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2 Dévissage, Convergence, Equivariance and Regularity Conditions

# 2 Dévissage, Convergence, Equivariance and Regularity Conditions

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Dévissage of a Poisson Boundary Under Equivariance and Regularity Conditions

203

2.3.1 On the Dévissage Condition

The starting point of the dévissage method is that the state space E of the original

diffusion can be written as X G (resp. X G=K) in an appropriate coordinate

system, where the corresponding first projection .xt /t 0 is a sub-diffusion of

.xt ; gt /t 0 (resp. .xt ; yt /t 0 ). This “splitting property” occurs in a large number of

situations, in particular when considering diffusion processes on manifolds that

show some symmetries.

For example, any left invariant diffusion .zt /t 0 with values in a semi-simple Lie

group H can be decomposed in Iwasawa coordinates as zt D nt at kt where nt 2 N,

at 2 A, kt 2 K take values in Lie subgroups and .kt /t 0 and .at ; kt /t 0 are subdiffusions. In other words, the state space can be decomposed as the product of

X D A K and G D N. Under some regularity conditions (see e.g. [11]), it can

be shown that the Poisson boundary of the sub-diffusion .at ; kt / is trivial and that nt

converges almost-surely to a random variable n1 2 N when t goes to infinity. Thus,

our results ensure that the Poisson boundary of the full diffusion .zt /t 0 is generated

by the single random variable n1 .

Another typical situation where the dévissage condition is fulfilled is the case

of standard Brownian motion on a Riemannian manifold with a warped product

structure, a very representative example being the classical hyperbolic space Hd seen

in polar coordinates .r; Â/ 2 RC Sd 1 , i.e. X D RC and G=K D SO.d/=SO.d

1/. In that case, the radial component .rt /t 0 is a one-dimensional transient subdiffusion whose Poisson boundary is trivial and the angular component .Ât /t 0 is a

time-changed spherical Brownian motion on Sd 1 that converges almost surely to a

random variable Â1 2 Sd 1 . Again, the dévissage method ensures that the Poisson

boundary of the full diffusion is generated by the single random variable Â1 . This

example generalizes to the case of a standard Brownian motion on a rotationally

symmetric manifold, see Sect. 4.1.

The hypothesis that the first projection .xt /t 0 is a sub-diffusion of the full diffusion .xt ; gt /t 0 (resp. .xt ; yt /t 0 ) is convenient and easy to check when considering

examples. Nevertheless it is not necessary in the sense that there are cases where

the couple .xt ; gt /t 0 does not a priori satisfy the dévissage condition, but where a

simple change of coordinates allows to implement the method, see Remark 1 below

for such an example.

Finally, remark that the absolute continuity condition required when Inv..xt /t 0 /

is non-trivial, is ensured for example if the infinitesimal generator of the diffusion

process .xt /t 0 is hypoelliptic. Moreover, without loss of generality, we can suppose

in that case that the measure on .S; G / is a probability measure, see [10].

2.3.2 On the Equivariance Condition

The main hypothesis that allows to implement the dévissage scheme is the third

one i.e. the equivariance condition. To emphasize its role, let us first consider the

following example where the diffusion process .xt ; gt /t 0 with values X G D R R

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J. Angst and C. Tardif

is solution of the stochastic differential equations system:

dxt D dt C e

x2t

dBt ;

dgt D e

xt

dt;

.x0 ; g0 / 2 R

R;

(1)

where .Bt /t 0 is a standard real Brownian motion. The infinitesimal generator L of

the diffusion is hypoelliptic, so that Hypothesis 4 is fulfilled. Naturally, the process

.xt /t 0 is a one dimensional sub-diffusion of .xt ; gt /t 0 and from the Lemma 1 below,

Hypothesis 1 is also fulfilled.

Lemma 1 There exists a process .ut /t 0 that converges P.x;g/ -almost surely to a

random variable u1 in R when t goes to infinity such that for all t 0

xt D x0 C t C u t :

Moreover, the invariant sigma field Inv..xt /t 0 / is trivial.

Rt

2

Proof For all t

0, we have xt D x0 C t C ut ; where ut WD 0 e xs dBs :

R t 2x2

The martingale ut satisfies huit D 0 e s ds Ä t so that from the law of iterated

logarithm, we have almost surely xt

t=2 for t sufficiently large. In particular,

hui1 < C1 almost surely and ut is convergent. Since xt goes almost surely

to infinity with t, standard shift-coupling arguments apply and we deduce that

Inv..xt /t 0 / is trivial. Note however that the tail sigma field of .xt /t 0 i.e. the

invariant sigma field of the space-time process Inv..t; xt /t 0 / is not trivial. Indeed,

x0 C u1 D limt!C1 .xt t/ is a non-trivial shift invariant random variable.

