A Poincaré cone condition in the Poincaré group

0.25 MB·English

by  Camille Tardif

A Poincar´e cone condition in the Poincar´e group
Tardif Camille ∗
Abstract
In [7], Ben Arous and Gradinaru described the singularity of the Green function of a general
2 sub-elliptic diffusion. In this article we first adapt their proof to the more general context of
1
a hypoelliptic diffusion. In a second time, we deduce a Wiener criterion and a Poincar´e cone
0
condition for a relativistic diffusion with values in the Poincar´e group (i.e the group of affine
2
direct isometries of the Minkowski space-time).
n
Key words: Green function. Wiener test. Poincar´e cone condition. Relativistic diffusion.
a
J Hypoelliptic operator.

2
2
Contents

]
R
1 Introduction 1
P
. 2 Estimates of the Green function for a general hypoelliptic diffusion. 3
h
t 2.1 Notation and hypothesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
a
2.2 Homogeneous norm and estimations of [23] . . . . . . . . . . . . . . . . . . . . . . 5
m
2.3 Rescaled diffusions and tangent process . . . . . . . . . . . . . . . . . . . . . . . . 6
[
2.4 Proof of Theorem 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1
v 3 A Wiener criterion and a Poincar´e cone condition in the Poincar´e group. 11
5 3.1 Geometric framework. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
7
3.2 Relativistic diffusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5
3.3 A Wiener criterion for thinness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4
3.4 Capacities of small compact sets and Poincar´e’scone condition. . . . . . . . . . . . 14
.
1
0
2 1 Introduction
1
:
v In [12], Dudley introduced a relativistic Brownian motion whose trajectories are time-like and
Xi which is invariant in law under Lorentz transformations. The space of states of this relativistic
Brownian motion is the unitary tangent bundle of the Minkowski space-time, and the diffusion
r
a consistsinaBrownianmotioninHd (the hyperboloidmodelofthe hyperbolicspace)anditstime
integralin the Minkowskispace-time R1,d. By consideringthe Eells-Elworthyconstruction of the
Brownian motion on Hd, we obtain Dudley’s diffusion by projecting a diffusion in the Poincar´e
groupG:SO (1,d)⋉R1,d onto Hd×R1,d. The groupG is identified with the orthonormalframe
0
bundle of the Minkowskispace-time and an element (g,ξ) of G is made of a matrix g ∈SO (1,d)
0
seen as an orthonormal frame above a point ξ ∈ R1,d. The diffusion {(g ,ξ )} in G that we
t t t≥0
consider in this paper has generator
σ2 d
L= V2+H ,
2 i 0
i=1
X
whichsatisfies the weakHo¨rmanderhypoellipticity condition. This means,inparticular,thatthe
drift H is needed to obtain the full rank of Lie(G).
0
∗[email protected]. Institut deRechercheMath´ematique Avanc´ee. Universit´ede Strasbourg.
1
Intuitively,{(g ,ξ )} describesthetime-liketrajectoryofasmallrigidobjectintheMinkowski
t t t≥0
space-time and consists in a stochastic perturbation, at the velocities level, of a geodesic trajec-
tory. This G-valued diffusion, considered as a Lorentzian analogue to the Euclidian Brownian
motion, was studied by Bailleul in [5] where he determined its Poisson boundary by providing a
comprehensivedescriptionoftheasymptoticbehaviorwhich,afterwards,wascompletedin[26]by
the study of the Lyapunov spectrum. In the present work we are interested in understanding the
infinitesimalsmall-timebehaviorandweaimatdescribing,bymeansofa(local)Wienercriterion,
the thinness of sets with respect to this diffusion. This kind of result is based on the knowledge
of the geometry of the level sets of the Green function, and more generally on the knowledge of
