Hyperbolic random walks
Although the hyperbolic r.w. defined on a regular hyperbolic planar grid satisfies an invariance principle, as we shall see, the picture radically differs from the Euclidean setting: the infinite grid is the whole space when the step is too small. We also
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Summary. Although the hyperbolic r.w. defined on a regular hyperbolic planar grid satisfies an invariance principle, as we shall see, the picture radically differs from the Euclidean setting: the infinite grid is the whole space when the step is too small. We also give a radial discretization of Bochner’s subordinated hyperbolic Brownian motions. Key words: Hyperbolic plane, random walk, invariance principle, non Fuchsian group, hypergroup, stable process
1 Introduction In hyperbolic space, we construct a family of discrete (both in time and in space) random planar walks which functionally converge towards hyperbolic Brownian motion (Theorem 1). The previously known construction (e.g. Gangolli [21]) only dealt with the isotropic case. Our method is to proceed by fixed geodesic steps and independent random choices of directions, the change of direction being chosen inside a regular polygon, then parallelly transported. The steps to which we apply the normalization are completely described by the length of the geodesic jumps. Let us stress that, unlike the nilpotent case, when the steps are sufficiently small, the subgroup generated by the associated isometries is dense in P SL2 (R). Hence the strong analogy between Poincar´e half-plane and discrete skeletons such as the regular trees is definitively not the whole story. In the last part, assuming the process to be radial, we construct random walks which converge towards a subordinated hyperbolic Brownian motion, alias a pseudo stable hyperbolic process of the second kind.
280
J.-C. Gruet
2 The isotropic random walk Let M be a complete Riemannian manifold with Ricci curvature bounded √2 ( n ) , k ∈ N) with from below by a constant. The isotropic random walk (W √k2 ( ) step d := n2 is inductively constructed by requiring Wk+1n to be uniformly √ ( 2) distributed on the geodesic sphere with radius n2 centered at Wk n . Then, a continuous interpolation (ξn (t), t ≥ 0) of this process √ is defined √ by letting ξn (
move with constant speed along the geodesic Wk time interval [k, k + 1].
2 n)
(
2
)
→ Wk+1n during the
Proposition 1. (Jørgensen [26], Pinsky [33] and Blum [11]). The processes (ξn (nt), t ≥ 0) converge in law towards the M -Brownian motion. Remark. Note that these authors consider the Poissonized (discontinuous) jump process (with independent exponential holding times) to simplify their proofs. The name isotropic transport process is due to Pinsky. There have been some misunderstandings about Jørgensen’s work since its third part introduces in the greatest generality laws which are invariant by parallel transport. Moreover, when M is a symmetricspace of rank one (such as the Poincar´e half-plane H2 ), the distance process Rn = d(Wn , o), n ∈ N from any given origin o is a Markov chain. This reduction to a single real process was exploited by the hypergroup approach [10,40,41]. In hypergroup theory, the convolution of two probabilities is conventionally defined by linearity from the convolution of two Dirac measures, which is a mixture of Dirac measures
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