Sara Brofferio
email: sara.brofferio@math.u-psud.fr

AND Maura Salvatori
email: maura.salvatori@unimi.it

Wolfgang Woess
email: woess@tugraz.at

BROWNIAN MOTION AND HARMONIC FUNCTIONS ON Sol(p, q)

Keywords: Mathematics Subject Classification. 58J65, 31C12, 60J50 Sol-group, hyperbolic plane, horocyclic product, Laplacian, Brownian motion, central limit theorem, rate of escape, boundary, positive harmonic functions

The Lie group Sol(p, q) is the semidirect product induced by the action of R on R 2 which is given by (x, y) → (e pz x, e -qz y), z ∈ R. Viewing Sol(p, q) as a 3-dimensional manifold, it carries a natural Riemannian metric and Laplace-Beltrami operator. We add a linear drift term in the z-variable to the latter, and study the associated Brownian motion with drift. We derive a central limit theorem and compute the rate of escape. Also, we introduce the natural geometric compactification of Sol(p, q) and explain how Brownian motion converges almost surely to the boundary in the resulting topology. We also study all positive harmonic functions for the Laplacian with drift, and determine explicitly all minimal harmonic functions. All this is carried out with a strong emphasis on understanding and using the geometric features of Sol(p, q), and in particular the fact that it can be described as the horocyclic product of two hyperbolic planes with curvatures -p 2 and -q 2 , respectively.

Introduction

Sol(p, q) is the group of all matrices of the form The parameters p and q are positive real numbers. It will be useful to think separately of Sol(p, q) as a Lie group and as a manifold. In the latter situation, we shall often write z = (x, y, z) or also x or y for its elements, instead of g. Its natural length element is (1.2) ds 2 = d p,q s 2 = e -2pz dx 2 + e 2qz dy 2 + dz 2 , which is invariant under the left action of Sol(p, q) on itself as an isometry group. If we identify the element g of (1.1) with (x, y, z), then Sol(p, q) is R 3 topologically (but of course not metrically). In those coordinates, the group product is (1.3) (a, b, c) • (x, y, z) = ( e pc x + a, e -qc y + b, c + z ) .

The manifold/Lie group Sol(1, 1) with the above metric has been attracting attention as one of the eight model geometries of Thurston's geometrisation (ex-)conjecture, see Thurston [START_REF] Thurston | The Geometry and Topology of Three-Manifolds[END_REF], [START_REF] Thurston | Three-dimensional Geometry and Topology[END_REF] and Scott [START_REF] Scott | The geometries of 3-manifolds[END_REF].

The purpose of this case study is to describe the behaviour of Brownian motion in space and time, and to determine all positive harmonic functions on Sol(p, q) with respect to its Laplace-Beltrami operator and the variant where a "vertical" drift term (in z) is added to the latter. More precisely, we shall derive a central limit theorem for Brownian motion with drift, describe convergence of this process to the natural geometric boundary at infinity, and we shall determine all positive eigenfunctions of those Laplacians. The experienced reader will know how intimately such stochastic and potential theoretic features are linked with each other.

Before we can explain the results, we need some details. To start, let H = {x + i w : x ∈ R , w > 0} be hyperbolic upper half plane with the standard length element w -2 (dx 2 + dw 2 ). We can pass to the logarithmic model by substituting z = log w , and in those coordinates the length element becomes e -2z dx 2 + dz 2 . Now we also change curvature by modifying the length element into

ds 2 = d p s 2 = e -2pz dx 2 + dz 2 .
We write H(p) for the hyperbolic plane with this parametrization and metric. Then we have the natural projections (1.4) π 1 : Sol(p, q) → H(p) , (x, y, z) → (x, z) π 2 : Sol(p, q) → H(q) , (x, y, z) → (y, -z) .

The horocycle at level z in H(p) is the set {(x, z) : x ∈ R}, and we write π(x, z) = z. Thus, we get another natural projection π : H(p) → R. We also consider π as a projection of Sol(p, q) onto R, where π(x, y, z) = z. We shall write d for each of the metrics induced by the respective length elments; it will usually be evident from the context to which of the underlying spaces this refers -or else, that space will appear in the index. (On R we then have d R (z 1 , z 2 ) = |z 1z 2 |.) Note that our projections preserve distances in the following sense:

(1.

) = |z 1 -z 2 | .
A main structural feature is that the manifold Sol(p, q) is made up by two hyperbolic planes (with respective curvatures -p 2 and -q 2 ) that are glued together by identifying opposite horocycles: it can be seen as the horocyclic product of H(p) and H(q), (1.6) Sol(p, q) = {(u, v) ∈ H(p) × H(q) : π(u) + π(v) = 0} , with its metric arising naturally from those two hyperbolic planes. We remark here that there are various different types of horocyclic products. Sol(p, q) has two sister structures. One is the Diestel-Leader graph DL(p, q), which is the horocyclic product of two regular trees with degrees p + 1 and q + 1, respectively, where p, q ≥ 2 are integer. One of its interesting features is that when p = q, it is a Cayley graph of the lamplighter group (Z/pZ) ≀ Z. Random walks and harmonic functions on DL(p, q) have been studied intensively by Bertacchi [9], Woess [START_REF] Woess | Lamplighters, Diestel-Leader graphs, random walks, and harmonic functions. Combinatorics[END_REF], Bartholdi and Woess [START_REF] Bartholdi | Spectral computations on lamplighter groups and Diestel-Leader graphs[END_REF] and Brofferio and Woess [START_REF] Brofferio | Green kernel estimates and the full Martin boundary for random walks on lamplighter groups and Diestel-Leader graphs[END_REF], [START_REF] Brofferio | Positive harmonic functions for semi-isotropic random walks on trees, lamplighter groups, and DL-graphs[END_REF]. The other sister structure is treebolic space HT(p, q), which is the horocyclic product of H(p) and the tree with degree q + 1, where p > 0 (real) and q ≥ 2 (integer). When p = q, the Baumslag-Solitar group ⟨a, b | ab = b q a⟩ acts on HT(q, q) with compact quotient. The study of potential theory and Brownian motion on treebolic space is harder than on Sol and on DL (where random walk replaces Brownian motion), first of all because of the conceptual and technical difficulty in constructing the right Laplacian(s) on the 2-dimensional complex HT. This is ongoing work of Bendikov, Saloff-Coste, Salvatori and Woess [START_REF] Bendikov | The heat semigroup and Brownian motionon strip complexes[END_REF], [START_REF] Bendikov | Positive harmonic functions and Brownian motion on treebolic spaces[END_REF].

Brownian motion and random walks on Sol(1, 1) made a brief appearance in the work of Lyons and Sullivan [START_REF] Lyons | Function theory, random paths and covering spaces[END_REF]. Harmonic functions for random walks on Sol(1, 1) also appear in Raugi [START_REF] Raugi | Fonctions harmoniques positives sur certains groupes de Lie résolubles connexes[END_REF]Exemple 2,p. 677].

HT, DL and Sol are also objects of great interest in relation with geometric group theory. Quasi-isometries of those spaces have been studied by Farb and Mosher [START_REF] Farb | A rigidity theorem for the solvable Baumslag-Solitar groups. With an appendix by Daryl Cooper[END_REF] (for HT(p, p)) and by Eskin, Fisher and Whyte [START_REF] Eskin | Quasi-isometries and rigidity of solvable groups[END_REF], [START_REF] Eskin | Coarse differentiation of quasi-isometries I: spaces non quasiisometric to Cayley graphs[END_REF] (for DL and Sol). The last two papers also contain a good description of several aspects of the geometry of Sol. The instructive article of Troyanov [START_REF] Troyanov | L'horizon de SOL[END_REF] contains a detailed study of various geometric features of Sol(1, 1), including curvature (which is positive in some directions and negative in others), the isometry group and in particular, the geodesic rays and their asymptotic classes. See also Kim [30], who basically provides a subset of the material of [START_REF] Troyanov | L'horizon de SOL[END_REF].

The Laplace operator with vertical drift parameter a ∈ R on Sol(p, q) is

(1.7) L a = L Sol(p,q) a = 1 2
( e 2pz ∂ 2 ∂x 2 + e -2qz ∂ 2 ∂y 2 + ∂ 2 ∂z 2 ) + a ∂ ∂z .
The Laplace-Beltrami operator arises for a = (qp)/2. As a matter of fact, this involves a small abuse of terminology: in differential geometry, the "true" Laplace-Beltrami operator would be twice the one which we are using. Here, we are following the probabilistic habits: with the factor 1 2 , in the standard Euclidean situation, the Laplacian is the infinitesimal generator of standard Brownian motion. The situation is similar here.

Under the projection π 1 , the operator L a projects onto the operator on H(p) given by

(1.8) L H(p) a = 1 2
( e 2pz ∂ 2 ∂x 2 + ∂ 2 ∂z 2 ) + a ∂ ∂z .
By "projects" we mean that for a

C 2 -function f 1 on H(p), one (obviously) has L a (f 1 •π 1 ) = (L H(p) a f 1 ) • π 1
. This is the Laplace-Beltrami operator on H(p) when a = -p/2. Analogously, under the projection π 2 (where the sign of z is changed), L a projects onto the operator on H(q) given by (1.9) L

H(q) -a = 1 2 ( e 2qz ∂ 2 ∂y 2 + ∂ 2 ∂z 2 ) -a ∂ ∂z .
And finally, L a projects under π onto the operator on R given by (1.10)

L a = 1 2 d 2 dz 2 + a d dz .
Coming back to the outline of the contents of this paper, some basic preliminaries are laid out in §2. Our first, probabilistic object of study is then Brownian motion with drift Z t = (X t , Y t , Z t ) t>0 on Sol(p, q), i.e., the diffusion process whose infinitesimal generator is L a . The projections of Z t on H(p), H(q) and R are the diffusions whose infinitesimal generators are the respective projected operators defined above.

In §3, we describe this process in terms of stochastic integrals and first derive a central limit theorem for (X t , Y t , Z t ). Combining this with estimates from §2 for the metric of Sol, we also obtain a central limit theorem for d(Z t , o). Its form for the case a = 0 is somewhat different from what happens for a ̸ = 0. As a corollary, we get the linear rate of escape:

d(Z t , o) t → |a| almost surely, as t → ∞ ,
where o = (0, 0, 0). This is the same as the rate of escape for the projected ("vertical") Brownian motion with drift (Z t ) t>0 on R, so that the lateral motion in the x-and yvariables does not contribute to that rate.

Since the Sol-group has exponential growth, our process is always transient, that is, with probability 1 it eventually leaves each compact set. §4 adds more details to the description of how our process tends to infinity in space. Namely, Sol(p, q) has a natural geometric compactification: since Sol(p, q) is a subset of the product of two hyperbolic planes (or equivalently, hyperbolic disks), it embeds naturally into the product of two closed unit disks, and the closure of Sol(p, q) in this bi-disk is the compactification. Topologically, the resulting boundary at infinity has the shape of a filled number "8", that is, two full closed disks glued together at a single (glueing) point. It is not a "visibility" boundary: neither the glueing point nor any of the interior points of the two disks come up as a limit of some geodesic ray in Sol; all the other boundary points are limits of geodesic rays.

It is a rather straighforward, but nevertheless informative task to verify that Brownian motion tends almost surely in the topology of that compactification to a limit random variable that lives on the boundary at infinity of Sol(p, q). If a = 0 then this limit is the glueing point deterministically. Otherwise, that limit random variable lies on one of the two circles that make up the "8" (not their interiors) and its distribution is continuous. Thus, when a ̸ = 0, we (almost surely) have the geodesic ray from the origin to the random limit point. If γ = ( γ(t)

)
t≥0 is that limit geodesic, then we show that for a ̸ = 0, the deviation of Z t from that ray is at most logarithmic, that is, there is c > 0 such that lim sup t→∞ d(Z t , γ)/ log t ≤ c almost surely. This result comprises the analogous one for Brownian motion with drift on the hyperbolic plane. The latter is a special case of a theorem stated by Ancona [1, Théorème 7.3]. There is a correponding theorem for random walks on free groups, resp. trees, that was shown by Ledrappier [START_REF] Ledrappier | Some asymptotic properties of random walks on free groups[END_REF]; see Woess [START_REF] Woess | Denumerable Markov chains. Generating functions, boundary theory, random walks on trees[END_REF]Thm. 9.59] for a general and simple proof.

The second main body of this work concerns positive harmonic functions. These are the positive C 2 -functions that are anihilated by the respective Laplacian. We can also handle positive eigenfunctions.

We start in §5 by displaying some of the potential theoretic, resp. analytic ingredients that are needed. Then we prove in §6 that every positive L a -eigenfunction on Sol(p, q) has the form h(x, y, z

) = h 1 (x, z) + h 2 (y, -z) ,
where h 1 is a non-negative L

H(p) a

-eigenfunction on H(p) and h 2 is non-negative L

H(q)

-aeigenfunction on H(q), both with the same eigenvalue as h.

This decomposition is not unique, but we can also see where non-uniqueness comes from, namely, harmonic functions that only depend on the "height" z. What we do is indeed to describe all minimal positive eigenfunctions, based on ideas from the discrete setting of Diestel-Leader graphs, see [START_REF] Brofferio | Positive harmonic functions for semi-isotropic random walks on trees, lamplighter groups, and DL-graphs[END_REF].