its behavior near the singularities. Since we dispose (see [5]) of a geometric description of the
Poissonboundary of Dudley’s diffusion, we can expect a Wiener criterion describing the thinness
at the boundary. This question is in the spirit of the works of Ancona who provides,for example
in[1],aanswerinthe caseofaBrownianmotionina Cartan-Hadamardmanifold. Unfortunately
the method used in this paper cannot provide such asymptotic result and we look, here, only at
the infinitesimal behavior.
Let us briefly perform a short survey on this subject. In the elliptic case, see for example [6],
the Wiener criterion is classical and the fine topology does not depend on the particular elliptic
operator. For sub-elliptic diffusions one finds analytic proofs in [24] and probabilistic methods
in [10] (for the Heisenberg Laplacian) and in [7] (for a general sub-elliptic diffusion under the
strong Ho¨rmander condition). The proofs rely on either explicit expression of Green functions or
estimates given in [23] or [25].
In the parabolic context the situation is more delicate due to the difficulty to understand the
geometry of level sets of Green functions; moreover the results are less generic. For the heat
operator, a probabilistic proof of Wiener criterion can be found in [27] and an analytic one in
[13]. The result is extended to the case of variable coefficients in [17] and to the case of the heat
operator on the Heisenberg group in [19]. The proofs are based on either explicit expression of
Green functions or strong Gaussian estimates.
Here the diffusion {(g ,ξ )} is neither sub-elliptic nor (purely) parabolic and the Green
t t t≥0
functionisnotknownexplicitly. Nevertheless,consideringtheDudley’sdiffusion(i.etheprojection
of {g } onto Hd×R1,d), by means of some time-changes, we are able to recoverto a parabolic
t t≥0
situation with a generator of type
d ∂2 ∂ ∂ ∂
ai,j(x) +ai(x) +bi(x) + ,
∂x ∂x ∂x ∂y ∂t
i j i i
i,j=1
X
at (x,y,t) in a local chart U ⊂Rd×Rd×R of Hd×R1,d, where ai,j (definite positive), ai and bi
are given explicitely.
ThisgeneratorisakindofgeneralizationofaKolmogorovparabolictype operators(forwhich
bi(x)=x ). Intherecentsworks[20],[11]and[22]onecanfindsomelocalGaussianestimatesfor
i
the Greenfunctionsofthoseoperators. But,eveninthesimplestcasewhereai,j areconstantand
the Green function is explicit, no Wiener criterion is known. However let us mention [18] where
the authors studied level sets of the Green function and prove an Harnack inequality in the case
of Kolmogorovoperators with constant coefficients.
Thus,itseemsthatwearefarfromreachingafullWienercriterionfor{(g ,ξ )} ,orevenfor
t t t≥0
Dudley’sdiffusion. Inthisarticle,weneverthelessprovideaweakWienercriterionfor{(g ,ξ )} ,
t t t≥0
which concerns sets into some homogeneous cones and we deduce a Poincar´e cone sufficient con-
dition for thinness. We use the same technics as in [7], extracting the information contained in
the infinitesimal homogeneity, by comparing the diffusion to a scale invariant “tangent process”.
In Section 2, using the estimates of [23] and the stochastic Taylor formula of [9], we adapt the
proof of [7] to the context of a general diffusion satisfying the weak Ho¨rmander condition. We
obtain Theorem 1, which provides some information about the singularity of the Green function.
There are not more technical difficulties than in [7], and the drift X is hidden in the notation.
0
EvenifthisresultisfarfromgivingafulldescriptionofthebehavioroftheGreenfunctiononthe
diagonal for a general hypoelliptic diffusion, it is nevertheless sufficient to obtain, in the Section
