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We give central lower estimates for the transition kernels corresponding to symmetric random walks on certain amenable p-adic groups.


Introduction
Let G be a locally compact amenable group.We shall denote by d r g (resp.d l g) the right (resp.left) Haar measure on G and by δ(g) = d r g/d l g the modular function on G normalized by δ(e) = 1 where e denotes the identity of G. let dµ(g) = ϕ(g)d r g ∈ P(G) be a probability measure on G where ϕ(g) ∈ L ∞ (G) is assumed to have a compact support or a fast decay at infinity.Let us assume that µ is symmetric (i.e. the involution g → g −1 stabilizes µ) and consider the random walk on G induced by µ, i.e. the G-valued process that evolves as follows: if X n = g is the position at time n then X n+1 = gh where h is chosen according to µ.We shall denote by dµ * n (g) = ϕ n (g)d r g the n th convolution power of µ and examine the behaviour of the decay of ϕ n (e) as n → ∞.
If we restrict ourselves to unimodular real Lie groups then the answer lies in the behaviour of the volume growth of G. Let us recall that if G is a locally compact group that is generated by some symmetric compact neighbourhood Ω ⊂ G of the identity in G, the volume growth function is (cf.[9]) γ(t) = Vol(B t (e)), t = 1, 2, ...
where the volume is taken with respect to d r g (or d l g) and where B t (e) is the ball of radius t centred on e defined by B t (e) = Ω...Ω, t times.
For x ∈ G the distance from e is defined by |x| = inf{t, x ∈ B t (e)} and a left invariant distance can be defined on G by setting d(x, y) = | y −1 x|, x, y ∈ G.If Ω 1 , Ω 2 are two neighbourhoods of e as above it is not difficult to check that there exists C > 0 such that C −1 ≤ |.| 2 /|.| 1 ≤ C and that the corresponding growth functions satisfy the obvious equivalence γ 1 ≈ γ 2 , i.e.
In the first case we say that G is of polynomial growth and in the second case we say that G is of exponential growth and the answer to our problem in the case of unimodular amenable real Lie groups was given by Varopoulos and is the following ϕ n (e) ≈ e −n 1/3 ⇐⇒ γ(t) ≈ e t (cf.[1], [6], [7], [10], [28]).
Varopoulos showed that the e −n 1/3 versus polynomial behaviour extends to the non-unimodular amenable case depending on wether the Lie group G is (C) or (N C).This classification introduced in [24] can be expressed in terms of the roots of the ad-action of the radical of the Lie algebra of the group on its nilradical.
The discret case is more complicated (cf.[1], [11], [16], [17], [22], [23]).First there is no dichotomy in the volume growth (cf.[8]).On the other hand if we suppose that the group G is of exponential growth then one can claim only the upper bound (cf.[11]) In general, the matching lower bound fails.Ch.Pittet and L. Saloff-Coste showed (cf.[16]) that there are soluble groups with exponentional volume growth for which the heat kernel decays as exp(−cn α ) with α ∈ (0, 1) which can be taken arbitrarly close to 1. This, as mentionned above, can not happen in the case of real Lie groups.
In a recent paper (cf.[17]) Ch.Pittet and L. Saloff-Coste established the lower bound ϕ 2n (e) ≥ c exp(−C n 1/3 ), n = 1, 2, ... for the large times asymptotic behaviours of the probabilities of return to the origin at even times 2n, for random walks associated with finite symmetric generating sets of solvable groups of finite Prüfer rank.They asked in this paper (cf.[17], §8) if a similiar lower bound is available in the case of analytic p-adic groups.An answer to this problem was given in [15] (cf.also [14]).The aim of this paper is to show that the e −n 1/3 lower bound obtained in [14] can be substantially improved for a large class of amenable p-adic groups.