Rt

From Lemma 1 again, the second projection gt D g0 C 0 e xs ds converges P.x;g/ almost surely when t goes to infinity to a random variable g1 in R and Hypothesis 2

is satisfied. Finally, considering the action of G D .R; C/ on itself by translation,

Hypothesis 3 is also satisfied since, for f 2 C.R R; R/ and .x; g; h/ 2 R3 we have

1

L .h f /.x; g/ D .@x f /.x; g C h/ C e

2

x2

.@2x f /.x; g C h/ C e x .@g f /.x; g C h/

D h .L f /.x; g/:

Hence, from Theorem A, the invariant sigma field Inv..xt ; gt /t 0 / coincide with

Inv..xt /t 0 /_ .g1 / D .g1 / up to P.x;g/ -negligeable sets i.e. the dévissage scheme

applies. Let us now consider a very similar process, namely the diffusion process

.xt ; gt /t 0 with values X G D R R which is solution of the new following

stochastic differential equations system:

dxt D dt C e

x2t

dBt ;

dgt D gt dt;

.x0 ; y0 / 2 R

R;

(2)

where .Bt /t 0 is again a standard real Brownian motion. With a view to apply the

dévissage method, the context seems favorable because the infinitesimal generator

L of the diffusion is hypoelliptic, .xt /t 0 is a one dimensional sub-diffusion of

Dévissage of a Poisson Boundary Under Equivariance and Regularity Conditions

205

.xt ; gt /t 0 , and gt D g0 e t converges (deterministically) to g1 D 0 when t goes to

infinity. In particular, the sigma field Inv..xt /t 0 / _ .g1 / is trivial. Nevertheless,

we have the following proposition:

Proposition 1 Let .x; g/ 2 R R with g Ô 0, then the sigma field Inv..xt ; gt /t 0 /

differs from Inv..xt /t 0 / _ .g1 / by a P.x;g/ -non-negligeable set.

Proof If g Ô 0, the sigma field Inv..xt ; gt /t 0 / is not trivial under P.x;g/ because the

process xt C log.jgt j/ converges P.x;g/ -almost surely to x0 C log.jg0 j/ C u1 which,

from the proof of Lemma 1, is a non-trivial invariant random variable.

The reason for which the dévissage method does not apply in this last example

is that Hypothesis 3 i.e. the equivariance condition is not fulfilled. Indeed, the

generator of the full diffusion writes

1

L D @x C e

2

and in general, for f 2 C.R

x2 2

@x

g@g ;

R; R/ and .x; g; h/ 2 R3 we have

1

L .h f /.x; g/ D .@x f /.x; g C h/ C e

2

Ô

1

h .L f /.x; g/ D .@x f /.x; g C h/ C e

2

x2

.@2x f /.x; g C h/

g.@g f /.x; g C h/

x2

.@2x f /.x; g C h/

.g C h/.@g f /.x; g C h/:

Remark 1 The equivariance condition is relatively strong and forces .xt /t 0 to be a

sub-diffusion (which is already supposed in Hypothesis 1). Indeed, since a function

f W X G ! R does not depend on its second variable if and only if g f D f for all

g 2 G, the equivariance condition implies that L maps C1 .X/ onto C1 .X/ (and

thus .xt / is a sub-diffusion). Nevertheless, some cases where this assumption is not

fulfilled can be solved by the dévissage method. For example, consider the diffusion

process .xt ; gt /t 0 solution of following system of stochastic differential equations

8

ˆ

ˆ

< dxt D

ˆ

ˆ

: dgt D

xt g2t

x2t Cg2t

Á

C gt dt C gt dBt ;

(3)

g3t

dt;

x2t Cg2t

where, clearly, there is no equivariance. It is yet possible to show that, almost

surely, xt escapes to infinity with t, gt converges to a random variable g1 and that

Inv..xt ; gt /t 0 / D .g1 / almost surely. Indeed the invariant sigma-field of .xt ; gt /

coincides with the one of .ut ; vt / WD .xt =gt ; log.gt // (since the map is bijective). But

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J. Angst and C. Tardif

now .ut ; vt / solves the system

8

ˆ

< dut D dt C dBt ;

ˆ

: dvt D

(4)

1

dt;

1Cu2t

and one can easily check that, for this new diffusion, all the hypotheses of the

dévissage method are now fulfilled. Therefore, applying Theorem A, we can

conclude that Inv..xt ; gt /t 0 / D Inv..ut ; vt /t 0 / D .v1 / D .g1 /.