3, a weak Wiener criterion and a Poincar´econe condition for {(g ,ξ )} in the Poincar´egroup.
t t t≥0
2
2 Estimates of the Green function for a general hypoelliptic
diffusion.
This sectionis written in a generalsetting and we consider a diffusion in anopen set ofa smooth
manifold Mn of dimension n, solution of the SDE
m
dx = X (x )◦dBi+X (x )dt, (1)
t i t t 0 t
i=1
X
and generated by
m
1
X2+X .
2 i 0
i=1
X
The vector fields (X ,X ,··· ,X ) will satisfy the weak Ho¨rmander conditions for hypoellip-
0 1 m
ticity:
∀x∈Mn, Lie(X ,...,X ,X )| =T Mn, (H)
1 m 0 x x
This insures the local existence of a smooth Green function, denoted by G(x,y).
We shall show the following result, which is the theorem (1.9) of [7] written for a general
hypoelliptic diffusion (non necessarily sub-Riemanian):
Theorem 1. Let x be fixed. We have
1
lim sup G(x,y)|y|Q(x)−2− g(x) 0,θ (y) =0.
ε→0|y|x<ε(cid:12) x Jx(0) x (cid:12)
(cid:12) (cid:0) (cid:1)(cid:12)
(cid:12) (cid:12)
Here Q(x) is some homogen(cid:12)eous dimension and |·|x some homog(cid:12)eneous norm associated to
(X ,X ,··· ,X )atx. Theangularvariableθ (y)istheprojectionofyontotheunithomogenous
0 1 m x
sphere centered at x and the non null Jacobian J (0) appears with a change of variable. The
x
function gx(0,·) is the Green function of some “tangent process” associated to the diffusion.
When the diffusion satisfies the strong Ho¨rmander condition, as in [7], this function is strictly
positive but in the general case it can vanish on a part of the unit homogeneous sphere. This is
the case for the diffusion in the Poincar´egroup.
Letusbriefly sketchanidea ofthe proof. Thewayhowthe {X } generateT Mn yields
i i=0···m x
a family of dilatations of T Mn, denoted by T for λ > 0. Then we consider some rescaled
x λ
diffusions in T Mn ≃ Rn defined by v(x,ε) := T (x ). This amounts to zooming x more and
x t 1/ε ε2t t
more as ε → 0. Then, using the stochastic Taylor formula of Castell ([9]), a “tangent process”
u(x) appearsinthe firstterm ofthe Taylorexpansionatx; we haveindeedv(x,ε) =u(x)+εR(ε,t)
t t t
where the term R(ε,t) is bounded in probability when ε → 0. We finish by using the estimates
obtainedby Nagel,SteinandWaingerin [23](Corollaryp114),toshow thatthe Greenfunctions
G(x,ε) of the rescaled diffusions v(x,ε) converge uniformly on compact set, when ε → 0, towards
the Green function gx of u(x):
t
Proposition 1. For all compact set K ⊂Rn\{0} we have:
sup G(x,ε)(0,u)−g(x)(0,u) −→0.
u∈K ε→0
(cid:12) (cid:12)
(cid:12) (cid:12)
In fine, Theorem 1 is obtained(cid:12) by taking ε = |y| an(cid:12)d expressing G(x,ε)(0,u) in terms of
x
G(x,y).
2.1 Notation and hypothesis.
We fix the notation and provide assumptions on the geometry induced by the {X } .
i i=0···m
For every multi-index J ∈{0,...,m}l, we denote by:
• |J| the length l of J.
• kJk the weight of J:
kJk:=|J|+Number of zeros in J.
3
• XJ :=[X ,[X ,[...,[X ,X ]]...] et X(j) =X (for |J|=1) .
j1 j2 jl−1 jl j
• For B = (B1,...,Bl) a Rl-Brownian motion, we set B0 = t and denote by BJ the
t t t t t
Stratonovich iterated integral:
BJ := ◦dBj1···◦dBjl,
t t1 tl
Z∆lt
where ∆l ={(t ,...,t ); 0<t <···<t <t}.
t 1 l 1 l
• For σ a permutation of {1,··· ,l}, we set J ◦σ = (j ,...,j ) and denote by e(σ) =
σ(1) σ(l)
Card{j ∈{1,...,l−1};σ(j)>σ(j+1)} the number of errors in ordering σ.
• We denote by cJ the linear combination of Stratonovich iterated integrals:
t
cJ := (−1)e(σ) BJ◦σ−1.
t |J|2Ce(σ) t
σ∈XS|J| |J|−1
• For some smooth vector field X, we denote by exp(sX)(x ) the solution at time s of the
0
ordinary differential equation :
du
=X(u(s))
ds