Amenable p-adic groups
In this section G will denote an algebraic connected amenable group over p the field of p-adic numbers; U ⊂ Q ⊂ G will denote the radical and the unipotent radical (cf.[3], [5]).Amenability of G is equivalent to the fact that the semi-simple group G/Q is compact (cf.[19]).Let S denote a fixed levi subgroup S of G (cf. [3]).The group G can then be written as a semi-direct product: where A is abelian and can be identified to the direct product of a finite group and a d-torus T ∼ = ( * p ) d (cf.[3], [5]).Here * p denotes the multiplicative group of the field p .Since ( * p ) d ∼ = d × K (where K is compact, cf.[2], [4]), by considering the projection we can fix π 1 , ..., π d ∈ A so that each z ∈ A admits a unique decomposition where K denotes a compact subgroup of A. Let = Lie(U) denote the Lie algebra of U. Let p denotes a finite extension of p which contains all the eigenvalues defined by The Ad-action of A on extends to ⊗ p p and it follows from the proof of the Zassenhaus lemma (cf.[12]) that we have a decomposition of ⊗ p p into a direct sum where the subspaces W j (1 ≤ j ≤ r) are invariant by Ad(π i ), i = 1, ..., d, and such that the restriction of Ad(π i ) to W j is the sum of the scalar λ j (π i ) and a nilpotent endomorphism.
Let χ j : A −→ p * defined by Here 1 denotes the homomorphism A → p identically equal to 1.This assumption garantees the fact that the group G is compactly generated (cf.[3]).Observe that the group G is then of exponential volume growth (cf.[18]).
Let γ j,1 , ..., γ j,d ∈ the integers defined by We shall denote by L j , j = 1, ..., s, the linear forms on d defined by (n 1 , ..., n d ) ∈ d and by L j the linear forms on d induced by the L j 's.
Let now dµ(g) = ϕ(g)d r g ∈ P(G) denote a symmetric probability measure on G.The density ϕ(g) is assumed to be a continuous compactly supported function on G.To avoid unnecessary complications we shall assume that there exists e ∈ Ω = Ω −1 ⊂ G such that the last condition in (7) implies that supp(µ) generates the group G.
denote the projection that we obtain from the identifications (1) and let where π denotes the canonical projection (2).It follows from ( 7) that there exists a choice of coordinates on d for which the covariance matrix of μ satisfies (9) We shall assume that d is equipped with the Euclidean structure associated to these coordinates.
From now on we shall assume that the L j 's induce a "Weyl chamber".More precisely we suppose that Π = {x ∈ d , L j (x) > 0, j = 1, 2, ..., s} ⊂ d define a nonempty convex cone in d .This condition is the analogue of the (N C)-condition introduced by Varopoulos in [24] in the setting of real amenable Lie groups.We shall prove that, under this condition, we have a lower bound of the form ϕ n (e) ≥ cn −ν .The argument follows the approach introduced by Varopoulos in [24].
The exact value of ν is defined as in the real case and is expressed in terms a parameter λ = λ(d, ) that is defined as follows.In the the rank one case (i.e.d = 1) we shall set In the case d ≥ 2 let us denote by Σ = {x ∈ d , |x| = 1} the unit sphere in d .Let ∆ Σ be the corresponding spherical Laplacian.Let Π Σ = Σ ∩ Π ⊂ Σ.We then set i.e. λ is the first Dirichlet eigenvalue of the region Π Σ .The scalar product and the L 2 -norm in (10 b ) are taken with respect to the Euclidean volume element on Σ.
The following comments may be helpful in placing the above theorem in its proper perspective.
(i) It is enough to to prove the estimate (11) when n is even since ϕ satisfies (7).
(ii) Observe that the group G is automatically non-unimodular, for otherwise we have (cf.[11]): The upper estimate .. is known to hold for general non-unimodular locally compact groups (cf.[26]).This shows that the index 3/2 cannot be improved in the rank one case.
(iv) We will show (cf.§4 below) that in the case of metabelian p-adic groups, the lower bound (11) can be complemented with a similar upper bound.
Throughout the remainder of the paper C denotes a positive constant which is not always the same, even in a given line.