Since the positive L H(p) a

-eigenfunctions are known explicitly as integrals of modified Poisson kernels, the above result leads to a complete description of all positive L aeigenfunctions on Sol(p, q). Thus, the positive eigenfunctions of the Laplacian on Sol(p, q) can be described fully in terms of modified Poisson kernels on each of the two hyperbolic planes that make up our space.

The computations undertaken here are related with the study of Martin compactifications of symmetric spaces, although the manifold Sol(p, q) has both positive and negative sectional curvatures. The reader is referred to the book by Guivarc'h, Ji and Taylor [START_REF] Guivarc'h | Compactifications of Symmetric Spaces[END_REF] and the survey by Kaimanovich [26] plus the references given there. In particular, we point out the work of Karpelevich [START_REF] Karpelevich | The geometry of geodesics and the eigenfunctions of the Beltrami-Laplace operator on symmetric spaces[END_REF] and Taylor [START_REF] Taylor | The Iwasawa decomposition and the limiting behaviour of Brownian motion on a symmetric space of noncompact type[END_REF]. For more general homogenous manifolds, see e.g. Babillot [START_REF] Babillot | Comportement asymptotique du mouvement brownien sur une variété homogène courbure négative ou nulle[END_REF].

We get close to answering the question of Lyons and Sullivan [START_REF] Lyons | Function theory, random paths and covering spaces[END_REF] to determine the Martin boundary of Sol; we find the minimal boundary and have a clear idea what the Martin compactification has to be.

Besides the work mentioned already, there is a rich literature on issues close to the ones studied here regarding Lie groups that are semidirect products of R (or other Abelian groups) and Abelian or nilpotent groups. See Molchanov [START_REF] Molchanov | Martin boundary for invariant Markov processes on a solvable group[END_REF], Malliavin and Malliavin-Brameret [START_REF] Malliavin | Factorisations et lois limites de la diffusion horizontale au-dessus d'un espace riemannien symétrique[END_REF], Damek and Hulanicki [START_REF] Damek | Boundaries for left-invariant subelliptic operators on semidirect products of nilpotent and abelian groups[END_REF], or Damek, Hulanicki and Urban [START_REF] Damek | Martin boundary for homogeneous Riemannian manifolds of negative curvature at the bottom of the spectrum[END_REF], to name just a few.

We want to underline that the main spirit of this paper is to study the outlined issues via strong use of the geometry of Sol(p, q) in terms of the two hyperbolic planes and their horocyclic product.

Acknowledgements. We warmly thank M. Yor, who suggested to the first of the three of us the approach used for proving the central limit via stochastic integration. Our own approach would have been random walk based, following the spirit of Grincevičjus [START_REF] Grincevičjus | A central limit theorem for the group of linear transformations of the real axis[END_REF]. Also, we thank A. Grigoryan for precious input regarding the Harnack inequality used in Proposition 5.9. Finally, we acknowledge instructive hints by V. A. Kaimanovich as well as the referees concerning the literature.

Basic facts

The first part of this section contains some basic facts regarding Sol(p, q) that are quite straightforward. They are included here for the sake of the completeness of the picture; most proofs are omitted. We refer to [START_REF] Troyanov | L'horizon de SOL[END_REF] for a careful study of geometry and group properties of Sol(1, 1), most of which carries over to Sol(p, q).

(2.1) Lemma. A natural base of the Lie algebra of the Lie group Sol(p, q) is given by the vector fields

X = e pz ∂ x , Y = e -qz ∂ y , Z = ∂ z .
The Lie brackets are

[X, Y ] = 0 , [Z, X] = pX , [Z, Y ] = -qY.
With X, Y, Z as orthonormal frame, the associated left invariant metric is the one of (1.2).

(2.2) Lemma. The sectional curvatures of the manifold Sol(p, q) in terms of the vector fields X, Y, Z of Lemma 2.1 are given by

K(X, Z) = -p 2 , K(Y, Z) = -q 2 , K(X, Y ) = pq.
The computations are precisely as done in [START_REF] Troyanov | L'horizon de SOL[END_REF] in the case p = q = 1.

(2.3) Lemma. The Riemannian volume element of the Sol-manifold is dz = e (q-p)z dx dy dz .

This is also the left Haar measure of Sol(p, q) as a group. The modular function on this group is ∆(g) = e (q-p) π(g) , where g is parametrized by (x, y, z) as in (1.1) and π(g) = z.

The group is unimodular if and only if p = q.

Next, consider the group Aff(p) of all matrices of the form (2.4)

( e pz x 0 1 ) , x, z ∈ R .
This is nothing but the group of orientation preserving affine transformations of hyperbolic plane, again parametrized by the logarithmic model and substituting the habitual upper left term e z with e pz . We can identify the group Aff(p) with the surface H(p) in the same way as we identified Sol as a group with Sol as a manifold. By left multiplication, Aff(p) acts isometrically on H(p). We recall the following.

(2.5) Lemma. (a) The Riemannian area element of H(p) is e -pz dx dz . This is also the left Haar measure on the group Aff(p), and the modular function on Aff(p) is ∆(g) = e p π(g) , where g =

( e pz x 0 1
) and π(g) = z.

Let D(2) be the group of reflections (x, y, z) → (±x, ±y, z). They all are isometries of the manifold Sol(p, q). When p = q, also the reflection (x, y, z) → (y, -x, -z) is an isometry (it is not when p ̸ = q), and we write D(4) for the (dihedral) group generated by D and this mapping.

(2.6) Lemma. When p ̸ = q, the full isometry group of the manifold Sol(p, q) is the semidirect product of D(2) with the group Sol(p, q).

When p = q, the full isometry group is the semidirect product of D(4) with the group Sol(p, q). For the case p = q = 1, see [START_REF] Troyanov | L'horizon de SOL[END_REF].

We can interpret the projections π 1 and π 2 of (1.4) as homomorphisms from the group Sol(p, q) onto Aff(p) and Aff(q), respectively. In the same way, π is a homomorphism onto the additive group R.

(2.7) Lemma. (a) The Laplacian L a on Sol(p, q) is reversible (self-adjoint) with respect to the measure m a (dz) = e (2a+p-q)z dz = e 2az dx dy dz . (c) The Laplacian L a on R is reversible with respect to the measure

e 2az dz .
Proof (hint). For proving (a) one has to show that for compactly supported C 2 -functions f, g on Sol, one has

∫∫∫ f (x, y, z) L a g(x, y, z) e 2az dx dy dz = ∫∫∫ L a f (x, y, z) g(x, y, z) e 2az dx dy dz .
This is straightforward by partial integration. (b) and (c) are analogous.

Our Laplacian is invariant under the group action of Sol. Let g 0 = (a, b, c) be a group element, and define the translate of a function f on Sol as τ g 0 f (g) = f (g 0 g) , that is,

(2.8) τ g 0 f (x, y, z) = f ( e pc x + a, e -qc y + b, c + z ) .
(2.9) Lemma. For any g 0 ∈ Sol(p, q),

L a (τ g 0 f ) = τ g 0 (L a f ) .
The proof is completely elementary, using (2.8).

We shall need the following observations on the metric. Regarding our hyperbolic planes in the logarithmic model, let us remark here that the metric of H(p) is linked with the standard one of H = H(1) by the formula

(2.10) d H(p) ( (x, z) , (x ′ , z ′ ) ) = 1 p d H(1) ( (px, pz) , (px ′ , pz ′ ) ) .
While for Diestel-Leader graphs, there is an explicit formula for the graph metric in terms of the two underlying trees [START_REF] Bertacchi | Random walks on Diestel-Leader graphs[END_REF], we do not have such a formula on Sol. However, we have at least the following distance estimates.

(2.11) Proposition. For all z = (x, y, z) ∈ Sol(p, q), with x, y ̸ = 0 in (iv),

d Sol (o, z) ≥ |z| , (i) d Sol (o, z) ≥ 2 log |x| p + 2 log |y| q -|z| - ( 1 p + 1 q ) log d Sol (o, z) , (ii) d Sol (o, z) ≤ d H(p) ( (x, z) , (0, 0) ) + d H(q) ( (y, -z) , (0, 0) ) -|z| , (iii) ≤ c + 2 log(1 + |x|) p + 2 log(1 + |y|) q + |z| d Sol (o, z) ≤ c ′ + log |x| p + log |y| q (iv) + min { log |x| p + log |y| q + z , log |x| p -z + log |y| q } ,
where c, c ′ > 0.

Proof. Inequality (i) is clear.

For (ii), Let z(t) = ( x(t), y(t), z(t) ) t∈[0,d] be a geodesic path in Sol from o to z, where d = d Sol (o, z). Let M = max{z(t) : t ∈ [0, d]} and m = min{z(t) : t ∈ [0, d]} , so that M ≥ 0 and m ≤ 0. Then d Sol (z , o) = ∫ d 0 √ e -2pz(t) ẋ(t) 2 + e 2qz(t) ẏ(t) 2 + ż(t) 2 dt ≥ e -pM ∫ d 0 √ ẋ(t) 2 + ż(t) 2 dt ≥ e -pM √ x 2 + z 2 Thus pM ≥ log √ x 2 + z 2 -log d Sol (o, z) , and analogously -qm ≥ log √ y 2 + z 2 -log d Sol (o, z) .
Now let z M and z m be points on the geodesic from 0 to z with heights M and m, respectively. Then, according to which of the two "comes first", (i) yields that either

d Sol (o, z) = d Sol (o, z M ) + d Sol (z M , z m ) + d Sol (z m , z) ≥ M + (M -m) + (z -m) , or d Sol (o, z) = d Sol (o, z m ) + d Sol (z m , z M ) + d Sol (z M , z) ≥ -m + (M -m) + (M -z) .
We see that

d Sol (o, z) ≥ 2(M -m) -|z| ,
and combining this with the above, we obtain (ii).

For proving the first part of (iii), we may suppose without loss of generality that z ≥ 0. Note that in the logarithmic model of H(p), any geodesic arc is either vertical (i.e., of the form t → (x 0 , t), where x 0 is fixed and t varies in an interval), or else it can be realised as t → ( t, z(t) )

, where z(t) is a strictly concave function of t varying in an interval. Let (x ′ , z) be the first ("leftmost") point on the geodesic arc from (0, 0) to (x, z) in H(p) with second coordinate z, and let (y ′ , 0) be the last ("rightmost") point on the geodesic arc from (0, 0) to (y, -z) in H(q) with second coordinate -z.

We may have x ′ = x or y ′ = 0, but in any case, the geodesic arc from (0, z) to (x ′ , z) in H(p) is strictly increasing in both coordinates, while the geodesic arc from (y ′ , 0) to (y, -z) in H(q) is strictly inreasing in the first and strictly decreasing in the second variable. That is, these two arcs can be parametrised, respectivley, as

t → ( x(t), t ) and t → ( y(t), -t ) ,
where t ∈ [0 , z] and ẋ(t), ẏ(t) > 0. Now we can "synchronise" the two in order to get the curve

t → ( x(t), y(t), t ) , t ∈ [0 , z] ,
that connects (0, y ′ , 0) with (x ′ , y, z) in Sol(p, q). The length of this curve majorises the distance between these two points in Sol(p, q) and is

∫ z 0 √ e -2pt ẋ(t) 2 + e 2qt ẏ(t) 2 + 1 dt ≤ ∫ z 0 ( √ e -2pt ẋ(t) 2 + 1 + √ e 2qt ẏ(t) 2 + 1 -1 ) dz = d H(p) ( (0, 0) , (x ′ , z) ) + d H(q) ( (y ′ , 0) , (y, -z) ) -z . Now, by (1.5), d Sol (o , z) ≤ d Sol ( (0, 0, 0) , (0, y ′ , 0) ) = d H(q) ( (0, 0) , (y ′ , 0) ) + d Sol ( (0, y ′ , 0) , (x ′ , y, z) ) + d Sol ( (x ′ , y, z) , (x, y, z) ) = d H(p) ( (x ′ , z) , (x, z) )
We insert the upper bound for the middle term that we derived above. Since

d H(p) ( (0, 0) , (x ′ , z) ) + d H(p) ( (x ′ , z) , (x, z) ) = d H(p) ( (0, 0) , (x, z) ) and d H(q) ( (0, 0) , (y ′ , -z) ) + d H(q) ( (y ′ , -z) , (y, -z) ) = d H(q) ( (0, 0) , (y, -z) ) ,
the proposed inequality follows. For the second part of (iii), we use (2.10):

d H(p) ( (x, z) , (0, 0) ) ≤ |z| + 1 p d H(1) ( (px, 0) , (0, 0) ) = |z| + 1 p log √ (px) 2 + 4 + |px| √ (px) 2 + 4 -|px| ≤ |z| + 1 p log ( (px) 2 + p|x| + 1 ) ≤ 2 log(1 + p) p + 2 log(1 + |x|) p .
Combining this with the analogous bound for d H(q) ( (y, -z) , (0, 0) ) , the inequality follows.

For proving (iv), first note that for all x ̸ = 0,

d H(p) ( ( 0, log |x| p ) , ( x, log |x| p ) )
= c p depends only on p. Then, using (1.5),

d Sol (o , z) ≤ d Sol ( o , ( 0, 0, log |x| p ) ) + d Sol ( ( 0, 0, log |x| p ) , ( x, 0, log |x| p ) ) + d Sol ( ( x, 0, log |x| p ) , ( x, 0, -log |y| q ) ) + d Sol ( ( x, 0, -log |y| q ) , ( x, y, -log |y| q ) ) + d Sol ( ( x, y, -log |y| q ) , z ) = log |x| p + c p + log |x| p + log |y| q + c q + log |y| q + z .
Exchanging the roles of x and y, as well as of p and q, the inequality follows.

Central limit theorem and rate of escape

Let Z t = (X t , Y t , Z t ), t ≥ 0, be the continuous diffusion on Sol(p, q) ≡ R 3 whose infinitesimal generator is L a . If the starting point is o = (0, 0, 0), then Z t is given by the stochastic integrals (2) s ,

(3.1)                Z t = a t + W t , X t = ∫ t 0 e pZ s dW (1) s Y t = ∫ t 0 e -qZ s dW
where (W t , W

t , W

t ) t≥0 are three independent standard Brownian motions. (We do not attach a superscript to the one that defines the coordinate Z t , because this is the most important one that determines the behaviour of all three.) See for instance Revuz and Yor [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF] or Protter [START_REF] Protter | Stochastic Integration and Differential Equations[END_REF], and compare, in particular, with Baldi, Casadio Tarabusi, Figà-Talamanca and Yor [START_REF] Baldi | Non-symmetric hitting distributions on the hyperbolic half-plane and subordinated perpetuities[END_REF].

For the following central limit theorem, let

N = W 1 , M = min{W t : 0 ≤ t ≤ 1} and M = max{W t : 0 ≤ t ≤ 1} ,
so that N has standard normal distribution.

(3.2) Theorem. (i) If a > 0, then as t → ∞ 1 √ t ( log |X t | -pa t , log |Y t | , Z t -a t ) → (p N , 0 , N ) in law. (ii) If a < 0, then as t → ∞ 1 √ t ( log |X t | , log |Y t | + qa t , Z t -a t ) → (0 , q N , N ) in law. (iii) If a = 0, then as t → ∞ 1 √ t ( log |X t | , log |Y t | , Z t ) → (pM , -q M , N ) in law. Proof. For α ∈ R, set V t (α) = ∫ t 0 e 2αZs ds,
so that the quadratic variations of X t and Y t are V t (p) and V t (-q), respectively. Then by a theorem of Dambis, Dubin and Schwartz [START_REF] Dambis | On decomposition of continuous submartingales[END_REF], [START_REF] Dubins | On continuous martingales[END_REF], see also [41, p. 173], there exist two standard Brownian motions (B

t ) t≥0 and (B

t ) t≥0 such that

(3.3) X t = B (1) Vt(p)
and Y t = B

(2)

Vt(-q) .
By a theorem of Knight [START_REF] Knight | A reduction of continuous square-integrable martingales to Brownian motion[END_REF], see [41, p. 175], the processes (B

t ) t≥0 and (B

t ) t≥0 are independent in our case.

By the scaling property of Brownian motion, for i = 1, 2 and α = p, resp. α = -q, log B

(i) Vt(α) /√ V t (α) √ t = log |B 1 | √ t → 0 in law.
In the following computations we use frequently the following simple fact.

(3.4)

If A t → A and C t → 0 in law then (A t , C t ) → (A, 0) in law, as t → ∞ . Thus (3.5) lim t→∞ 1 √ t ( log |X t | -p a t , log |Y t | , Z t -a t ) = lim t→∞ 1 √ t ( log √ V t (p) -p a t , log √ V t (-q) , W t ) in law.
Case (i): a > 0. First observe that for all α > 0 and β ∈ R,

lim t→∞ ∫ t 0 e -αs+βWs ds = ∫ ∞ 0
e -αs+βWs ds ∈ (0, +∞) almost surely, since lim s→∞ log(e -αs+βWs )/s = -α < 0. Therefore

(3.6) 1 √ t log √ V t (-α) → 0 in law, when α > 0.
Using (3.4), we get (in law) that

lim t→∞ 1 √ t ( log √ V t (p) -p a t , log √ V t (-q) , W t ) = lim t→∞ 1 √ t ( log √ V t (p) -p a t , 0 , W t ) = lim t→∞ 1 √ t ( 1 2 log ( ∫ t 0 e 2p(a(s-t)+W s) ds ) , 0 , W t ) = lim t→∞ 1 √ t ( 1 2 log ( e 2pW t ∫ t 0 e 2p(a(s-t)+Ws-Wt) ds ) , 0 , W t ) Since (W t-s ) s≤t = (W t -W s ) s≤t in law, lim t→∞ 1 √ t log ( ∫ t 0 e 2p(a(s-t)+W s-Wt) ds ) = lim t→∞ 1 √ t log ( ∫ t 0 e -2p(a(t-s)-W t-s ) ds ) = lim t→∞ 1 √ t log ( ∫ t 0 e -2p(a(s)-Ws) ds ) = 0
and using (3.4) once more, we get that

lim t→∞ 1 √ t ( log |X t | -p a t , log |Y t | , Z t -a t ) = lim t→∞ 1 √ t ( p W t , 0 , W t ) = (p N , 0 , N ) in law.
Case (ii): a < 0. This is obtained from Case (i) by exchanging the roles of the x-and y-coordinates.

Case (iii): a = 0. We take up (3.5) and continue to compute, with all identities holding in law

lim t→∞ 1 √ t ( log √ V t (p) , log √ V t (-q) , W t ) = lim t→∞ 1 2 √ t ( log
( ∫ t 0 e 2pWs ds ) , log
( ∫ t 0 e -2qWs ds ) , 2W t ) = lim t→∞ 1 2 √ t ( log ( t ∫ 1 0 e 2pWst ds ) , log ( t ∫ 1 0 e -2qWst ds ) , 2W t ) = lim t→∞ 1 2 √ t ( log ( t ∫ 1 0 e 2p √ t Ws ds ) , log ( t ∫ 1 0 e -2q √ t Ws ds ) , 2 √ t W 1
)

[setting τ = 2 √ t ] = lim τ →∞ ( log ( ∫ 1 0 e τ pWs ds ) 1/τ , log ( ∫ 1 0 e -τ qW s ds ) 1/τ , W 1 ) since (W st ) 0≤s≤1 = ( √ t W s ) 0≤s≤1 in law. Now recall that the L τ -norm on C([0 , 1
]) converges to the L ∞ -norm as τ → ∞. We apply this to the functions s → e pWs and s → e -qWs , respectively, and then take logarithms. Thus, almost surely log

( ∫ 1 0 e τ pWs ds ) 1/τ → p • max{W s : 0 ≤ s ≤ 1} and log ( ∫ 1 0 e -τ qWs ds ) 1/τ → -q • min{W s : 0 ≤ s ≤ 1}.
This leads to statement (iii).

Next, with N , M and M as above, we deduce the following central limit theorem for the distance of Brownian motion to the origin.

(3.7) Theorem. If a ̸ = 0 then d Sol (Z t , o) -|a| t √ t → N in law, as t → ∞ . If a = 0 then d Sol (Z t , o) √ t → 2(M -M ) -|N | in law, as t → ∞ .
Proof. We start with a > 0. Combining Theorem 3.2(i) with Proposition 2.11(iv), we obtain in law

lim t→∞ d Sol (Z t , o) -a t √ t ≤ lim t→∞ log |Xt| p + log |Yt| q + log |Yt| q + log |Xt| p -Z t -at √ t = lim t→∞ log |X t| p -at + ( log |Xt| p -at ) - ( Z t -at ) √ t = lim t→∞ log |Xt| p -at √ t (since Pr[log |X t | < 0] → 0) = lim t→∞ log |Xt| p -at √ t = N .
On the other hand

lim t→∞ d Sol (Z t , o) -a t √ t ≥ lim t→∞ |Z t | -at √ t = N in law.
When a < 0, the result follows once more by exchanging the roles of the x-and ycoordinates.

Now consider the case when a = 0. Combining Theorem 3.2(iii) with Proposition 2.11(iv), we obtain in law

lim t→∞ d Sol (Z t , o) √ t ≤ M -M + min { M + N -M , M + M -N } = 2 ( M -M ) -|N |.
This upper bound together with the fact that d Sol (Z t , o) → ∞ almost surely yields that

log d Sol (Z t , o) √ t → 0
in law (and in fact almost surely). We can combine this with Theorem 3.2(iii) and Proposition 2.11(i), and get the required lower bound in the case a = 0.

Compare this with the analogous result of [START_REF] Bertacchi | Random walks on Diestel-Leader graphs[END_REF] for simple random walk with drift on Diestel-Leader graphs. We next observe the following (denoting expectation by E).

(3.8) Lemma. Let U n = max { d Sol (Z n , Z n+t ) : 0 ≤ t ≤ 1 } . Then E(U n ) < ∞ , and
lim n→∞ 1 log n max { d Sol (Z n , Z n+t ) : 0 ≤ t ≤ 1 } = 0 almost surely.
Proof. The random variables U n , n ≥ 0, are i.i.d. Let

X * = max{|X t | : 0 ≤ t ≤ 1} , Y * = max{|Y t | : 0 ≤ t ≤ 1} , Z * = max{|Z t | : 0 ≤ t ≤ 1} .
Then by Proposition 2.11(iii),

U 0 ≤ c + ( 1 + log(1 + |X * |) + log(1 + |Y * |) + |Z * |
) .

Observe that by the Burkholder-Davis-Gundy inequality [41, pag. 161], we have for every r > 2 that

E(X r * ) ≤ c 1 E ( (∫ 1 0 e 2pZ s ds ) r/2 ) ≤ c 1 ∫ 1 0 E ( e prZ s ) ds < ∞ ,
where c 1 > 0. The same holds for Y * . For Z * , observe that by duality

Pr[Z * > z] = Pr[max{W t -at : 0 ≤ t ≤ 1} > z or max{-W t + at : 0 ≤ t ≤ 1} > z] ≤ 4Pr[W 1 > z -|a| ] . Thus E ( e rZ * ) = ∫ ∞ 0 Pr[e rZ * > u] du ≤ 4 ∫ ∞ 0 Pr[e r(W 1 +|a|) > u] du = E ( e r(W 1 +|a|) ) < ∞ .
We find that for all r > 0,

E ( e rUn ) = E ( e rU 0 ) ≤ e cr E ( (1 + X * ) cr (1 + Y * ) cr e crZ * ) < ∞ .
By the law of large numbers, e rU n /n → 0 almost surely, whence lim sup n→∞ U n / log n < 1/r almost surely, for all r > 0.

Given the (left) action on Sol(p, q) on itself by isometries and the group-invariance of our Laplacian (Lemma 2.8), we get that along any time interval [s , t], the increment of our Brownian motion Z t = (X t , Y t , Z t ) of (3.1) satisfies (3.9) Z -1 s Z t = Z t-s in law, and for an arbirary number of time intervals which do not overlap (i.e., they meet at most at the endpoints), the associated increments are independent. We now also get the rate of escape for our Brownian motion with drift. Proof. In view of Lemma 3.8 and the spatial homogeneity (3.9), the subadditive ergodic theorem of Kingman [START_REF] Kingman | Subadditive ergodic theory[END_REF] implies that d Sol (Z t , o)/t converges almost surely to a constant. (Compare with Derriennic [START_REF] Derriennic | Quelques applications du théorème ergodique sous-additif[END_REF] for the case of discrete time.) Theorem 3.7 implies that the limit is a in probability, whence also with probability 1.

Convergence to the boundary at infinity, and the deviation from the limit geodesic

The natural geometric compactification of the hyperbolic plane, in the unit disk model, is just the closed (Euclidean) disk. In the upper half plane model H(p), the boundary at infinity ϑH(p) of the compactification H(p) is obtained by adding the bottom line ϑ * H(p) = R and the point at infinity, denoted here by ϖ p . In the logarithmic model, convergence to the boundary is as follows: we have that (x, z) → ξ ∈ ϑ * H(p) when z → -∞ and x → ξ, and (x, z) → ϖ p if |x| + e z → ∞. Now Sol(p, q) embeds into H(p) × H(q) via (1.6). Therefore the most natural geometric compactification Sol(p, q) of Sol(p, q) is its closure in the compact bidisk H(p) × H(q). ("Bidisk" because when we use the unit disk model of hyperbolic plane, this is just the direct product of two closed unit disks.) We assemble a brief description of convergence to the boundary in the next lemma; no proof is required. We underline once more the analogy with Diestel-Leader graphs [START_REF] Woess | Lamplighters, Diestel-Leader graphs, random walks, and harmonic functions. Combinatorics[END_REF] and treebolic space [START_REF] Bendikov | Positive harmonic functions and Brownian motion on treebolic spaces[END_REF]. As pointed out in the Introduction, the boundary at infinity is topologically a filled number "8", that is, two closed disks glued together at a single point. This sheds some light on the observations made by Lyons and Sullivan [START_REF] Lyons | Function theory, random paths and covering spaces[END_REF].