2.3.3 On the Regularity Condition

As already noticed in the examples of the last section, the regularity condition is

automatically satisfied for a large class of diffusion processes, namely when the

infinitesimal generator L is elliptic or hypoelliptic. The role of this assumption will

be clear at the end of the proof of Theorems 1 and 2, since it allows to go to the limit

in the regularization procedure. In a more heuristical way, the regularity condition

can be seen as a mixing hypothesis which prevents pathologies that may occur when

considering foliated dynamics.

To be more precise on the kind of pathologies we have in mind, consider

the following discrete and deterministic example that was suggested to us by S.

Gouëzel. The underlying space is the product space X Y D S1 S1 where S1 is

identified to R=Z. Fix ˛ … Q, and define the transformation T W X Y ! X Y

such that T.x; y/ WD .x C ˛; y/. Now let X.x; y/ WD x and Y.x; y/ WD y be the first and

second projections and for n 0 define Xn WD X ı T n i.e. Xn .x; y/ D .x C n˛; y/ and

Yn WD Y ı T n Á Y. In this discrete time context, the resulting sequence .Xn ; Yn /n 0

plays the role of .xt ; yt /t 0 in the framework described in Sect. 2.1. The dynamics of

.Xn / does not depend on .Yn /, which is constant, and thus converges when n goes to

infinity. It is thus natural to ask if the devissage method applies or not in this context.

The answer is negative in general. To see this, for y 2 S1 , consider the probability

measure

y

WD C

X

n2Z

1

ıyCn˛ ;

1 C n2

where C is a normalizing constant and define a measure P on X

ÄZ

Z

Z

h.x; y/P.dx; dy/ WD

X Y

y2S1

Â

x2S1

h.x; y/

1

1

y .dx/ C

2

2

Y such that

Ã

yC1=2 .dx/

dy:

R

Note that the first marginal PX . / D Y P. ; dy/ of P is the Lebesgue measure hence

the invariant sigma field Inv..Xn /n 0 / is trivial under P. Since Y is T-invariant, the

invariant sigma field Inv..Yn /n 0 / is composed of events that do not depend on the

Dévissage of a Poisson Boundary Under Equivariance and Regularity Conditions

207

first coordinate x. Thus, under P, the sigma field Inv..Xn /n 0 / _ Inv..Yn /n 0 / is only

composed of events that do not depend on the first coordinate. Now consider the

sets A WD f.y C n˛; y/; y 2 S1 ; n 2 Zg and B WD f.y C 1=2 C n˛; y/; y 2 S1 ; n 2 Zg.

Both sets are invariant by the dynamics but they do depend on the first coordinate.

Hence, the global invariant sigma field Inv..Xn ; Yn /n 0 / differs from Inv..Yn /n 0 /

by some P-non negligeable events and the dévissage method does not apply here:

Inv..Xn ; Yn /n 0 / Ô Inv..Xn /n 0 / _ Inv..Yn /n 0 /:

3 Proof of the Main Result

We now give the proof of our results. To highlight the main ideas behind the proof,

we first consider the simplest case when the invariant sigma field of .xt /t 0 is trivial

and when Y D G is a finite dimensional Lie group. Then, we extend the result in the

case where the invariant sigma field of .xt /t 0 is non-trivial and finally, we consider

the homogeneous case.

3.1 Starting from a Trivial Poisson Boundary

Let us first prove the following result:

Theorem 1 Suppose that the full diffusion .xt ; gt /t 0 satisfies Hypotheses 1–4.

Suppose moreover that for all .x; g/ 2 X G, the invariant sigma field Inv..xt /t 0 /

is trivial for the measure P.x;g/ . Then the two sigma fields

Inv..xt ; gt /t 0 / and

.g1 /

coincide up to P.x;g/ -negligeable sets. Equivalently, if H is a bounded L -harmonic

function, then there exists a bounded measurable function on G such that H can

be written as H.x; g/ D E.x;g/ Œ .g1 /, for all .x; g/ 2 X G.

Proof (Proof of Theorem 1) The first step of the proof is the following lemma,

which is valid under Hypotheses 1–4 (the triviality of Inv..xt /t 0 / is not required

here). From Hypothesis 2, for all .x; g/ 2 X G, the process .gt /t 0 converges

P.x;g/ -almost surely to a random variable g1 D g1 .!/ in G.