u(0)=x .
 0
We need to consider:

C (x)=vect XJ(x); kJk≤i .
i
Since the {Xi}i=0···m satisfy the Ho¨rmander c(cid:8)ondition (H), we(cid:9)denote by r(x) the smaller integer
such that C (x)=T Mn.
r(x) x
Denote by B = (J ,...,J ) a family of multi-indexes such that (XJ) is a trian-
1 n J∈B
gular basis of T Mn. This means that for j ≤ r, {XJ; J ∈ B, kJk ≤ j} is a basis of C and
x j
thus dimC =Card{k,kJ k≤j}.
j k
For any multi-index L there exist smooth functions aL(x) such that:
J
XL = aLXJ.
J
J∈B
X
We denote by Q(x) the graded dimension at x:
r(x)
Q(x)= i×(dimC (x)−dimC (x))
i i−1
i=1
X
r(x)
= i×Card{k,kJ k=i}.
k
i=1
X
We will make the following assumptions on the family {X }:
i
i )WeassumethatthegeometryoftheLiebracketsisconstant,thismeansthatthedimC (x)
i
are locally constant for i∈N∗. Thus r et Q are locally constant too.
ii ) r ≥2
iii ) dimC −dimC (=Card{k; kJ k=i})≥1, ∀i=2,··· ,r
i i−1 k
iv ) dimC (=Card{k; kJ k=1})≥2.
1 k
Hypothesis ii) iii) and iv) are technical and ensure that the dimensions are large enough so that
x leaves any bounded domain within an almost surely finite time. This is needed to justify the
t
finiteness of integrals which appear in the proof of Proposition 1. Moreover, it is easy to check
that this is satisfied for the relativistic diffusion in the Poincar´e group.
There exists a neighborhood W of 0 in Rn such that the map:
n
ϕ : u7→exp u XJi (x)
x i
!
i=1
X
4
is a smooth diffeomorphism from W onto ϕ (W).
x
Let be U ⊂V ∩ϕ (W) a neighborhood of x. For y ∈U we define by
x
1
Q Q
r 2k
|y| := u2 ,
x   i 
Xk=1 i,kXJik=k
 
   
the homogeneous norm at x of y = ϕ (u). For (u ) ∈ Rn, we set |u| := |ϕ (u)| , and
x i i=1···n n x x
denote by kuk is the euclidian norm.
During the proof we will use homogenous norm at different points near of the reference point
x. For this, we use that ϕ depends continuously on x, and we can take U small enough so that
x
every ϕ−1 :U →ϕ−1(U), where y ∈U, be a diffeomorphism.
y y
Thus, |z| is well defined for z,y ∈U.
y
This norm is homogenous with respect to the dilatations Tε :u7→(εkJikui)i=1...n; this means
that:
|ϕ ◦T (u)| =ε|ϕ (u)| .
x ε x x x
2.2 Homogeneous norm and estimations of [23]
WebrieflyrecalltheresultsobtainedbyNagel,SteinandWaingerin[23]relatingtotheestimation
of the Green function, in terms of the pseudo-metric ρ associated to the X defined by:
i
ρ(y,z):=inf{δ >0; ∃ϕ∈C(δ) tq ϕ(0)=y et ϕ(1)=z}, (2)
where C(δ) is the set of absolutely continous functions such that almost everywhere:
d
ϕ′(t)= a (t)XJi(ϕ(t)),
i
i=1
X
with |ai(t)| < δkJik. The Corollary p 114 of [23] provides that, when n ≥ 2, for any K compact
set in U ×U, there exists C >0 such that :
ρ2(y,z)
∀y,z ∈K, |G(y,z)|≤C (3)
Vol(B(y,ρ(y,z)))
and for J =(j ,··· ,j ):
1 k
ρ2−kJk(y,z)
∀y,z ∈K, |X ···X G(x,y)|≤ . (4)
j1 jk Vol(B(y,ρ(y,z)))
We have the following property:
Proposition 2 (Equivalence between homogenous norm and pseudo-metric). There exists a
neighborhood U˜ ⊂ U of x such that for every compact set K ⊂ U˜, we can find C ,C > 0
1 2
such that:
∀y,z ∈K, C ρ(y,z)≤|z| ≤C ρ(y,z). (5)
1 y 2
Proof. Set:
ρ (x,y)=inf{δ >0; ∃ϕ∈C (δ) tq ϕ(0)=x etϕ(1)=y}, (6)
2 2
where C (δ) is the set of smooth functions ϕ such that:
2
n
ϕ′(t)= a XJi(ϕ(t)), (7)
i
i=1
X
with constants ai such that |ai|≤δkJik.
5
Let z ∈U; there exists a unique vector (u ) such that:
i i=1···n
n
z =exp u XJi (y),
i
!
i=1
X
and, if ϕ(t)=exp ni=1tuiXJi (y) we have:
(cid:0)P (cid:1) n
ϕ′(t)= u XJi(ϕ(t))
i
i=1
X
and |u |≤|z|kJik. So ρ (y,z)≤|z| .
i y 2 y
Moreover,for some z,y ∈U, there is a unique function ϕ such that ϕ′(t)= ni=1aiXJi(ϕ(t))
withThϕu(s0,)i=f δy<an|zd|yϕ,(t1h)e=n δzk:Jtikhi<s i|suti|7→anedxpC(2(δni)=i1steumiXptJyi)a(yn)d. we have ρ2(y,z)P=|z|y . To finish
P
the proof, we use Theorem 2 of [23] which shows that ρ and ρ are locally equivalent.
2
Using (3) and the previous proposition we obtain:
Proposition 3 (a priori estimates). For any small enough compact set U, we can find C > 0
such that:
C
∀y 6=z ∈U, |G(y,z)|≤ .
|z|Q−2
y
Moreover for J =(j ,··· ,j ) we have:
1 k
C
∀y 6=z ∈U, |X ···X G(y,z)|≤
j1 jk |z|Q−2+kJk
y
Proof. This is immediate, remarking that:
Vol(B(y,|z| ))= du=|z|Q du.
y y
Z|u|d<|z|y Z|u|d<1
Proposition 4 (Triangular inequality and comparaison with a euclidian norm). For any small
enough compact set U, we can find c >0 such that for any t,y,z ∈U:
0
|y| ≤c (|z| +|z| ). (8)
t 0 t y
Moreover, there exist constant c′,c′′ >0 such that for any y,z ∈U:
c′kϕ−1(z)k≤|z| ≤c′′kϕ−1(z)k1/r. (9)
y y y
Proof. This is a consequence of the local equivalence between ρ and |·| , and of the proposition
x
(1.1) p 107 and of (iii’) p 109 in [23].
2.3 Rescaled diffusions and tangent process
Zooming on the trajectory x in a neighborhood U of x we introduce the rescaled diffusions
t
v(x,ε) := T ◦ϕ−1(x ). It is a Rn-diffusion which lies in a neighborhood U˜ε := T ◦ϕ−1(U)
t 1/ε x ε2t 1/ε x
of 0 and is defined up to the time τ :=τ/ε, τ being the exittime for x fromU. We denote by
ε t
G(x,ε)(u,v) theGreen function ofv(x,ε) ,definedforu6=v ∈U˜ε. WedenotebyJ :=|Jac(ϕ )|
t x x
the Jacobian of ϕ on U˜ε. A direct computation shows that:
x
G(x,ε)(u,v)=εQ−2J (T (v))G(ϕ ◦T (u),ϕ ◦T (v)). (10)
x ε x ε x ε
6
This can be rewriten for y ∈U as:
1
G(x,y)= G(x,ε)(0,T ◦ϕ−1(y)). (11)
J (ϕ−1(y))εQ−2 1/ε x
x x
We can take U small enough so that it be a chart of Mn and we consider, via some choice
of coordinates, x and X as vectors in Rn. We can use Theorem 4.1 of [9] and write the Taylor
t i
expansion:
r
x =exp εkcLXL +εr+1R˜(ε,t)
ε2t  t 
kX=1L,kXLk=k
 
r d
=exp εkcL aLXJi +εr+1R˜(ε,t)
 t Ji 
kX=1L,kXLk=k Xi=1
 
n
=exp εkLkaLcL XJi +εr+1R˜(ε,t),1 (12)
  Ji t 
Xi=1 L,kJikX≤kLk≤r
   
where the remainder term R˜(ε,t) is bounded in probability. Then, composing by T ◦ϕ−1 we
1/ε x
obtain:
v(x,ε) =T ◦ϕ−1(x ) (13)
t 1/ε x ε2t
= εkLk−kJikaLcL +εR¯(ε,t) (14)
 Ji t
L,r≥kXLk≥kJik i=1···n
 
=u(x)+ε εkLk−kJik−1aLcL +R¯(ε,t) , (15)
t  Ji t 
L,r≥kXLk>kJik i=1···n
  
where u(x) is a Rd-valued process, called tangent process, defined by:
t
u(x) := aLcL .
t  Ji t
L,kLXk=kJik i=1···n
 
Considering the free r-nilpotent Lie algebra with m-generators we can show, as in Proposition
(3.2) of [7], that u(x) is a linear projection of an hypoelliptic diffusion in the associated nilpotent
t
Lie group. In [7] the authors deduce (corollary (3.6)) that u(x) admits a smooth density with
t
respect to the Lebesgue measure, but this does not necessarily hold in our context. Nevertheless
we can prove, using the linear projection, that u(x) admits a smooth Green function, denoted by
t
g(x)(0,u). This means that we have, for every φ∈C0(Rn):
b
+∞
E φ(u(x))dt = φ(u)g(x)(0,u)du.
0 t
(cid:20)Z0 (cid:21) Zu∈Rd
Moreoversince u(x) has the same law as T (u(x)) we deduce that (since v =T (u)⇒dv =εQdu):
ε2t ε t ε
g(x) 0,T (u) =εQ−2g(x)(0,u). (16)
1/ε
For ε=1/|y| we obtain: (cid:0) (cid:1)
x
1
g(x)(0,ϕ−1(y))= g(x)(0,θ (y)). (17)
x |y|Q−2 x
x
Here θ (y):=T (ϕ−1(y)) is an angular variable in the unit sphere S of T Mn for
x 1/|y|x x x x
the homogenous norm |·| .
n
1Recall that (XJ)J∈B is a triangular basis and that we haveaLJ =0 when kLk<kJk.
7
2.4 Proof of Theorem 1
Considering (17) and (11) and taking K = S and ε = |y| , Theorem 1 follows easily from
x x
Proposition 1.
Now we presentthe proofofProposition1. As in [7] we begin by showinga weak convergence
of G(x,ε)(0,·) to gx(0,·):
Proposition 5. Let K be a compact set in Rn\{0} and f be a smooth function supported on K.
We have:
lim f(u)G(x,ε)(0,u)du− f(u)g(x)(0,u)du =0.
ε→0
(cid:12)Z Z (cid:12)
(cid:12) (cid:12)
Proof. By definition of the(cid:12)Green functions we have: (cid:12)
(cid:12) (cid:12)
τε +∞
f(u)G(x,ε)(0,u)du− f(u)g(x)(0,u)du = E f(v(x,ε))dt− f(u(x))dt .
t t
(cid:12)Z Z (cid:12) (cid:12) (cid:20)Z0 Z0 (cid:21)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
No(cid:12)w fixing T >0, we can decompose the term(cid:12) (cid:12)τεf(v(x,ε))dt− +∞f(u(x))dt to: (cid:12)
0 t 0 t
T R τε R +∞
1 f(v(x,ε))−f(u(x)) dt+1 f(v(x,ε))dt− f(u(x))dt
T≤τε t t T≤τε t t
Z0 (cid:16) (cid:17) ZT ZT
τε T
+1 f(v(x,ε))dt−1 f(u(x))dt.
T>τε t T>τε t
Z0 Z0
Thus we have the inequality:
τε +∞ T
E f(v(x,ε))dt− f(u(x))dt ≤E 1 |f(v(x,ε))−f(u(x))|dt
t t T≤τε t t
(cid:12) (cid:20)Z0 Z0 (cid:21)(cid:12) " Z0 #
(cid:12) (cid:12)
(cid:12) τε +(cid:12)∞
(cid:12)+E 1 |f(v(x,ε))|dt +E (cid:12) |f(u(x))|dt +2kfk TP(T ≥τ ).
T≤τε t t ∞ ε
(cid:20) ZT (cid:21) (cid:20)ZT (cid:21)
Now,takingfirstthe limsup andsecondlythe limsup ,Proposition5followsfromthe
ε→0 T→∞
following facts:
Fact 1. For T >0 fixed,
T
limE 1 |f(v(x,ε))−f(u(x))|dt =0.
ε→0 " T≤τεZ0 t t #
Fact 2.
+∞
lim E |f(u(x))|dt =0.
t
T→+∞ (cid:20)ZT (cid:21)
Fact 3.
τε
liminflimsupE 1 |f(v(x,ε))|dt =0.
T→+∞ ε→0 (cid:20) T≤τεZT t (cid:21)
• The proof of the fact 1 is very similar to the proof of Proposition (2.12) in [7].
We use the Taylor expansion (15) which can be written v(x,ε) =u(x)+εR(x)(ε,t), where:
t t
R(x)(ε,t):= εkLk−kJik−1aLcL +R¯(ε,t).
 Ji t
L,r≥kXLk>kJik i=1···n
 