(p, q) of Sol(p, q) is ( ϑ * H(p) R ×{ϖ q } ) ∪ ( {ϖ p } × ϑ * H(q) R ) ∪ ( H(p) × {ϖ q } ) ∪ ( {ϖ p } × H(q) ) ∪ { (ϖ p , ϖ q ) } .
Convergence to the boundary is as follows. In general,

x = (x, y, z) → (ξ, η) ∈ ϑSol(p, q) , if (x, z) → ξ in H(p) and (y, -z) → η in H(q).
This means that

x = (x, y, z) → (ξ, ϖ q ) ∈ ϑ * H(p) × {ϖ q } , if x → ξ and z → -∞ , x = (x, y, z) → (ϖ p , η) ∈ {ϖ p } × ϑ * H(q) , if y → η and z → ∞ , x = (x, y, z) → ( (x 0 , z 0 ), ϖ q ) ∈ H(p) × {ϖ q } , if x → x 0 , z → z 0 and |y| → ∞ , x = (x, y, z) → ( ϖ p , (y 0 , -z 0 ) ) ∈ {ϖ p } × H(q) , if y → y 0 , z → -z 0 and |x| → ∞ , x = (x, y, z) → (ϖ p , ϖ q ) , if |x| + e z → ∞ and |y| + e -z → ∞ in R .
A geodesic ray is a continuous mapping γ : [0 , ∞) → Sol (or to any of our other spaces) such that d ( γ(t) , γ(s)

)
= |t -s| for all s, t. Its starting point is γ(0). For any (x 0 , z 0 ) ∈ H(p) and ξ ∈ ϑH(p), there is a unique geodesic ray ( x(t), z(t) ) that starts at (x 0 , z 0 ) and converges to ξ. In the case when ξ = ϖ p then this is the upwards going vertical half-line t → (x 0 , z 0 + t) in H(p).

For x = (x 0 , y 0 , z 0 ) ∈ Sol(p, q) and a boundary point (ϖ p , η) ∈ {ϖ p } × ϑ * H(q) of Sol(p, q), we can consider the (unique) upwards geodesic ray starting at x given by γ η x (t) = ( x 0 , y(t), z(t)

)

, where ( y(t), -z(t) ) t≥0 is the geodesic ray from (y 0 , -z 0 ) to η in H(q). Analogously, for a boundary point (ξ, ϖ q ) ∈ ϑ * H(p) × {ϖ q }, we have the (unique) downwards geodesic ray starting at x given by γ ξ x (t) = ( x(t), y 0 , z(t) )

, where

( x(t), z(t) ) t≥0
is the geodesic ray from (x 0 , z 0 ) to ξ in H(p). All those geodesics converge to their defining boundary points, as t → ∞, and any two geodescis that converge to the same boundary point are at bounded Hausdorff distance. This is true because it holds in the hyperbolic plane.

In the first of the above two cases, it will be most convenient to use the initial point x = (0, η, 0), and omit the index x in that case. Thus, we can parametrise by z ≥ 0 and get γ η (z) = (0, η, z). Analogously, in the second case, we use the standard initial point x = (ξ, 0, 0) and get the corresponding geodesic ray γ ξ (z) = (ξ, 0, -z), again parametrised by z ≥ 0. We call these the (upwards, resp. downwards) vertical geodesic rays. We remark that there is no geodesic ray in Sol from any starting point that converges to (ϖ p , ϖ q ).

Compare with [START_REF] Eskin | Quasi-isometries and rigidity of solvable groups[END_REF], [START_REF] Eskin | Coarse differentiation of quasi-isometries I: spaces non quasiisometric to Cayley graphs[END_REF] for further details on the geometry. Let us return to our Brownian motion Z t = (X t , Y t , Z t ) of (3.1). (2) s almost surely.

(4.2) Proposition. (i) If a > 0 then lim t→∞ Y t = Y ∞ = ∫ ∞ 0 e -qZs dW
That is, Z t → (ϖ p , Y ∞ ) ∈ ϑSol(p, q) almost surely in the topology of Sol(p, q). (1) s almost surely.

(ii) If a < 0 then lim t→∞ X t = X ∞ = ∫ ∞ 0 e pZ s dW
That is, Z t → (X ∞ , ϖ q ) almost surely in the topology of Sol(p, q).

In both cases (i) and (ii), the respective limiting random variable is a.s. finite.

(iii) If a = 0 then

|X t | + e Zt → ∞ and |Y t | + e -Zt → ∞ almost surely.
That is, Z t → (ϖ p , ϖ q ) almost surely in the topology of Sol(p, q).

Proof. (i) and (ii) are immediate from the representation (3.1) via [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF]Prop. 1.26]. For (iii), consider X t = (X t , Z t ) as a process on the affine group Aff(p) of (2.4). It also satisfies (3.9) and (3.8). We consider our process at discrete times:

(4.3) X n = (X n , Z n ) = X 1 • (X -1 1 X 2 ) • • • (X -1 n-1 X n )
is a right random walk on Aff(p). We can apply a result of Brofferio [START_REF] Brofferio | How a centred random walk on the affine group goes to infinity[END_REF]. In the notation of [START_REF] Brofferio | How a centred random walk on the affine group goes to infinity[END_REF], A 1 = e pZ 1 and B 1 = X 1 . Since the expectation of log A 1 is 0, and all moment conditions of [10, Thm. 1] are satisfied, X n → ϖ p almost surely in H(p), as n → ∞ in Z. By Lemma (3.8), also X t → ϖ p almost surely, as t → ∞ in R.

In the same way, (Y t , -Z t ) → ϖ q almost surely, as t → ∞ in R. Statement (iii) follows.

(4.4) Remark. When a > 0, the distribution of the random variable Y ∞ is well understood; compare with [START_REF] Baldi | Non-symmetric hitting distributions on the hyperbolic half-plane and subordinated perpetuities[END_REF]. Indeed, by (3.3),

Y ∞ = B (2) V ∞(-q) in law = √ V ∞ (-q) B (2) 1 ,
and the latter is a product of two independent random variables. Now

4q 2 V ∞ (-q) in law = 4q 2 ∫ ∞ 0 e -2qa t-W 4q 2 t dt in law = ∫ ∞ 0 e -a 2q
s-Ws ds is Dufresne's integral with c = a/2q. By the formula of Dufresne [START_REF] Dufresne | The distribution of a perpetuity, with applications to risk theory and pension funding[END_REF] -see also

Bailleul [3] -the density of √ V ∞ (-q) is computed as u → 2 (2q 2 ) κ/2 Γ(κ/2) × e -1/(2q 2 u 2 ) u 1+κ , u > 0 , where κ = κ(q) = 2|a|/q .
The density

f ∞ = f (a,q) ∞
of Y ∞ can now be obtained as the multiplicative convolution of this function with the standard Gaussian density:

f ∞ (y) = ( (κ -1)/2 1/2 ) q 2 (1 + q 2 y 2 ) (κ+1)/2 , y ∈ R .
The same applies of course to X ∞ when a < 0, with p in the place of q.

Thus, when a > 0, we have the vertical limit geodesic γ Y∞ to whose limit point our Brownian motion converges, and when a < 0 we have to replace this by γ X∞ . In order to simplify notation, we just write γ ∞ for the respective limit geodesic in each of those cases.

We now prove that when a ̸ = 0, the convergence of Z t to its boundary limit is very straight, in the sense that its deviation from γ ∞ is of the order of log t.

( Z t , γ ∞ ) = inf { d ( Z t , γ ∞ (u) ) : u ≥ 0 } .
Proof. Once more, it is sufficient to consider only the case a > 0.

For each t ≥ 0, the point (0, Y ∞ , Z t ) lies on the geodesic γ ∞ = γ Y∞ . We shall show that for integer n,

(4.6) lim sup n→∞ 1 log n d ( Z n , (0, Y ∞ , Z n ) ) ≤ 2 a almost surely,
Together with Lemma 3.8, this will yield the result. The metric d Sol is invariant under the left action of the group Sol(p, q). Using the product formula (1.3) and subsequently Proposition 2.11(iii), we find

d Sol ( Z t , (0, Y ∞ , Z t ) ) = d Sol ( (e -pZ t X t , e qZt (Y t -Y ∞ ), 0), o ) ≤ c + 2 p log ( 1 + e -pZt |X t | ) + 2 q log ( 1 + e qZt |Y t -Y ∞ | ) .
We have

e qZ t (Y ∞ -Y t ) = e q(W t+at) ∫ ∞ t e -q(W s+as) dW (2) s = ∫ ∞ t e -q((Ws-Wt)+a(s-t)) dW (2) s in law = ∫ ∞ 0 e -q(Ws+as) dW (2) s = Y ∞ . By Remark 4.4, Pr[ |Y ∞ | > y ] ≍ y -κ(q) as y → ∞ . Now take δ > 1/κ(q). Then ∞ ∑ n=2 Pr [ log ( 1 + e qZn |Y n -Y ∞ | ) > δ log n ] = ∞ ∑ n=2 Pr [ log ( 1 + |Y ∞ | ) > δ log n ] = ∞ ∑ n=2 Pr [ |Y ∞ | > n δ -1] ≍ ∞ ∑ n=2 (n δ -1) -κ(q) < +∞ .
Thus by the Borel-Cantelli Lemma lim sup

n→∞ 1 log n log ( 1 + e qZn |Y n -Y ∞ | ) ≤ q 2a almost surely.
We now consider the first coordinate. For fixed t observe that e -pZ t X t = e -p(W t+at) ∫ t 0 e p(W s+as) dW (1) s = ∫ t 0 e -p((Wt-Ws)+a(t-s)) dW (1) s

in law = ∫ t 0 e -p(Ws+as) dW (1) s =: X t ,
and X ∞ = lim t→∞ X t exists almost surely. As above, one finds that

e pZt ( X ∞ -X t ) = ∫ ∞ t e -p((Ws-Wt)+a(s-t)) dW (1) s =: X t,∞ ,
where X t,∞ is independent from (Z s ) 0≤s≤t and has the same law as X ∞ . Thus, for some constant C > 0 and for any x > 0,

Pr [ e -pZt |X t | > x ] = Pr [ | X n | > x , | X ∞ -X n | ≤ x 2 ] + Pr [ | X n | > x , | X ∞ -X n | > x 2 ] ≤ Pr [ | X ∞ | > x 2 ] + Pr [ e -pZn |X n,∞ | > x 2 ] ≤ C ( x 2 ) κ(p) + E ( Pr [ |X n,∞ | > x 2 e pZn Z n ] ) ≤ C ( x 2
) -κ(p) + C E ( ( x 2 e pZn ) -κ(p) ) ≤ C ( x 2
) -κ(p) + C ( x 2
) -κ(p) E ( e -κ(p)pZ 1 ) = 1 n = 2C ( x 2
) -κ(p)
Proceeding as above, the Borel-Cantelli Lemma implies that lim sup

n→∞ 1 log n log ( 1 + e -pZn |X n | ) ≤ p 2a almost surely.
As mentioned in the introduction, the analogous result for Brownian motion with nonzero vertical drift on the hyperbolic plane is a special case of the result stated in [1, Théorème 7.3]. The latter is based on subtle potential theoretic arguments. By combining this with properties of the metric of Sol(p, q), one can also deduce the qualitative statement of Theorem 4.5. Our proof uses much more direct computations and leads to the explicit bound 2/|a| for the log-deviation.

Elements of potential theory

If L is any of our different Laplacians on Sol, H, or R, and λ ∈ R, then we denote by H(L, λ) the space of all functions h on our space which satisfy Lh = λ • h. The positive cone H + (L, λ) contains non-zero functions if and only if λ ≥ λ min (L), the bottom of the positive spectrum. Below we shall clarify what the values of λ min are in each of our cases. In any case, λ min ≤ 0, since the space of harmonic functions H(L) = H(L, 0) contains all constant functions.

By the minimum principle, every non-zero function in H + (L, λ) must be strictly positive in each point.

A function

h in H + (L, λ) is called minimal if h(0) = 1 and whenever h ≥ f ∈ H + (L, λ) then f /h is constant.
A basic fact in classical potential theory of Riemannian manifolds says that every function in H + (L, λ) can be expressed uniquely as an integral over the minimal harmonic functions with respect to a finite Borel measure on the latter set.

shall specify this in more detail in our cases below. Let us now recall what happens in the case of the standard Laplacian We have λ min (L H -1/2 ) = -1/8, and the minimal elements in H + (L H -1/2 , λ) are the functions (5.2) λ) , where ξ ∈ ϑH and α(λ

L H -1/2 = 1 2 ( e 2z ∂ 2 ∂x 2 + ∂ 2 ∂z 2 - ∂ ∂z ) on standard hyperbolic plane H = H(1) = {x + i e z : x, z ∈ R} in
P (•, ξ) α(
) = 1 + √ 1 + 8λ 2 .
Next, let us turn our attention to H + (L

H(p) a , λ). (5.3) Lemma. A function f on H(p) is in H ( L H(p) a
, λ ) if and only if the function on

H(1) given by ( e (a+p/2)z f ) • θ is in H ( L H(1) -1/2 , 8λ+4a 2 -p 2 8p 2
)

, where θ(x, z) = (x/p, z/p). In particular,

λ min (L H(p) a ) = -a 2 /2 ,
and for λ ≥ -a 2 /2, the minimal elements in

H + ( L H(p) a , λ ) are the functions P p,a,λ ( (x, z), ϖ p ) = e αz and P p,a,λ ( (x, z), ξ ) = e αz ( (ξ 2 + 1) (ξ -p x) 2 + e 2pz ) β , ξ ∈ ϑ * H(p) ≡ R , where α = α(λ, a) = √ a 2 + 2λ -a and β = β(λ, a, p) = 1 2 + √ a 2 +2λ p .
Proof. First of all, it is a straightforward computation that

( L H(p) a f ) • θ = p 2 L H(1) a/p (f • θ) . Therefore f is in H(L H(p) a , λ) if and only if f = f • θ is in H(L H(1)
a/p , λ/p 2 ) .

