Spherical and Hyperbolic Fractional Brownian Motion

We define a Fractional Brownian Motion indexed by a sphere, or more generally by a compact rank one symmetric space, and prove that it exists if, and only if, 0 < H ≤ 1/2. We then prove that Fractional Brownian Motion indexed by an hyperbolic space exists if, and only if, 0 < H ≤ 1/2. At last, we prove that Fractional Brownian Motion indexed by a real tree exists when 0 < H ≤ 1/2.


Introduction
Since its introduction [10,12], Fractional Brownian Motion has been used in various areas of applications (e.g.[14]) as a modelling tool.Its success is mainly due to the self-similar nature of Fractional Brownian Motion and to the stationarity of its increments.Fractional Brownian Motion is a field indexed by R d .Many applications, as texture simulation or geology, require a Fractional Brownian Motion indexed by a manifold.Many authors (e.g.[13,8,1,7,2]) use deformations of a field indexed by R d .Self-similarity and stationarity of the increments are lost by such deformations: they become only local self-similarity and local stationarity.We propose here to build Fractional Brownian Motion indexed by a manifold.For this purpose, the first condition is a stationarity condition with respect to the manifold.The second condition is with respect to the self-similar nature of the increments.Basically, the idea is that the variance of the Fractional Brownian Motion indexed by the manifold should be a fractional power of the distance.Let us be more precise.The complex Brownian motion B indexed by R d , d ≥ 1, can be defined [11] as a centered Gaussian field such that: can be defined [10,12] as a centered Gaussian field such that: The complex Brownian motion B indexed by a sphere S d , d ≥ 1, can be defined [11] as a centered Gaussian field such that: where O is a given point of S d and d(M, M ) the distance between M and M on the sphere (that is, the length of the geodesic between M and M ).Our first aim is to investigate the fractional case on S d .We start with the circle S 1 .We first prove that there exists a centered Gaussian process (called Periodical Fractional Brownian Motion, in short PFBM) such that: where O is a given point of S 1 and d(M, M ) the distance between M and M on the circle, if and only if, 0 < H ≤ 1/2.We then give a random Fourier series representation of the PFBM.We then study the general case on S d .We prove that there exists a centered Gaussian field (called Spherical Fractional Brownian Motion, in short SFBM) such that: where O is a given point of S d and d(M, M ) the distance between M and M on S d , if and only, if 0 < H ≤ 1/2.We then extend this result to compact rank one symmetric spaces (in short CROSS).
Let us now consider the case of a real hyperbolic space H d .We prove that there exists a centered Gaussian field (called Hyperbolic Fractional Brownian Motion, in short HFBM) such that: where O is a given point of H d and d(M, M ) the distance between M and M on H d , if, and At last, we consider the case of a real tree (X, d).We prove that there exists a centered Gaussian field such that: where O is a given point of X, for 0 < H ≤ 1/2.
2. Assume 0 < H ≤ 1/2.Let us parametrize the points M of the circle S 1 of radius r by their angles x.B H can be represented as: where and (ε n ) n∈Z is a sequence of i.i.d.complex standard normal variables.
Proof of Theorem 2.1 Without loss of generality, we work on the unit circle S 1 : r = 1.Let M and M be parametrized by x, x ∈ [0, 2π[.We then have: The covariance function of B H , if there exists, is: Let us expand the function x → d H (x, 0) in Fourier series: We will see that the series n∈Z |f n | converges.It follows that equality (7) holds pointwise. Since We can therefore write, no matter if x − x is positive or negative: .
We now prove that R H is a covariance function if and only if 0 < H ≤ 1/2.p i,j=1 Let us study the sign of f n , n ∈ Z .Since f −n = f n , let us only consider n > 0.
Using the concavity/convexity of the functions x → x 2H , one sees that 1. H ≤ 1/2 All the f n are negative and (8) is positive.
2. H > 1/2.We check that, if B H exists, then we should have: All the f n , with n even, are positive, which constitutes a contradiction.
In order to prove the representation (5), we only need to compute the covariance: 3 Spherical Fractional Brownian Motion

Proof of Theorem 3.1
Let us first recall the classification of the CROSS, also known as two points homogeneous spaces [9,17]: spheres S d , d ≥ 1, real projective spaces P d (R), d ≥ 2, complex projective spaces P d (C), d = 2k, k ≥ 2, quaternionic projective spaces P d (H), d = 4k, k ≥ 2 and Cayley projective plane P 16 .[6] has proved that Brownian Motion indexed by CROSS can be defined.The proof of Theorem 3.1 begins with the following Lemma, which implies, using [6], the existence of the Fractional Brownian Motion indexed by a CROSS for 0 < H ≤ 1/2.Lemma 3.1 Let (X, d) be a metric space.If the Brownian Motion B indexed by X and defined by: exists, then the Fractional Brownian Motion B H indexed by X and defined by: Proof of Lemma 3.1 For λ ≥ 0, 0 < α < 1, one has: We then have, for 0 < H < 1/2: Let us remark that: , so that: Denote by R H (M, M ) the covariance function of B H , if exists: Let us check that R H is positive definite: (9) is clearly positive and Lemma 3.1 is proved.
We now prove by contradiction that the Fractional Brownian Motion indexed by a CROSS does not exist for H > 1/2.The geodesic of a CROSS are periodic.Let G be such a geodesic containing 0. Therefore, the process B H (M ), M ∈ G is a PFBM.We know from Theorem 2.1 that PFBM exits if, and only if, 0 < H ≤ 1/2.

Proof of Corollary 3.1
Let φ be the isometric mapping between M and the CROSS and let d (resp.d) be the metric of M (resp.the CROSS).Then, for all M, M ∈ M, one has: Let O be a given point of M and O = φ( O).Denote by R H (resp. R H ) the covariance function of the Fractional Brownian Motion indexed by M (resp.the CROSS).
It follows that R H is positive definite if and only if, R H is positive definite.Corollary 3.1 is proved.

Hyperbolic Fractional Brownian Motion
Let us consider real hyperbolic spaces H d : with geodesic distance: where The HFBM is the Gaussian centered field such that: where O is a given point of H d .

Real trees
A metric space (X, d) is a real tree (e.g.[3]) if the following two properties hold for every x, y ∈ X.
• There is a unique isometric map f x,y from [0, d(x, y)] into X such that f x,y (0) = x and f x,y (d(x, y)) = y.
Theorem 5.1 The Fractional Brownian Motion indexed by a real tree (X, d) exists for 0 < H ≤ 1/2.

Corollary 4. 1
Let (M, d) be a complete Riemannian manifold such that M and H d are isometric.Then the Fractional Brownian Motion indexed by M and defined by:B H ( O) = 0 (a.s.) , E|B H (M ) − B H (M )| 2 = d 2H (M, M ) M, M ∈ M , exists if, and only, if 0 < H ≤ 1/2.The proof of Corollary 4.1 is identical to the proof of Corollary 3.1.
Let (M, d) be a complete Riemannian manifold such that M and a CROSS are isometric.Then the Fractional Brownian Motion indexed by M and defined by: