) and (4.3) follows by taking δ ↓ 0.
We claim that in fact The upper bound is obvious. Now observe that for any 0 < < ε 0 we can find an n with 1/(n + 1) < < 1/n. Then and the lower bound follows since h d (n −1 )/h d ((n + 1) −1 ) → 1.

Recall that
The statement (1.4) will follow immediately from the next result.

Lemma 4.2. For any analytic set
Proof of Lemma 4.2: If dim p (E) < a, then by regularization we can represent E as a countable union E = ∪ n E n with dim m (E n ) < a. It follows immediately from (4.2) that E n ∩ Late ≥a = ∅ a.s., and therefore E ∩ Late ≥a = ∅ a.s. Now assume that dim p (E) > a. By [10], we can find a closed E ⊂ E, such that for all open sets V , whenever E ∩ V = ∅, then for some δ > 0 Define the open sets A a (k) = x ∈ M : Since A a ⊆ Late ≥a it suffices to show that with probability one, A a ∩ E = ∅. Define the open sets B a (n) := ∪ ∞ k=n A a (k), n ≥ 1. We claim that for all n ≥ 1, the relatively open set B a (n) ∩ E is a.s. d-dense in (the complete metric space) E . If so, Baire's category theorem implies that E ∩ ∞ n=1 B a (n) is dense in E and in particular, nonempty. Since A a = ∩ n B a (n), the result follows. Fix an open set V such that V ∩ E = ∅. Using (4.9) and (4.7) with E replaced by V ∩ E we see that A a (n) ∩ V ∩ E = ∅ for infinitely many n, a.s. Thus B a (n) ∩ V ∩ E = ∅ for all n a.s.; by letting V run over a countable base for the open sets, we conclude that B a (n) ∩ E is a.s. d-dense in E .
Proof of Theorem 1.3: The upper bound has already been proven in Section 3. Set A + a = ∩ ∞ m=1 A a−1/m and B + a = A + a − ∪ ∞ m=1 Late ≥a+1/m . It is easy to see that It follows from the argument in the proof of Corollary 3.3 in [5] that for any analytic set E ⊆ M with dim p (E) > a, we have B + a ∩ E = ∅ a.s. For the convenience of the reader we reproduce the short proof. Let Λ m (n) := ∪ ∞ k=n A a−1/m (k). Since dim p (E) > a, by [10] there exists a closed E ⊂ E such that dim p E ∩ V = a for any open set V such that E ∩ V = ∅, implying as in the last proof that Λ m (n) ∩ E is a.s. dense in the complete metric space E . Consequently, by Baire's theorem it follows that E ∩ (∩ ∞ n=1 ∩ ∞ m=1 Λ m (n)) is dense in E , a.s., and in particular is non-empty. Obviously, dim p (E ) = a, so by Lemma 4.2, Late ≥a+1/m ∩ E = ∅, a.s.. It follows that B + a ∩ E = ∅, a.s. as claimed. We now wish to conclude the proof of the lower bound of Theorem 1.3 by appealing to Lemma 3.4 of [11]. However, the statement and proof of that Lemma are for sets in [0, 1] d , whereas our sets are in M . This can be easily remedied as follows. Let φ : M → [0, 1] d be a diffeomorphism from some subset M ⊆ M . We can choose φ, M so that for some δ > 0 for all x ∈ M and r sufficiently small. This is enough to show that for any K ⊆ M we have dim(φ(K)) = dim(K) where the first dimension is computed in [0, 1] d using the Euclidean distance while the second is computed in M using the Riemannian distance. Our theorem now follows from the above cited Lemma 3.4 of [11].
Proof of Theorem 1.1: In view of (1.3) it suffices to show that ≥ dκ M a. s.