For the moment, set ā = a/p and λ = λ/p 2 . Then we compute

L H(1) -1/2 ( e (ā+1/2)z f ) = e (ā+1/2)z ( L H(1) ā f + 4ā 2 -1 8 f ) . Therefore f is in H(L H(1) ā , λ) if and only if e (ā+1/2)z f is in H(L H (1)
-1/2 , λ + 4ā 2 -1 8 ). Combining these computations, the statements follow.

Thus, every function

h ∈ H + (L H(p) a
, λ) has a unique integral representation (5.4) h(x, z) =

∫ ϑH P p,a,λ ( (x, z), ξ ) dν(ξ) ,
where ν is a (finite, positive) Borel measure on ϑH . (This includes ξ = ϖ p .)

(5.5) Remark. When λ = 0, that is, when we consider ordinary harmonic functions, we see that the constant function 1 is minimal harmonic if and only if a ≥ 0. This can be stated also by saying that L H(p) a has the (weak) Liouville property, i.e., all bounded harmonic functions are constant, precisely when a ≥ 0.

Everything that we have said so far in this section is very well known; see e.g. Helgason [START_REF] Helgason | Groups and Geometric Analysis[END_REF], or many other sources.

Let us now turn our attention to Sol. The following is immediate.

(5.6) Lemma.

If the function h 1 on H(p) is such that h = h 1 • π 1 is minimal in H + ( L Sol(p,q) a , λ ) , then h 1 is also minimal in H + ( L H(p) a , λ
) . In the same way, if the function h 2 on H(q) is such that h

= h 2 • π 2 is minimal in H + ( L Sol(p,q) a , λ ) , then h 2 is also minimal in H + ( L H(q)
-a , λ

) .

We now need a part of the Martin boundary theory for elliptic operators on manifolds. The reader is referred to Ancona [START_REF] Ancona | Théorie du potentiel sur les graphes et les variétés[END_REF] and Taylor [START_REF] Taylor | The Martin compactification associated with a second order strictly elliptic partial differential operator on a manifold M[END_REF] for the necessary backgroud material. See also [24, Chapter VI]. In the following propositions, we subsume the necessary material without all proofs.

(5.7) Proposition. The Markov semigroup H t = H a t = exp(tL a ), t > 0, admits a symmetric, bounded kernel h t (x, z) = h a t (x, z) with respect to the measure m a of Lemma 2.7(a), such that

H t f (x) = ∫ Sol h t (x, z)f (z) dm a (z) .
For each z = (x, y, z) ∈ Sol, the function h t (•, z) is in C 2 (Sol). Furthermore, its kernel with respect to the volume element dz of the Sol-manifold, p t (x, z) = h t (x, z) e (a+p-q)z , is stochastic and invariant under the action of the group Sol(p, q).

(5.8) Proposition. The associated Green kernel

g a (x, z|λ) = g(x, z|λ) = ∫ ∞ 0 e -λt p t (x, z) dt (x, z ∈ Sol , x ̸ = z)
is strictly positive and finite for each λ ≥ λ min (L a ).

We remark that finiteness at λ = λ min (L a ) can be seen in different ways.

(1) It is well known that the Green kernel of L H(p) a exists (i.e., it is finite off the diagonal) at λ = λ min , see e.g. Korányi and Taylor [START_REF] Korányi | Fine convergence and admissible convergence for symmetric spaces of rank one[END_REF]Proposition 1.2]. Via the projection π 1 , that kernel is obtained by integrating g a (x, z|λ min ) over the y-variable of z. Thus, g a (x, z|λ min ) must also be finite.

(2) It is a general fact known from potential theory that in the case when g a (•, •|λ min ) ≡ ∞, the cone of positive eigenfunctions must be one-dimensional. This is not the case here, since we already have distinct positive eigenfunctions with value 1 at the origin on H(p), as one can see from Lemma 5.3 combined with (5.1) and (5.2).

(3) One may also invoke an observation Guivarc'h [23, p. 85], formulated for random walks. It implies, in particular, that on a connected Lie group like ours the Green kernel at λ min can be infinite only when that group carries a recurrent random walk, which by Baldi [START_REF] Bailleul | Une preuve simple d'un résultat de Dufresne[END_REF] can only happen the group contains a co-compact lattice which is Abelian of rank 1 or 2.

(5.9) Proposition. For each d > 0 and λ ≥ λ min , the Green kernel satisfies the Harnack inequality g(x,

z ′ |λ) g(x, z|λ) ≤ C d (λ) and g(z ′ , x|λ) g(z, x|λ) ≤ C d (λ) , whenever d(z, z ′ ) ≤ d and min{d(z, x), d(z ′ , x)} ≥ 10(d + 1), where C d (λ) > 1 is such that C d (λ) → 1 when d → 0. Furthermore, every function h in H + ( L Sol(p,q) a , λ ) satisfies h(z ′ ) h(z) ≤ C d (λ) for all z, z ′ ∈ Sol with d(z, z ′ ) ≤ d.
Proof (outline). In the case L a is the Laplace-Beltrami operator of Sol(p, q) , one can apply well-known Harnack inequalities of Li and Yau, see [START_REF] Li | On the parabolic kernel of the Schrödinger operator[END_REF], because the Riemannian structure is invariant under a group action and thus the Ricci curvature is bounded below.

For arbitrary values of a, our operator is obtained by adding to the Laplace-Beltrami operator a multiple of ∂ ∂z , which leads just to conjugating our functions with an exponential in z, compare with the proof of Lemma 5.3. Thus, the inequalities hold with any drift term a.

The Martin kernel is (5.10) k a (x, z|λ) = k(x, z|λ) = g a (x, z|λ) g a (0, z|λ) , z ̸ = 0, x .
The Martin compactification is the smallest compactification of the underlying space Sol (i.e., a Hausdorff space into which Sol embeds homeomorphically and densely) such that each function k a (x, •|λ) has a continuous extension in the second variable. The Martin boundary M(λ) = M(L a , λ) is the ideal boundary added to the space in that compactification. The extended kernel is also denoted k a (•, •|λ)

(5.11) Proposition. Every minimal eigenfunction h in H + ( L Sol(p,q) a , λ ) , λ ≥ λ min , is of the form h(x) = k a (x, ζ|λ), where ζ ∈ M(λ) .
That is, there is a (suitable) sequence

(z n ) in Sol with d(0, z n ) → ∞, such that h(x) = lim n→∞ k(x, z n |λ) .
The minimal Martin boundary M min (λ) = M min (L a , λ) consists of all ζ ∈ M(λ) for which k a (•, ζ|λ) is minimal. It is a Borel subset of M(λ). The Poisson-Martin representation theorem says the following.

(5.12) Proposition. For every function h ∈ H + ( L

Sol(p,q) a , λ ) , there is a unique Borel measure ν h on M min (λ) such that h(x) = ∫ M min (λ) k(x, •|λ) dν h for every x ∈ Sol.
All this is of course true for more general manifolds and elliptic operators; see [START_REF] Taylor | The Martin compactification associated with a second order strictly elliptic partial differential operator on a manifold M[END_REF]. While we are not able to determine the whole Martin compactification, that is, the directions of convergence of the Martin kernels, we shall determine precisely the minimal positive λ-eigenfunctions for each λ ≥ λ min .

Positive harmonic functions on Sol(p, q)

We now show that every positive eigenfunction of our Laplacian on Sol(p, q) splits as a sum of two eigenfunctions that live on the two respective hyperbolic planes which make up Sol, and we determine precisely all minimal positive eigenfunctions. The first step is the following. , λ ) be minimal, where λ ≥ -a 2 /2. Then h(x, y, z) = h 1 (x, z), where h 1 is minimal in H + ( L Sol(p,q) a , λ ) , or h(x, y, z) = h 2 (y, -z), where h 2 is minimal in H + ( L

H(q)

-a , λ

) .

Proof. Let h be a minimal eigenfunction in

H + ( L a , λ ) . Then h = lim n→∞ k(•, z n |λ). Write z n = (x n , y n , z n ). Claim. (a) If inf n z n > -∞ , then for each a ∈ R, h(x + a, y, z) = h(x, y, z) for all (x, y, z) ∈ Sol . (b) If sup n z n < +∞ , then for each b ∈ R, h(x, y + b, z) = h(x, y, z) for all (x, y, z) ∈ Sol .
To prove part (a) of this claim, let a ∈ R and consider the group element

g a = (a, 0, 0) = ( 1 a 0 0 1 0 0 0 1 ) ∈ Sol(p, q) .
We abbreviate τ a = τ ga . Let x = (x, y, z) ∈ Sol. Then, by (2.8), τ a h(x) = h(g a x) = h(x + a, y, z), and Lemma 2.9 tells us that τ a h is in H + ( L a , λ ) . Now by (1.5)

d(g a z n , z n ) = d Sol ( (x n + a, y n , z n ), (x n , y n , z n ) = d H(p) ( (x n + a, z n ), (x n , z n ) ) .
Elementary properties of the hyperbolic metric imply that

d H(p) ( (x n + a, z n ), (x n , z n ) ) = d H(p) ( (a, z n ), (0, z n ) ) ≤ d H(p) ( (a, c), (0, c) ) = d a ,
where c = inf n z n . Let C da (λ) be the corresponding Harnack constant in Lemma 5.9. Then, using that g(

•, •|λ) is Sol(p, q)-invariant, k(g a x , z n |λ) = g(g a x , z n |λ) g(g a x , g a z n |λ) g(g a x , g a z n |λ) g(0 , z n |λ) ≤ C da k(x , z n |λ) .
Letting n → ∞, we obtain

τ a h(x) = h(g a x) ≤ C da h(x) for all x ∈ Sol . Now minimality of h implies that the function τ a h/h is constant. For x = (x, y, z) ∈ Sol, d Sol (g a x, x) = d H(p) ( (x + a, z), (x, a) ) → 0 , if z → +∞ .
Therefore the second statement in Lemma 5.9 implies that h(g a x)/h(x) → 1 as z → +∞ , and we conclude that τ a h/h ≡ 1 . This proves statement (a) of the claim, and statement (b) follows by exchanging the roles of the x-and y-coordinates and changing the sign of z. Now (z n ) must have a subsequence which converges to a limit in [-∞ , +∞]. We may assume without loss of generality that (z n ) itself converges.

Case 1. z n → ∞ . Then we can apply part (a) of the Claim, and conclude that h depends only on (y, z). By Lemma 5.6, there is a function h 2 on H(q) which is minimal in

H + ( L H(q) -a , λ )
, such that h(x, y, z) = h 2 (y, -z) for all x = (x, y, z) ∈ Sol. , λ ) , so that -by Lemma 5.3 -we can only have the "+" sign, that is, α = α(λ, a). We shall see below that the corresponding function can really be a minimal λ-eigenfunction on Sol only when λ = λ min . -a , λ

) such that for all x = (x, y, z) ∈ Sol(p, q), h(x, y, z) = h 1 (x, z) + h 2 (y, -z).

Proof. We see from Theorem 6.1 that the set of all minimal λ-eigenfunctions on Sol(p, q) is contained in the union of the sets of minimal λ-eigenfunctions on H(p) and H(q), with a change of the sign of z for the latter, according to the above cases. Thus, taking into account Remark 6.2, M min (λ) can be parametrised by a subset of the disjoint union ϑH(p) ∪ ϑ * H(q) ∼ = ( ϑH(p) × {ϖ q }

) ∪ ( {ϖ p } × ϑ * H(q)

) ⊂ Sol(p, q), or in other terms, of the "8"-shaped outer part of the geometric boundary of Sol (without the interiors of the two disks). By Proposition 5.12, for every function h ∈ H + ( L Sol(p,q) a , λ ) , there is a Borel measure ν = ν h on M min (λ) that yields the integral representation of h. Now let ν 1 be the restriction of ν to ϑH(p) and ν 2 the restriction to ϑ * H(q). Then we get for every x = (x, y, z) ∈ Sol P q,-a,λ ( (y, -z), η ) dν 2 (η) = h 1 (x, z)+h 2 (y, -z) , as proposed.

(6.4) Corollary. The Laplacian L Sol(p,q) a has the (weak) Liouville property, i.e., all bounded harmonic functions on Sol are constant, if and only if the rate of escape a vanishes.