Theorem 4.1 and Proposition P and P in [9] ensure that there exist α>0 and c>0 such
1 2
that for all R>c:
Rα
limP sup kR(x)(ε,s)k>R, T <τ ≤exp − .
ε
ε→0 (cid:18)0≤s≤T (cid:19) (cid:18) cT (cid:19)
8
Hence for η >0 we can find ε >0 and R>0 such that for all ε<ε
0 0
η
P sup kR(x)(ε,s)k>R, T <τ ≤ .
ε
2Tkfk
(cid:18)0≤s≤T (cid:19)
So, decomposing the expectation on the event sup kR(x)(ε,s)k>R, T ≤τ we obtain
ε
(cid:26)0≤s≤T (cid:27)
that:
T
E 1 |f(v(x,ε))−f(u(x))|dt ≤η+εTRkDfk .
T≤τε t t ∞
" Z0 #
Hence for all η >0 we have:
T
limsupE 1 |f(v(x,ε))−f(u(x))|dt ≤η,
T≤τε t t
ε→0 " Z0 #
and Fact 1 is proved.
• To prove Fact 2, it suffices to show that:
+∞
E 1 (u(x))dt <∞, (18)
B(0,ρ) t
(cid:20)Z0 (cid:21)
whereB(0,ρ)istheballofradiusρforthehomogenousnorm|·| . Theproofof (18)issimilar
n
totheproofofProposition(3.11)of[7]anditremainstoshowthat 1 du˜<∞
p−x1(B(0,ρ)) |u˜|Q˜−2
N
whereN isthedimensionofthefreer-nilpotentLiealgebrawithmR-generatorsG(m,r),and
p the linear projection which maps the diffusion in G(m,r) to the tangent process and Q˜
x
the homogenous dimension of the diffusion in G(m,r). The hypotheses i), ii) and iii) on the
dimension ensure the finiteness of the integral, using Lemma (A.7) of [7].
• The proof ofFact 3 is very similar to the proofof Proposition(4.1) of [7], andwe provetwo
lemmas to conclude. Denote by µ(x,ε) the measure whose density is 1 with respect to
T T<τε
the law of v(x,ε). By the Markov property we have:
T
τε
E 1 |f(v(x,ε))|dt = dµ(x,ε)(u) G(x,ε)(u,v)|f(v)|dv.
0(cid:20) T<τεZT t (cid:21) ZT1/ε◦ϕ−x1(U) T ZT1/ε◦ϕ−x1(U)
For u,v ∈T ◦ϕ−1(U), we set ux :=ϕ ◦T (u) and vx :=ϕ ◦T (v). 2
1/ε x ε x ε ε x ε
By (10) we have:
G(x,ε)(u,v)=εQ−2J (T (v))G(ux,vx),
x ε ε ε
and using Proposition 3 we can find C >0 such that for all ε>0:
CεQ−2
G(x,ε)(u,v)|f(v)|dv ≤ dv,
ZT1/ε◦ϕ−x1(U) ZB(0,ρ) |vεx|uQxε−2
where ρ is large enough, so that the support of f be included in B(0,ρ).
As in [7] we show:
Lemma 1. For all R>0 there exist ε >0 and c>0 such that for all ε<ε and u∈Rn
0 0
such that kuk≥R we have:
εQ−2
dv ≤c.
ZB(0,ρ) |vεx|uQx−2
ε
Moreover,
εQ−2
lim dv =0,
kuk→∞ZB(0,ρ) |vεx|uQx−2