Proof of Theorem 1
Let G, dµ(g) = ϕ(g)d r g ∈ P(G) and dµ * n (g) = ϕ n (g)d r g be as in Theorem 1.Let ξ 1 , ξ 2 ,...∈ G be a sequence of independent equidistribued random variables of law dµ(g) and let X n = ξ 1 ξ 2 ...ξ n , n = 1, 2, ... denote the corresponding random walk starting at X 0 = e.Let B ⊂ G a borel subset, we have The symmetry of dµ(g) implies that We have also On the other hand we have Schwarz inequality applied to (12) gives then Let us further observe that the group G (resp.G/U) decomposes as a semi-direct (resp.direct) product where S is compact.This follows from (1).
Let us write where ξ j = u j z j with u j ∈ U and z j ∈ A × S, j = 1, 2, .... Using the interior automorphisms x y = y x y −1 , x, y ∈ G, we rewrite ( 14) We shall use the exponential map and identify U to its Lie algebra (cf.[5], [20]) and write each u j in the above expression (16) We have therefore Let us fix e 1 , e 2 , ..., e m a basis of ⊗ p p adapted to the decomposition (4).For Since µ is compactly supported we can suppose that the v i 's in ( 16) satisfy where C > 0 is an appropriate positive constant.
Let us equip End p ⊗ p p with the norm It is clear that |. | satisfies the ultrametric property we have where (z) denotes an upper triangular matrice and where On the other hand it is clear that where |.| d denotes the Euclidean norm on d .We have also Combining ( 20), ( 21), (22) we deduce that the triangular matrice (z) that appears in (20) satisfies Let now z 1 , z 2 , ..., z k ∈ A × S and 1 ≤ l ≤ r.We have Using the fact that in the last sum all the terms corresponding to indexes α > n vanish and combining this with ( 21), ( 23) and the ultrametric property (19) we deduce that If we use the linear forms L l defined by ( 5) and ( 6) we then deduce By the above considerations we have finally proved that
Let us consider, for n = 1, 2, ..., the event E n defined by where C > 0 denotes an appropriate large constant.Using (15), Campbell-Hausdorff, (24) and the ultrametric property (19) we see that the event E n verifies where B n is defined by It is easy to see that d r g shows that It remains to estimate the probability of the event (25).Let Π denote the polyhedral region in d defined by where C denotes the same constant as in (25).Let h n (x, y), n = 1, 2, ..., x, y ∈ Π, denote the transition kernel corresponding to the random walk S j j∈N with killing outside of Π. Precise lower estimates for the kernel h n (x, y) can be obtained thanks to the results of [26] and [27].To write down these estimates we need the following notations.Let us assume that d ≥ 2 and let 0 < u 0 (σ), σ ∈ Π Σ , denote the eigenfunction corresponding to the first Dirichlet dirichlet eigenvalue of the region Π Σ defined by (10).Let u(x) be the function defined on Π by and where (r, σ) denote the polar coordinates on * + × Σ.The function u defined in this way is a positive function harmonic in Π which vanishes on ∂ Π (cf.[24]).In the case where d = 1, the function u is defined by u(x) ≡ x, x ∈ * + .It follows easily from [27] (cf.estimate (2) p. 359) and [26] (cf.Theorem 1 and estimate (0.3.4)) that there exists C > 0 such that We have therefore Using the homogeneity of u we deduce then that The lower estimate ( 11) is an immediate consequence of ( 13), ( 26), ( 27) and ( 28).This completes the proof of Theorem 1.

Metabelian p-adic groups
Our aim in in this section is to show that in the case of metabelian groups the lower estimate ( 11) can be complemented with a similar upper bound.We keep the notation of §2.We shall denote by * More precisely, we assume that the multiplication in G is given by We shall denote by the right and the left invariant Haar measure on G.The modular function δ(g) is given by Let dµ(g) = ϕ(g)d r (g) denote the symmetric, compactly supported, probability measure on G defined by The notation I A is used here to denote the indicator of a subset A and α > 0 denotes an appropriate positive constant.
We shall assume that the χ j 's in (30) verify the condition This garantees (cf.§2) that the group G defined by ( 29) is compactly generated and it is easy to see that supp(µ) is a generating set of the group G.We shall denote, as in §2, by L j , j = 1, ..., k, the k linear forms on l induced by the χ j 's and we shall assume that these linear forms induce a "Weyl chamber" Π = {x ∈ l , L j (x) > 0, j = 1, 2, ..., k} as in §2..We shall denote by Σ = {x ∈ l , |x| = 1} the unit sphere in l , by Π Σ = Σ ∩ Π and by λ be the first Dirichlet eigenvalue of the region Π Σ .
We have then the following: ϕ n (g)d r g denote the n th convolution power of µ.Then there exists C > 0 such that The first step to prove Theorem 2 consists in establishing an explicit formula for ϕ n (g), g ∈ G.
In what follows we shall use the notation

. By definition of convolution product we have:
An obvious change of variables combined with Fubini gives We also have where * denotes the usual convolution in p (cf. [21]).Taking Fourier transform (cf.[21]) we obtain p ...
Let us denote by S ζ j ( y 1 , ..., y n ) ⊂ p , y 1 , ..., y n ∈ * p the subset defined by With this notation we have Putting together (33), (36), (37), ( 38) and (39) and taking into account the fact that and that we finally deduce the formula This completes the proof of Theorem 2.