Proof. If a = 0, then all bounded harmonic functions are constant by Corollary 6.3 and Remark 5.5. Conversely, if a ̸ = 0, then again by Remark 5.5, one of L H(p) a and L H(q) -a has non-constant bounded harmonic functions, and they lift to harmonic functions on Sol(p, q). The last corollary, which was obtained in a very concrete, case-specific way, should be compared with the theorem of Karlsson and Ledrappier [START_REF] Karlsson | Propriété de Liouville et vitesse de fuite du mouvement Brownien[END_REF], which says that (under very general conditions) the weak Liouville property holds if and only if the rate of escape of Brownian motion is 0.

When a ̸ = 0, we have the following.

(6.5) Corollary. (i) If a > 0 then every bounded harmonic function for L Sol(p,q) a has the form (x, y, z) → h 2 (y, -z) , where h 2 is a bounded harmonic function for L H(q) -a . (ii) If a < 0 then every bounded harmonic function for L Sol(p,q) a has the form (x, y, z) → h 1 (x, z) , where h 1 is a bounded harmonic function for L H(p) a .

Proof. Let h be a bounded harmonic function on Sol(p, q). We may assume without loss of generality that it is non-negative. We decompose h(x, y, z) = h 1 (x, z) + h 2 (y, -z) according to Corollary 6.3. Then both h 1 and h 2 are bounded harmonic. When a > 0, Remark 5.5 tells us that h 1 must be constant, so that we can "incorporate" it into h 2 . Analogously, when a < 0, the function h 2 must be constant. , λ ) , λ ≥ λ min , are precisely the functions (x, y, z) → P p,a,λ ( (x, z), ξ) and (x, y, z) → P q,-a,λ ( (y, -z), η) , ξ, η ∈ R , and in addition, when λ = λ min , the function (x, y, z) → e -a z .

) , which is not the case by Lemma 5.3. Analogously, when λ > λ min , the function (x, y, z) → P q,-a,λ ( (y, -z), ϖ q )e -α(λ,-a)z cannot be minimal in H + ( L Sol(p,q) a , λ ) .

Finally, consider the case λ = λ min and the function (x, y, z) → e -az in H + ( L Sol(p,q) a , λ ) . We use a well-known trick, conjugating our operator with this exponential: suppose that e -az ≥ h(x, y, z), where h ∈ H + ( L Sol(p,q) a , λ ) . Then a straightforward computation shows that the function h(x, y, z) = e az h(x, y, z) is in H + ( L Sol(p,q) 0 , 0 ) , that is, it is bounded harmonic, and the new rate of escape is 0. By Corollary 6.4, h is constant. This proves minimality of (x, y, z) → e -az in H + ( L Sol(p,q) a , λ ) .

  , y, z ∈ R .

( b )

 b The Laplacian L H(p) a on H(p) is reversible with respect to the measure e 2az dx dz .

( 3 .

 3 10) Corollary. For any value of a, lim t→∞ d Sol (Z t , o) t = |a| almost surely.

( 4 . 1 )

 41 Lemma. The boundary at infinity ϑSol

( 4 . 5 )

 45 Theorem. If a ̸ = 0 then Brownian motion on Sol(p, q) satisfies

  the logarithmic model. The minimal harmonic functions are the Poisson kernels, which are parametrised by the (hyperbolic) boundary ϑH = R ∪ {ϖ}. (Recall that ϖ = ϖ 1 is the point at infinity.) In the logarithmic model the kernels are (5.1) P ( (x, z), ϖ ) = e z and P ( (x, z), ξ ) = (ξ 2 + 1)e z (ξx) 2 + e 2z , ξ ∈ R .

( 6 . 1 )

 61 Theorem. Let h ∈ H + ( L Sol(p,q) a

Case 2 .

 2 z n → -∞ . Then we can apply part (b) of the Claim, and again by Lemma 5.6, there is a function h 1 on H(p) which is minimal inH + ( L H(p) a , λ ), such that h(x, y, z) = h 1 (x, z) for all x = (x, y, z) ∈ Sol. Case 3. z n → z 0 ∈ R. Then we can apply both parts (a) and (b) of the claim, and there is a function h on R such that h(x, y, z) = h(z) for all x = (x, y, z) ∈ Sol. It must be minimal both as a function (x, z) → h(z) in H + ( L H(p) a , λ ) and as a function (y, z) → h(-z) in H + ( L H(q)-a , λ

( 6 . 3 )

 63 Corollary. If h ∈ H + ( L Sol(p,q) a , λ) , where λ ≥ -a 2 /2, then there are nonnegative functions h 1 ∈ H + ( L

( 6 . 6 )

 66 Theorem. The minimal eigenfunctions in H + ( L Sol(p,q) a

  depends only on z then it must arise by lifting a minimal element of h ∈ H + ( L a , λ ) from R to Sol. That is, we must have h(x, y, z) = e αz , where α = ± √ a 2 + 2λa. Furthermore, in this case, the function (x, z) → e αz must be minimal in H + ( L

	Sol(p,q) a	, λ	) is minimal and H(p) a

)

. Of course, it must also be a minimal element of

H + ( L R a , λ ) . (6.2) Remark. If h ∈ H + ( L

Proof. Combining Theorem 6.1 with Lemma 5.3, we see that each minimal λ-eigenfunction on Sol must be of the form (x, y, z) → P p,a,λ ( (x, z), ξ) , where ξ ∈ ϑH(p) , or (x, y, z) → P q,-a,λ ( (y, -z), η) , where η ∈ ϑH(q) .

We have to show that for ξ ̸ = ϖ p and for η ̸ = ϖ q , the respective functions are indeed all minimal. Furthermore, we have to show that for ξ = ϖ p and for η = ϖ q , the two resulting functions are not minimal λ-eigenfunctions on Sol, unless λ = λ min . In this last case both coincide and are equal to e -a z .

So first we show minimality of (x, y, z) → P p,a,λ ( (x, z), ξ) with ξ ∈ ϑ * H(p). Suppose that P p,a,λ ( (x, z), ξ) ≥ h(x, y, z) for all z = (x, y, z), where h ∈ H + ( L Sol(p,q) a , λ ) . We decompose h(x, y, z) = h 1 (x, z)+h 2 (y, -z) according to Corollary 6.3. By minimality of P p,a,λ

where ν is a Borel measure on ϑH(q). Setting y = z = 0, we get

If x → ∞, then we see from the formula for P q,-a,λ of Lemma 5.3 that the left hand side in the last inequality tends to 0. Therefore ν ( ϑH(q) ) = 0, whence h 2 ≡ 0, contradicting the assumption that c < 1.

The proof of minimality of (x, y, z) → P q,-a,λ ( (x, -z), η), where η ∈ ϑ * H(q), follows as usual by exchanging the roles of the x-and y-variables.

Next, let ξ = ϖ p and λ > λ min , so that we are considering the function (x, y, z) → P p,a,λ ( (x, z), ϖ p ) = e α(λ,a)z .

If it were minimal in H + ( L Sol(p,q) a , λ ) , then by Lemma 5.6, also the function (y, z) → e -α(λ,a)z would have to be minimal in H + ( L

-a , λ

Our results tell us that the Poisson boundary of Brownian motion with drift on Sol is the "outer" boundary

) ∪ { (ϖ p , ϖ q ) } together with the limit distribution provided by Proposition 4.2. Indeed, for a < 0, it is just the first of these three pieces, because the limit distributition is supported by that piece. For a > 0, it is just the second piece, and for a = 0, it is trivial, i.e., the singleton of the third piece. Here, we do not go into details regarding the construction of the Poisson boundary. (In short and informally, it is the smallest probability space that gives rise to an integral representation of all bounded harmonic functions and at the same time provides a model for the limit behavior of the process at infinity.) The reader is referred to the body of work of Kaimanovich, e.g. [START_REF] Kaimanovich | The Poisson formula for groups with hyperbolic properties[END_REF]. Regarding the Martin boundary (which is a metric space, while the Poisson boundary is a measure space), our results underline the evidence that M(λ min ) is the boundary in the geometric compactification that we have described in §5, while for λ > λ min it should be bigger: one first should consider the horocyclic compactification of H(p), which can be built from the usual one as follows. Replace the boundary point ϖ p by the set

which carries the topology of the extended real line. Furthermore, modify the topology by saying that in the new compactification, (x, z) → ϖ ζ p if |x| → ∞ and z → ζ. Then we expect that the Martin compactification of Sol(p, q) for λ > λ min is the closure of Sol in the direct product of the horocyclic compactifications of the two hyperbolic planes. This evidence comes from the strong analogy with the DL-graphs (the horocyclic product of two homogeneous trees), see [START_REF] Brofferio | Green kernel estimates and the full Martin boundary for random walks on lamplighter groups and Diestel-Leader graphs[END_REF]; the rigorous proof still has to be carried out.


Frederic Menous

Frédéric Patras

F R Éd

Éric Menous

Éric Patras

Logarithmic derivatives and generalized Dynkin operators

Motivated by a recent surge of interest for Dynkin operators in mathematical physics and by problems in the combinatorial theory of dynamical systems, we propose here a systematic study of logarithmic derivatives in various contexts. In particular, we introduce and investigate generalizations of the Dynkin operator for which we obtain Magnus-type formulas.

Introduction

Dynkin operators are usually defined as iterated brackettings. They are particularly popular in the framework of linear differential equations and the so-called continuous Baker-Campbell-Hausdorff problem (to compute the logarithm of an evolution operator). We refer to [START_REF] Reutenauer | Free Lie algebras[END_REF] for details and an historical survey of the field. Dynkin operators can also be expressed as a particular type of logarithmic derivatives (see Corollary 3 below). They have received increasingly more and more interest during the recent years, for various reasons.

(1) In the theory of free Lie algebras and noncommutative symmetric functions, it was shown that Dynkin operators generate the descent algebra (the direct sum of Solomon's algebras of type A, see [START_REF] Gelfand | Noncommutative symmetric functions[END_REF][START_REF] Reutenauer | Free Lie algebras[END_REF]) and play a crucial role in the theory of Lie idempotents.

Date: June 6, 2012.

(2) Generalized Dynkin operators can be defined in the context of classical Hopf algebras [START_REF] Patras | On Dynkin and Klyachko Idempotents in Graded Bialgebras[END_REF]. The properties of these operators generalize the classical ones. Among others (see also [START_REF] Ebrahimi-Fard | Rota-Baxter algebras and new combinatorial identities Letters in[END_REF][START_REF] Ebrahimi-Fard | A noncommutative Bohnenblust-Spitzer identity for Rota-Baxter algebras solves Bogoliubov's recursion[END_REF]), they can be used to derive fine properties of the renormalization process in perturbative quantum field theory (pQFT): the Dynkin operators can be shown to give rise to the infinitesimal generator of the differential equation satisfied by renormalized Feynman rules [START_REF] Ebrahimi-Fard | A Lie theoretic approach to renormalisation[END_REF]. This phenomenon has attracted the attention on logarithmic derivatives in pQFT where generalized Dynkin operators are expected to lead to a renewed understanding of Dyson-Schwinger-type equations, see e.g. [START_REF] Kreimer | An Etude in non-linear DysonSchwinger Equations[END_REF].

The present article was however originated by different problems, namely problems in the combinatorial theory of dynamical systems (often referred to as Ecalle's mould calculus, see e.g. [START_REF] Sauzin | Mould expansions for the saddle-node and resurgence monomials[END_REF], also for further references on the subject) and in particular in the theory of normal forms of vector fields. It appeared very soon to us that the same machinery that had been successfully used in the above mentioned fields and problems was relevant for the study of dynamical systems. However, the particular form of logarithmic derivatives showing up in this field requires the generalization of the known results on logarithmic derivatives and Dynkin operators to a broader framework: we refer to the section 2.1 of the present article for more details on the subject.

The purpose of the present article is therefore to develop further the algebraic and combinatorial theory of logarithmic derivatives, following various directions and point of views (free Lie algebras, Hopf algebras, Rota-Baxter algebras...), all of them known to be relevant for the study of dynamical systems but also to various other fields, running from the numerical analysis of differential equations to pQFT.

We limit the scope of the present article to the general theory and plan to use the results in forthcoming articles.

Twisted Dynkin operators on free Lie algebras

Let T (X) be the tensor algebra over an alphabet X = {x 1 , ..., x n , ...}. It is graded by the length of words: the degree n component T n (X) of T (X) is the linear span (say over the rationals) of the words y 1 ...y n , y i ∈ X. The length, n, of y 1 ...y n is written l(y 1 ...y n ). It is equipped with the structure of a graded connected cocommutative Hopf algebra by the concatenation product:

µ(y 1 ...y n ⊗ z 1 ...z m ) = y 1 ...y n • z 1 ...z m := y 1 ...y n z 1 ...z m
and the unshuffling coproduct:

∆(y 1 ...y n ) := n p=0 I J=[n] y i 1 ...y ip ⊗ y j 1 ...y j n-p ,
where 1 ≤ i 1 < i 2 < ... < i p ≤ n, 1 ≤ j 1 < j 2 < ... < j n-p ≤ n and I = {i 1 , ..., i p }, J = {j 1 , ..., j n-p }. The antipode is given by: S(y 1 ...y n ) = (-1) n y n ...y 1 , where y i ∈ X. We refer e.g. to [START_REF] Reutenauer | Free Lie algebras[END_REF] for further details on the subject and general properties of the tensor algebra viewed as a Hopf algebra and recall simply that the antipode is the convolution inverse of the identity of T (X): S * Id = Id * S = ε, where ε is the projection on the scalars T 0 (X) orthogonally to the higher degree components T n (X), n > 0, and the convolution product f * g of two linear endomorphisms of T (X) (and, more generally of a Hopf algebra with coproduct ∆ and product µ) is given by: f * g := µ • (f ⊗ g) • ∆. In particular, S * Id is the null map on T n (X) for n > 0.

