On the diffeomorphisms group generated by gaussian vector fields

This note gives a partial answer to a question asked during the workshop Mathematics on Shape Spaces. We will denote by Hσ the reproducing kernel Hilbert space of vector fields on R generated by a Gaussian kernel kσ(x, y) = e−‖x−y‖ /σ IdRd for a positive real parameter σ. Let us first recall an analytical characterization of the spaceHσ, denoting f̂ the Fourier transform of f ∈ L(R,R): Hσ = { f ∈ L(R,R) ∣∣∣ ‖f‖2Hσ = σ 2dπd/2 ∫ Rd |f̂(ω)| exp ( σ|ω| 4 ) dω <∞ } . (1) The group GHσ consists of all flows that can be generated by Hσ vector fields, GHσ = { φ(1) : φ(t) is the solution of (2) with u ∈ L([0, 1],Hσ) } . Given a time-dependent vector field u ∈ L([0, 1],Hσ), there exists a unique curve φ ∈ C([0, 1],Diff(Rd)) solving ∂tφ(t) = u(t) ◦ φ(t) , φ(0) = Id , (2) for t ∈ [0, 1] almost everywhere. Let us recall what is proved in [You10]: • Since the space Hσ can be continuously embedded in the space of H(R) vector fields (Sobolev space of order n ≥ 1), the flow is contained in Diff∞(Rd). • Since the kernel kσ is positive definite on R, the group GHσ acts n−transitively on R if d ≥ 2, i.e. for any two ordered sets of n distinct points (x1, . . . , xn) and (y1, . . . , yn) in R there exists an element φ ∈ GHσ such that for each i ∈ J1, nK, φ(xi) = yi. These groups are widely used in application such as diffeomorphic image matching [BMTY05, RVW11, SLNP11], although their understanding is less developed than the group of Sobolev diffeomorphisms [KLMP11, MP10, BV14]. A possible generalization of the previous property is the following question: Question 1. Does the group GHσ acts transitively on the space of compactly supported smooth densities Dens∞(Rd) := {ρ ∈ C∞(Rd,R) | ∫ Rd ρ(x) dx = 1}? Recall that this property is well-known in the case of smooth diffeomorphisms by the so-called Moser trick. As we will prove in this note, the answer to question 1 is negative. Indeed, we have: Proposition 1. The group GHσ is contained in the group of real analytic diffeomorphisms of R, GHσ ⊂ Diff (Rd). More precisely, any element of GHσ admits a holomorphic extension on a cylindrical open set of R in C, namely C(r) = {z ∈ C ∣∣ ∀i ∈ J1, dK | Im(zi)| ≤ r} for r > 0 sufficiently small. 1July 2014 in Bad Gastein, Austria