ε
uniformly with respect to ε.
2Be carefull: uxε is a vector of Rn and uε(x) denotes the tangent process at time ε.
9
Proof of Lemma 1. For u = 0, this means ux = x, εQ−2 = 1 = 1 . Lemma
ε |vεx|uQxε−2 |ϕx(v)|xQ−2 |v|nQ−2
(A-1) of [7] ensures that dv is bounded by a constant depending only on ρ. The
B(0,ρ) |v|Q−2
d
family of diffeomorphismsRϕ−y1 depends smoothly on y ∈U. So we can find a neighborhood
U of x, a constant C > 0 and some ε > 0 such that for all y ∈ U and for all ε < ε ,
x 0 x 0
B(0,ερ)⊂ϕ−1(U)∩ϕ−1(U) and for all w ∈B(0,ερ), |ϕ−1◦ϕ (w)| ≥C|w| .
x y y x n n
So we have:
εQ−2 εQ−2 εQ−2
sup dv = sup dv = sup dv
y∈UxZB(0,ρ) |vεx|yQ−2 y∈UxZB(0,ρ) |ϕ−y1(vεx)|nQ−2 y∈UxZB(0,ρ) |ϕ−y1◦ϕx◦Tε(v)|nQ−2
εQ−2 dv
≤C˜ dv ≤C˜ . (19)
ZB(0,ρ) |Tε(v)|nQ−2 ZB(0,ρ) |v|nQ−2
Now choose ε > 0 such that for all ε < ε and for all u ∈ Rn such that kuk ≤ R we have
1 1
ux ∈U . Using(19),thereisaconstantcsuchthatforallε<min(ε ,ε )andforallu∈Rn
ε x 0 1
with kuk≤R:
εQ−2
dv ≤c.
ZB(0,ρ) |vεx|uQx−2
ε
This is the first point of the lemma. The second point will follow from (8) and (9) of
Proposition 4. For kuk large enough such that 1|ux| −|vx| >0 we have:
c0 ε x ε x
εQ−2 εQ−2 1
dv ≤ dv ≤ dv
ZB(0,ρ) |vεx|uQxε−2 ZB(0,ρ) c10|uxε|x−|vεx|x Q−2 ZB(0,ρ) c10|ϕ−1(u)|x−|ϕ−1(v)|x Q−2
(cid:16) (cid:17) (cid:16) 1 (cid:17)
≤ dv.
Q−2
ZB(0,ρ) c′kuk−ρ
c0
(cid:16) (cid:17)
This inequality ensures that the convergence is uniform in ε.
Let return to the proof of Fact 3.
By the previous lemma we obtain:
τε CεQ−2
E 1 |f(v(x,ε))|dt ≤cµ(x,ε)(B(0,R))+ sup dv.
0(cid:20) T<τεZT t (cid:21) T u,kuk≥RZB(0,ρ) |vεx|uQx−2
ε
−→ 0
R→+∞
| {z }
To end the proof of Fact 3 it remains to prove:
Lemma 2. For all R>0,
liminflimsupµ(x,ε)(B(0,R))=0.
T
T→+∞ ε→0
Proof of Lemma 2. By definition of µ(x,ε) we have,
T
µ(x,ε)(B(0,R))=E 1 1 (v(x,ε)) .
T 0 T<τε B(0,R) T
h i
Let χ be a smooth function which is equals to 1 on B(0,R) and which is supported on
B(0,R+1). We have
µ(x,ε)(B(0,R))≤E 1 χ(v(x,ε)) .
T 0 T<τε T
h i
Using the Taylor expansion of v(x,ε), as in the proof of the fact 1, we obtain:
T
E 1 χ(v(x,ε)) −→E χ(u(x)) .
0 T<τε T ε→0 0 T
h i h i
10