We write δ an arbitrary derivation of T (X) (in particular δ acts as the null application on the scalars, T 0 (X)). The simplest and most common derivations are induced by maps f from X to its linear span: the associated derivation, written f , is then defined by f (y

1 ...y n ) = n i=0 y 1 ...y i-1 f (y i )y i+1 ...y n . Since for a, b ∈ T (X), f ([a, b]) = [f (a), b] + [a, f (b)]
, where [a, b] := abba, these particular derivations are also derivations of the free Lie algebra over X. These are the derivations we will be most interested in in practice (the ones that map the free Lie algebra over X to itself), we call them Lie derivations.

In the particular case f = Id, we also write Y for Ĩd: Y is the graduation operator, Y (y

1 ...y n ) = n • y 1 ...y n . When f = δ x i (f (x i ) = x i , f (x j ) = 0 for j = i), f
counts the multiplicity of the letter x i in words and is the noncommutative analog of the derivative with respect to x i of a monomial in the letters in X. Let us assume, by induction, that the identity holds in degrees < n. Then, with the same notation as the ones used to define the coproduct ∆:

S * δ(y 1 ...y n ) = n p=0
(-1) p y ip ...y i 1 δ(y j 1 ...y j n-p ) where I J = [n]. Notice that, if i p = n, j n-p = n (and conversely). Therefore, since δ is a derivation:

S * δ(y 1 ...y n ) = n-1 p=0 (-1) p (y ip ...y i 1 δ(y j 1 ...y j n-p-1 )y n +y ip ...y i 1 y j 1 ...y j n-p-1 δ(y n )) + n-1 p=0
(-1) p+1 y n y ip ...y i 1 δ(y j 1 ...y j n-p-1 ), where the sums run over the partitions

I J = [n -1].
The first and third term of the summation sum up to [S * δ(y 1 ...y n-1 ), y n ] which is, by induction, equal to [...[[δ(y 1 ), y 2 ], y 3 ]..., y n ]. The second computes S * Id(y 1 ...y n-1 )δ(y n ), which is equal to 0 for n > 1. The Proposition follows.

Corollary 2. For a Lie derivation δ, the map D δ maps T (X) to Lie(X), the free Lie algebra over X. Moreover, we have, for l ∈ Lie(X),

D δ (l) = δ(l).
The first part of the corollary follows from the previous proposition. To prove the second part, recall that l ∈ Lie(X) if and only if ∆(l) = l⊗1+1⊗l. Notice furthermore that, since D δ = S * δ, we have δ = Id * D δ . Therefore:

δ(l) = (Id * D δ )(l) = D δ (1) • l + D δ (l).
The proof follows since D δ (1) = 0.

We recover in particular the theorem of Dynkin [START_REF] Dynkin | Calculation of the coefficients in the Campbell-Hausdorff formula[END_REF], Specht [START_REF] Specht | Die linearen Beziehungen zwischen höheren Kommutatoren[END_REF], Wever [START_REF] Wever | Über Invarianten in Lieschen Ringen[END_REF] (case f = Id) and obtain an extension thereof to the case f = δ x i . We let the reader derive similar results for other families of Lie derivations.

Corollary 3. We have, for the classical Dynkin operator D = S * Y and an arbitrary element l in T n (X) ∩ Lie(X):

D(l) = n • l.
In particular, the operator D n is a projection from T n (X) onto T n (X) ∩ Lie(X).

The definition D = S * Y of the Dynkin operator seems due to von Waldenfels, see [START_REF] Reutenauer | Free Lie algebras[END_REF].

Let us write T i n (X) for the linear span of words over X such that the letter x i appears exactly n times. The derivation δx i acts as the multiplication by n on T i n (X). Corollary 4. We have for D x i := S * δx i and an arbitrary element l in

T i n (X) ∩ Lie(X): D x i (l) = n • l. In particular, the operator D n is a projection from T i n (X) onto T i n (X) ∩ Lie(X).

Abstract logarithmic derivatives

Quite often, the logarithmic derivatives one is interested in arise from dynamical systems and geometry. We explain briefly why on a fundamental example, the classification of singular vector fields (section 2.1). Although we settle our later computations in the general framework of Lie and enveloping algebras, the reader may keep that motivation in mind.

The second section (2.2) shows briefly how to extend the results on generalized Dynkin operators obtained previously in the tensor algebra to the general setting of enveloping algebras.

We show at last (section (2.3) how these results connect to the theory of Rota Baxter algebras, which is known to be the right framework to investigate the formal properties of derivations. Indeed, as we will recall below, Rota-Baxter algebra structures show up naturally when derivations have to be inverted. See also [START_REF] Ebrahimi-Fard | Rota-Baxter algebras and new combinatorial identities Letters in[END_REF][START_REF] Ebrahimi-Fard | A noncommutative Bohnenblust-Spitzer identity for Rota-Baxter algebras solves Bogoliubov's recursion[END_REF][START_REF] Bauer | Time ordering and a generalized Magnus expansion[END_REF] for further details on the subject of Rota-Baxter algebras and their applications.

2.1. An example from the theory of dynamical systems. Derivations on graded complete Lie algebras appear naturally in the framework of dynamical systems, especially when dealing with the formal classification (up to formal change of coordinates) of singular vector fields.

The reader can refer to [START_REF] Ilyashenko | Lectures on analytic differential equations[END_REF] for an overview and further details on the objects we consider (such as identity-tangent diffeomorphisms or substitution automorphisms) -let us also mention that the reader who is interested only in formal aspects of logarithmic derivatives may skip that section.

A formal singular vector field in dimension ν is an operator:

X = ν i=1 f i (x 1 , . . . , x ν ) ∂ ∂x i such that f i (0) = 0 for all i (that is f i ∈ C ≥1 [[x 1 , . . . , x ν ]]
). Such operators act on the algebra of formal series in ν variables. In practice, a vector field is given by a series of operators such as

x n 1 1 . . . x nν ν ∂ ∂x i with n 1 + . . . + n ν 1 that acts on monomials x m 1 1 . . . x mν ν : x n 1 1 . . . x nν ν ∂ ∂x i .x m 1 1 . . . x mν ν = m i • x m 1 +n 1 1 . . . x m i +n i -1 i . . . x mν +nν
ν so that the total degree goes from m 1 + . . . + m ν to m 1 + . . . + m ν + n 1 + . . . + n ν -1 and the graduation for such an operator is then n 1 + . . .

+ n ν -1.
The 0 graded component of a vector field X is called the linear part since it can be written X 0 = A ij x i ∂ ∂x j and a fundamental question in dynamical systems is to decide if X is conjugate, up to a change of coordinates, to its linear part X 0 . Notice that:

• The vector space L of vector fields without linear part (or without component of graduation 0) is a graded complete Lie algebra. • The exponential of a vector field in L gives a one to one correspondence between vector fields and substitution automorphisms on formal power series, that is operators F such that

F (A(x 1 , . . . , x ν )) = A(F (x 1 ), . . . , F (x ν ))
where (F (x 1 ), . . . , F (x ν )) is a formal identity-tangent diffeomorphism. • The previous equation also determines an isomorphism between the Lie group of L and the group of formal identity-tangent diffeomorphism G 1 .

These are essentially the framework (the one of graded complete Lie algebras) and the objects (elements of the corresponding formal Lie groups) that we will consider and investigate in our forthcoming developments. Consider now a vector field X = X 0 + Y ∈ L 0 ⊕ L and suppose that it can be linearized by a change of coordinates in G 1 , or rather by a substitution automorphism F in the Lie group of L. It is a matter of fact (see [START_REF] Ilyashenko | Lectures on analytic differential equations[END_REF], [START_REF] Écalle | Singularités non abordables par la géométrie[END_REF]) to check that the corresponding conjugacy equation reads:

X 0 F = F (X 0 + Y ) ⇐⇒ [X 0 , F ] = F Y ⇐⇒ ad X 0 (F ) = F Y
This equation, called the homological equation, delivers a derivation δ = ad X 0 on L that is compatible with the graduation. The linearization problem is then obviously related to the inversion of the logarithmic derivation

D δ (F ) := F -1 δ(F ).
In the framework of dynamical systems, the forthcoming theorem 15 ensures that if the derivation ad X 0 is invertible on L, any vector field X 0 + Y can be linearized. This is the kind of problems that can be addressed using the general theory of logarithmic derivatives to be developed in the next sections.

2.2. Hopf and enveloping algebras. We use freely in this section the results in [START_REF] Patras | On Dynkin and Klyachko Idempotents in Graded Bialgebras[END_REF] to which we refer for further details and proofs. The purpose of this section is to extend the results in [START_REF] Patras | On Dynkin and Klyachko Idempotents in Graded Bialgebras[END_REF] The ground field is chosen to be Q (but the results in the article would hold for an arbitrary ground field of characteristic zero and, due to the Cartier-Milnor-Moore theorem [START_REF] Milnor | On the structure of Hopf algebras[END_REF][START_REF] Patras | L'algèbre des descentes d'une bigèbre graduée[END_REF], for arbitrary graded connected cocommutative Hopf algebras).

The enveloping algebra U (L) is naturally provided with the structure of a Hopf algebra. We denote by ǫ :

Q = U (L) 0 → U (L) the unit of U (L), by η : U (L) → Q the counit, by ∆ : U (L) → U (L) ⊗ U (L) the coproduct and by µ : U (L) ⊗ U (L) → U (L) the product. An element l of U (L) is primitive if ∆(l) = l ⊗ 1 + 1 ⊗ l; the set of primitive elements identifies canonically with L. Recall that the convolution product * of linear endomorphisms of U (L) is defined by f * g = µ • (f ⊗ g) • ∆; ν := ǫ • η is the neutral element of * .
The antipode is written S, as usual.

Definition 5. element f of End(U (L)) admits F ∈ End(U (L)) ⊗ End(U (L)) as a pseudo-coproduct if F • ∆ = ∆ • f . If f admits the pseudo- coproduct f ⊗ ν + ν ⊗ f , we say that f is pseudo-primitive.
In general, an element of End(U (L)) may admit several pseudo-coproducts. However, this concept is very flexible, as shows the following result [16, Thm. 2]. Proposition 6. If f, g admit the pseudo-coproducts F, G and α ∈ F, then f + g, αf, f * g, f • g admit respectively the pseudo-coproducts F + G, αF, F * G, F • G, where the products * and • are naturally extended to End(U (L)) ⊗ End(U (L)).

An element f ∈ End(U (L)) takes values in P rim(U (L)) if and only if it is pseudo-primitive.

Let δ be an arbitrary derivation of L (∀l,

l ′ ∈ L, δ([l, l ′ ]) = [δ(l), l ′ ] + [l, δ(l ′ )]
). We also write δ for its unique extension to a derivation of U (L) and write D δ := S * δ. For an element l ∈ L, exp(l) is group-like (∆(exp(l)) = exp(l) ⊗ exp(l)), which it follows that: D δ (exp(l)) = S(exp(l))δ(exp(l)) = exp(-l)δ(exp(l)), the (noncommutative) logarithmic derivative of exp(l) with respect to δ. We call therefore D δ the logarithmic derivative of δ.

Proposition 7. The logarithmic derivative D δ is a pseudo-primitive: it maps U (L) to L. Indeed, S ⊗ S is a pseudo-coproduct for S (see [START_REF] Patras | On Dynkin and Klyachko Idempotents in Graded Bialgebras[END_REF]). On the other hand U (L) is spanned by products l 1 ...l n of elements of L. Since δ is a derivation, we get:

∆ • δ(l 1 ...l n ) = ∆( n i=1 l 1 ...δ(l i )...l n ) = (δ ⊗ Id + Id ⊗ δ) • ∆(l 1 ...l n ),
where the last identity follows directly from the fact that the l i are primitive, which implies that ∆(l 1 ...l n ) can be computed by the same formula as the one for the coproduct in the tensor algebra. In particular, δ ⊗ Id + Id ⊗ δ is a coproduct for δ. We get finally:

∆ • D δ = ∆ • (S * δ) = (S ⊗ S) * (δ ⊗ Id + Id ⊗ δ) • ∆ = (D δ ⊗ ν + ν ⊗ D δ ) • ∆,
from which the proof follows.

Proposition 8. For l ∈ L, we have δ(l) = D δ (l). In particular, when δ is invertible on L, D δ (l) is a projection from U (L) onto L.

The proof is similar to the one in the free Lie algebra. We have

D δ (l) = (S * δ)(l) = π • (S ⊗ δ) • ∆(l) = π • (S ⊗ δ)(l ⊗ 1 + 1 ⊗ l) = π(S(l) ⊗ δ(1) + S(1) ⊗ δ(l)) = δ(l), since δ(1) = 0 and S(1) = 1.

2.3.

