On the convex hull of symmetric stable processes

Let alpha \in (1, 2] and X be an R^d-valued alpha-stable process with independent and symmetric components starting in 0. We consider the closure S_t of the path described by X on the interval [0, t] and its convex hull Z_t. The first result of this paper provides a formula for certain mean mixed volumes of Z_t and in particular for the expected first intrinsic volume of Z_t. The second result deals with the asymptotics of the expected volume of the stable sausage Z_t+B (where B is an arbitrary convex body with interior points) as t \to 0.


Introduction and main results
For fixed α ∈ (1, 2] and fixed integer d ≥ 1 we consider an R d -valued stochastic process X ≡ (X(t)) t≥0 = (X 1 (t), . . . , X d (t)) t≥0 , defined on the probability space (Ω, A, P), such that the components X j := (X j (t)) t≥0 , j ∈ {1, . . . , d}, are independent α-stable symmetric Lévy processes with scale parameter 1 starting in 0. The characteristic function of X j (t) is given by cf. [8,Section 1.3]. This implies that X is self-similar in the sense that (X(st)) s≥0 d = t 1/α X for any t > 0, see [8,Example 7.1.3] and [4,Chapter 15]. We assume that X is rightcontinuous with left-hand limits (rcll). For t ≥ 0, let S t be the closure of the path S 0 t := {X(s) : 0 ≤ s ≤ t} and let Z t denote the convex hull of S t . These are random closed sets. We abbreviate Z := Z 1 . By self-similarity Z t d = t 1/α Z, t > 0. (1.2) If α = 2 then X is a standard Brownian motion. A classical result of [12] for planar Brownian motion says that where V 1 (K) denotes half the circumference of a convex set K ⊂ R 2 . Our first aim in this paper is to formulate and to prove such a result for arbitrary α ∈ (1, 2] and arbitrary dimension d. In fact we also consider more general geometric functionals. A convex body (in R d ) is a non-empty compact and convex subset of R d . We let V (K 1 , . . . , K d ), denote the mixed volumes of convex bodies K 1 , . . . , K d ⊂ R d [9, Section 5.1]. These functionals are symmetric in K 1 , . . . , K d and we have for any convex bodies K, B ⊂ R d The jth intrinsic volume V j (K) of a convex body K is given by where B d is the Euclidean unit ball in R d , and κ j is the j-dimensional volume of B j . In particular, For any p ≥ 1 define B p := {u ∈ R d : u p ≤ 1} as the unit ball with respect to the L p -norm (u 1 , . . . , u d ) p := (|u 1 | p + . . . + |u d | p ) 1/p . Finally we introduce the constant In the case of α < 2 we are not aware of an explicit expression for c α .
Remark 1.2. By the scaling relation (1.2) and the homogeneity property of mixed volumes [9, (5.1.24)] the identity (1.9) can be generalized to A similar remark applies to all results of this paper.
The proof of Theorem 1.1 relies on the fact that where S d−1 denotes the unit sphere, is the support function of a convex body K, and ·, · denotes the Euclidean scalar product on R d . The next corollary provides a direct generalization of (1.3).
In the case of Brownian motion it is possible to calculate the expectation of the second intrinsic volume V 2 (Z) of Z.
Our second theorem deals with the asymptotic behaviour of the expected volume of the stable sausage S t + B as t → 0, where B is a convex body. Our result complements classical results on the asymptotic behaviour of EV d (S t + B) as t → ∞, cf. [11] for the case of Brownian motion and [3] for the case of more general stable processes. Theorem 1.5. Let B be a convex body with non-empty interior and let α ′ be as in Theorem In the case α = 2 the random set S t + B is known as Wiener sausage. Even then Theorem 1.5 seems to be new: Corollary 1.6. Assume that X is a Brownian motion and let B be a convex body with non-empty interior. Then In the case B = B d the limit equals 2 √ 2π (d−1)/2 /Γ(d/2).
Remark 1.7. In the special case d = 3 and α = 2 we have (see [11]) for any r, t ≥ 0. The term constant in t clearly allows a geometric interpretation as V 3 (rB 3 ). Now we are able to give a geometric interpretation of the coefficient of √ t as well.
From (1.13) we get On the other hand from the proof of Theorem 1.5 one can see By (1.5) and the homogenity property of mixed volumes (see e.g. [9, (5.1.24)]) we have Altogether this is 4 √ 2πr 2 = r 2 κ 2 EV 1 (Z).

Proofs
We need the following measurability property of the closure S t of {X(s) : 0 ≤ s ≤ t} and its convex hull Z t , refering to [7,10] for the notion of a random closed set.
Lemma 2.1. For any t ≥ 0, S t and Z t are random closed sets.
Proof: To prove the first assertion it is enough to show that {S t ∩ G = ∅} is measurable for any open G ⊂ R d , see [10, Lemma 2.1.1]. But since X is rccl it is clear that S t ∩ G = ∅ iff X(u) / ∈ G for all rational numbers u ≤ t. The second assertion is implied by [10,Theorems 12.3.5,12.3.2].
The previous lemma implies, for instance, that V (K 1 , . . . , K d−1 , Z t ) and V d (S t + B) are random variables, see e.g. [9, p. 275 For any u ∈ S d−1 we have It follows directly from (1.1) that the process X, u has the same distribution as u α X 1 . where a + := max{0, a} denotes the positive part of a real number a. Since X 1 (1) has a symmetric distribution and P(X 1 (1) = 0) = 0 (stable distributions have a density) we have E|X 1 (1)| = 2EX 1 (1) + and it develops that Eh(Z, u) = c α u α , with c α given by (1.7). Inserting this result into (2.2) gives By [9, Remark 1.7.8], u α is the support function of the polar body Using the Hölder inequality, it is straightforward to check that B * α = B α ′ , where 1/α + 1/α ′ = 1. Using this fact as well as (2.1) in (2.3), we obtain the assertion (1.9).
Proof of Corollary 1.3: Since α = 2 we have α ′ = 2 and B α ′ = B d . By Theorem 1.1 and (1.5), where G 2 denotes the set of all 2-dimensional linear subspaces of R d , ν 2 is the Haar measure on G 2 with ν 2 (G 2 ) = 1 and Z|L denotes the image of Z under the orthogonal projection onto the linear subspace L. By Fubini's theorem, The spherical symmetry of Brownian motion implies that EV 2 (Z|L) does not depend on L. Assume that L = {(x 1 , x 2 , 0, . . . , 0) : x 1 , x 2 ∈ R}. Now it is clear from the definition of the d-dimensional Brownian motion that the random closed set Z|L is the convex hull of a Brownian path in L. By Remark (a) in [2, p. 149] (see also [6]) we have EV 2 (Z|L) = π/2. Therefore, and the result follows by a straightforward calculation.
Proof of Theorem 1.5: By self-similarity and the dominated convergence theorem, on whose conditions we will comment below, we have (2.5) In order to justify the application of the dominated convergence theorem, put As noted in the proof of Theorem 1.1, Y j has a finite expectation. Since −Ỹ j has the same distribution as Y j ,Ỹ j has a finite expectation as well. From (1.4) we obtain for all t ∈ (0, 1] that Furthermore, where e j denotes the jth unit vector. It follows that This is a product of independent random variables with finite expected values and hence has finite expected value. By