Integro-differential calculus. The notation are the same as in the previous section, but we assume now that the derivation δ is invertible on L and extends to an invertible derivation on U (L) + := n≥1 U (L) n . The simplest example is provided by the graduation operator Y (l) = n • l for l ∈ L n (resp. Y (x) = n • x for x ∈ U (L) ). This includes the particular case, generic for various applications to the study of dynamical systems, where L is the graded Lie algebra of polynomial vector fields spanned linearly by the x n ∂ x and δ := x∂ x acting on P (x)∂ x as δ(P (x)∂

x ) := xP ′ (x)∂ x .
We are interested in the linear differential equation

(1) δφ = φ • x, x ∈ L, φ ∈ 1 + U (L) + .
The inverse of δ is written R and satisfies, on U (L) + , the identity:

R(x)R(y) = R(R(x)y) + R(xR(y)),
that follows immediately from the fact that δ is an invertible derivation.

In other terms, U (L) + is provided by R with the structure of a weight 0 Rota-Baxter algebra and solving (1) amounts to solve the so-called Atkinson recursion:

φ = 1 + R(φ • x).
We refer to [START_REF] Ebrahimi-Fard | Rota-Baxter algebras and new combinatorial identities Letters in[END_REF][START_REF] Ebrahimi-Fard | A noncommutative Bohnenblust-Spitzer identity for Rota-Baxter algebras solves Bogoliubov's recursion[END_REF] for a detailed study of the solutions to the Atkinson recursion and further references on the subject. Perturbatively, the solution is given by the Picard (or Chen, or Dyson... the name given to the expansion depending on the application field) series:

φ = 1 + n≥1 R [n] (x),
where R [1] (x) = R(x) and R

[n] (x) := R(R [n-1] (x)x).
Since the restriction to the weight 0 is not necessary for our forthcoming computations, we restate the problem in a more general setting and assume from now on that R is a weight θ Rota-Baxter (RB) map on U (L) + , the enveloping algebra of a graded Lie algebra. That is, we assume that:

R(x)R(y) = R(R(x)y) + R(xR(y)) -θR(xy).
This assumption allows to extend vastly the scope of our forthcoming results since the setting of Rota-Baxter algebras of arbitrary weight includes, among others, renormalization in perturbative quantum field theory and difference calculus, the later setting being relevant to the study of diffeomorphisms in the field of dynamical systems. We refer in particular to the various works of K. Ebrahimi-Fard on the subject (see e.g. [START_REF] Ebrahimi-Fard | A noncommutative Bohnenblust-Spitzer identity for Rota-Baxter algebras solves Bogoliubov's recursion[END_REF][START_REF] Bauer | Time ordering and a generalized Magnus expansion[END_REF] for various examples of RB structures and further references). We assume furthermore that R respects the graduation and restricts to a linear endomorphism of L.

Lemma 9. The solution to the Atkinson recursion is a group-like element in 1 + U (L) + . In particular, S(φ) = φ -1 .

Recall from [START_REF] Patras | On Dynkin and Klyachko Idempotents in Graded Bialgebras[END_REF] and [START_REF] Ebrahimi-Fard | A Lie theoretic approach to renormalisation[END_REF] that the generalized Dynkin operator D := S * Y (the convolution of the antipode with the graduation map in U (L)) maps U (L) to L and, more specifically, defines a bijection between the set of group-like elements in Û (L) and L. The inverse is given explicitly by ([5, Thm. 4.1]):

D -1 (l) = 1 + n∈N * k 1 +...+k l =n l k 1 • ... • l k l k 1 (k 1 + k 2 )...(k 1 + ... + k l )
,

where l n is the component of l ∈ L in L n . According to [6, Thm. 4.3], when l = D(1 + n≥1 R [n] (x)
) we also have:

1 + n≥1 R [n] (x) = 1 + n∈N * k 1 +...+k l =n l k 1 • ... • l k l k 1 (k 1 + k 2 )...(k 1 + ... + k l ) , that is, since l ∈ L by Prop. 8, 1 + n≥1 R [n] (x) is a group-like element in U (L).
The last part of the Lemma is a general property of the antipode acting on a group-like element; the Lemma follows.

We are interested now in the situation where another Lie derivation d acts on U (L) and commutes with R (or equivalently with δ when R is the inverse of a derivation). A typical situation is given by Schwarz commuting rules between two different differential operators associated to two independent variables. Theorem 10. Let d be a graded derivation on U (L) commuting with the weight θ Rota-Baxter operator R. Then, for φ a solution of the Atkinson recursion as above, we have:

D d (φ) = φ -1 • d(φ) = n≥1 R [n] d (x) with I [1] d = d(x), R [1] d (x) = R(d(x)), I [n+1] d (x) = [R [n] d (x), x] + θx • I [n] d (x) and R [n+1] d (x) = R(I [n+1] d (x)).
Notice, although we won't make use of this property, that the operation x • y := [R(x), y] + θy • x showing up implicitly in this recursion is a preLie product, see e.g. [START_REF] Ebrahimi-Fard | Rota-Baxter algebras and new combinatorial identities Letters in[END_REF]. In particular, for y the solution to the preLie recursion:

y = d(x) + y • x, we have: D d (φ) = R(y).
The first identity

D d (φ) = φ -1 • d(φ)
follows from the definition of the logarithmic derivative D d := S * d and from the previous Lemma.

The second is equivalent to

d(φ) = φ • n≥1 R [n] d (x)
, that is:

d(R [n] (x)) = R [n] d (x) + n-1 i=1 R [i] (x)R [n-i] d (x).
For n = 1, the equation reads d(R(x)) = R(d(x)) and expresses the commutation of d and R.

The general case follows by induction. Let us assume that the identity holds for the components in degree n < p of D d (φ). We summarize in a technical Lemma the main ingredient of the proof. Notice that the Lemma follows directly from the Rota-Baxter relation and the definition of R Lemma 11. We have, for n, m ≥ 1:

R(R [m] (x) • ([R [n] d (x), x] + θx • I [n] d (x))) = R [m] (x)R [n+1] d (x) -R(R [m-1] (x) • x • R [n+1] d (x)) + θR(R [m-1] (x) • x • ([R [n] d (x), x] + θx • I [n] d (x)).
We have, for the degree p component of D d (φ), using the Lemma to rewrite R(R [m]

(x) • R [n] d (x) • x): d(R [p] (x)) = d(R(R [p-1] (x) • x)) = R((dR [p-1] (x)) • x + R [p-1] (x) • dx) = R(R [p-1] (x) • dx) + R((R [p-1] d (x) + p-2 k=1 R [p-1-k] (x) • R [k] d (x)) • x) = R(R [p-1] (x) • dx) + R(R [p-1] d (x) • x) + p-2 k=1 [R [p-1-k] (x) • R [k+1] d (x) -R(R [p-2-k] (x) • x • R [k+1] d (x)) + θR(R [p-2-k] (x) • x • ([R [k] d (x), x] + θx • I [k] d (x)) +R(R [p-1-k] (x) • x • R [k] d (x)) -θR(R [p-1-k] (x) • x • I [k] d (x))].
The fourth and sixth terms cancel partially and add up to R(R [p-2]

(x) • x • R [1] d (x)) -R(x • R [p-1] d (x)
). The fifth and last terms cancel partially and add up to θR(x • I

[p-1] d (x)) -θR(R [p-2] (x) • x • I [1]
d (x)). In the end, we get:

d(R [p] (x)) = p-2 k=1 R [p-1-k] (x) • R [k+1] d (x) + [R(R [p-2] (x) • x • R [1] d (x))+ R(R [p-1] (x)•dx)-θR(R [p-2] (x)•x•I [1] d (x)]+[R(R [p-1] d (x)•x)-R(x•R [p-1] d (x)) +θR(x • I [p-1] d (x))]
Using the RB identity for the expressions inside brackets, we get finally:

d(R [p] (x)) = p-2 k=1 R [p-1-k] (x) • R [k+1] d (x) + R [p-1] (x)R [1] d (x) + R [p] d (x),
from which the Theorem follows.

Magnus-type formulas

The classical Magnus formula relates "logarithms and logarithmic derivatives" in the framework of linear differential equations. That is, it relates explicitly the logarithm log(X(t)) =: Ω(t) of the solution to an arbitrary matrix (or operator) differential equation X ′ (t) = A(t)X(t), X(0) = 1 to the infinitesimal generator A(t) of the differential equation:

Ω ′ (t) = ad Ω(t) exp ad Ω(t) -1 A(t) = A(t) + n>0 B n n! ad n Ω(t) (A(t)),
where ad stands for the adjoint representation and the B n for the Bernoulli numbers.

The Magnus formula is a useful tool for numerical applications (computing the logarithm of the solution improves the convergence at a given order of approximation). It has recently been investigated and generalized from various points of view, see e.g. [START_REF] Ebrahimi-Fard | A Magnus-and Fer-type formula in dendriform algebras[END_REF], where the formula has been extended to general dendriform algebras (i.e. noncommutative shuffle algebras such as the algebras of iterated integrals of operators), [START_REF] Chapoton | Enveloping algebras of preLie algebras, Solomon idempotents and the Magnus formula[END_REF] where the algebraic structure of the equation was investigated from a purely preLie algebras point of view, or [START_REF] Bauer | Time ordering and a generalized Magnus expansion[END_REF] where a generalization of the formula has been introduced to model the commutation of time-ordered products with time-derivations.

The link with preLie algebras follows from the observation that (under the hypothesis that the integrals and derivatives are well-defined), for arbitrary time-dependent operators, the preLie product

M (t) N (t) := t 0 [N (u), M ′ (u)]du satisfies (M (t) N (t)) ′ = ad N (t) M ′ (t)
. The Magnus formula rewrites therefore:

Ω ′ (t) =   t 0 A(x)dx Ω exp(Ω) -1   ′
where Ω exp(Ω)-1 is computed in the enveloping algebra of the preLie algebra of time-dependent operators.

Here, we would like to go one step further and extend the formula to general logarithmic derivatives in the suitable framework in view of applications to dynamical systems and geometry, that is, the framework of enveloping algebras of graded Lie algebras and Lie derivations actions. Notations are as before, that is L is a graded Lie algebra and δ a graded Lie derivation (notice that we do not assume its invertibility on L or U (L)).

Lemma 12. For l ∈ L, k ≥ 1 and x ∈ U (L), we have:

x • l k = k i=0 k i l k-i • (-ad l ) i (x).
The proof is by induction on n. The formula holds for n = 1: xl = -[l, x] + lx. Let us assume that it holds for an arbitrary n < p. Then we have:

x • l p = (x • l p-1 ) • l = ( p-1 i=0 p -1 i l p-1-i • (-ad l ) i (x)) • l = p-1 i=0 p -1 i l p-1-i • (-ad l ) i+1 (x) + p-1 i=0 p -1 i l p-i • (-ad l ) i (x).
The identity follows then from Pascal's triangular computation of the binomial coefficients.

Theorem 13. For l ∈ L, we have: Since exp(l), being the exponential of a Lie element is group-like, D δ (exp(l)) = exp(-l)d(exp(l)) and the theorem follows.

Let us show as a direct application of the Thm (13) how to recover the classical Magnus theorem (other applications to mould calculus and to the formal and analytic classification of vector fields are postponed to later works).

Example 14. Let us consider once again an operator-valued linear differential equation X ′ (t) = X(t)λA(t), X(0) = 1. Notice that the generator A(t) is written to the right, for consistency with our definition of logarithmic derivatives D δ = S * d. All our results can of course be easily adapted to the case d * S (in the case of linear differential equations this amounts to consider instead X ′ (t) = A(t)X(t)), this easy task is left to the reader. Notice also that we introduce an extra parameter λ, so that the perturbative expansion of X(t) is a formal power series in λ. Although a direct consequence of the previous theorem (recall that any group-like element in the enveloping algebra U (L) can be written as an exponential), the following proposition is important and we state it also as a theorem.

Theorem 15. When δ is invertible on L, the logarithmic derivative D δ is a bijection between the set of group-like elements in U (L) and L.

Indeed, for h ∈ L the Magnus-type equation in L l = δ -1 ( -ad l exp(-ad l ) -1 (h))

has a unique recursive solution l ∈ L such that exp(l) = D -1 δ (h).

Proposition 1 .

 1 For arbitrary letters y 1 , ..., y n of X, we have, for D δ := S * δ: D δ (y 1 ...y n ) = [...[[δ(y 1 ), y 2 ], y 3 ]..., y n ]

  on the Dynkin operator to more general logarithmic derivatives. Let L = n∈N * L n be a graded Lie algebra, L = n∈N * L n its completion, U (L) = n∈N * U (L) n the (graded) enveloping algebra of L and Û (L) = n∈N * U (L) n the completion of U (L) with respect to the graduation.

D

  δ (exp(l)) = exp(-ad l ) -1 -ad l δ(l).Indeed, from the previous formula we get,d(exp(l)) = -1-i (-ad l ) i (d(l)) 1)!(n -1i)! l n-1-i (-ad l ) i (d(l)) = exp(l)( i≥0 1 (i + 1)! (-ad l ) i )d(v).

  Consider then the Lie algebra O of operators M (t) (equipped with the bracket of operators) and the graded Lie algebra L = n∈N * λ n O = O ⊗ λC[λ]. Applying the Thm (13), we recover the classical Magnus formula.