Lemma 2 Under Hypotheses 1–4, and for all starting points .x; g/ 2 X G and

h 2 G, the law of the process h:.xt ; gt /t 0 D .xt ; h:gt /t 0 under P.x;g/ coincides with

the law of .xt ; gt /t 0 under P.x;h:g/ . In particular,

1. the law of the limit g1 under P.x;h:g/ is the law of h:g1 under P.x;g/ ;

2. for all .g; g0 / 2 G2 , the push-forward measures of both P.x;g/ and P.x;g0 / under

the measurable map ! D .! X ; ! G / 7! hg11 :! D .! X ; hg11 .!/:! G / coincide.

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J. Angst and C. Tardif

Proof (Proof of Lemma 2) The result is an direct consequence of the equivariance

Hypothesis 3. Indeed, if f 2 C1 .X G; R/ is compactly supported, from Itô’s

formula, under P.x;g/ we have for all h 2 G:

Z

f .xt ; h:gt / D .h f /.xt ; gt / D .h f /.x; g/ C

Z

D f .x; h:g/ C

0

L .h f / .xs ; gs /ds C Mt

t

h .L f / .xs ; gs /ds C Mt

0

Z

D f .x; h:g/ C

t

t

0

.L f / .xs ; h:gs /ds C Mt ;

where Mt is a martingale vanishing at zero. Otherwise, under P.x;h:g/ we have:

Z

f .xt ; gt / D f .x; h:g/ C

t

0

.L f /.xs ; gs /ds C Nt ;

where Nt is again a martingale vanishing at zero. In other words, under P.x;g/

and P.x;h:g/ respectively, both processes h:.xt ; gt /t 0 and .xt ; gt /t 0 solve the same

martingale problem, hence their laws coincide.

Let us go back to the proof of Theorem 1. From Hypothesis 2, for all starting

points .x; g/ 2 X G, the process .gt /t 0 converges P.x;g/ -almost surely to a random

variable g1 D g1 .!/ in G. We define

.x;g/

˝0

WD f! 2 ˝; lim gt .!/ existsg;

t!C1

.x;g/

.x;g/

and consider gQ 1 such that gQ 1 WD g1 on ˝0 and gQ 1 WD IdG on ˝n˝0 . Let

H be a bounded L -harmonic function. By the standard duality between bounded

invariant random variables and bounded harmonic functions, see e.g. Proposition 3.4

p. 423 of [14], there exists a bounded variable Z W ˝ ! R which is measurable with

respect to Inv..xt ; gt /t 0 /, i.e. Z is F 1 -measurable and satisfies Z.Âs !/ D Z.!/ for

all ! 2 ˝, such that for all .x; g/ 2 X G:

H.x; g/ D E.x;g/ ŒZ:

Moreover, .x; g/ 2 X

G being fixed, for P.x;g/ -almost all paths !, we have:

Z.!/ D lim H.xt .!/; gt .!//:

t!C1

The first idea here is to use the Lie group structure to condition the diffusion to

escape at a prescribed point in G. Remark that standard conditioning methods such

as Doob h-transform can not be implemented here since the law of the limit g1 is

Dévissage of a Poisson Boundary Under Equivariance and Regularity Conditions

209

not known a priori. For h 2 G, consider the random variable

Z h .!/ WD Z.h:Qg11 :!/ D Z.! X ; hQg1 .!/ 1 ! G /:

This new variable Z h can be seen as modification of the initial variable Z so that

the value of Z h .!/ is the value of Z but conditioned by the event that the G-valued

component ! G of the sample path ! does not exit at the random point g1 .!/ but

at the fixed point h. This variable is again Inv..xt ; gt /t 0 /-measurable. Indeed, since

the constant function equal to h and the random variable Z are shift-invariant, we

have

Z.h:Qg11 .Âs !/:Âs !/ D Z.Âs .h:Qg11 :!// D Z.h:Qg11 :!/:

Since Z h is bounded and is measurable with respect to Inv..xt ; gt /t 0 /, the function

.x; g/ 7! E.x;g/ ŒZ h  is also a bounded L -harmonic function. But from the second

point of Lemma 2, for all starting points .x; g; g0 / 2 X G2 , we have

E.x;g/ ŒZ h  D E.x;g0 / ŒZ h :

In other words, the harmonic function .x; g/ 7! E.x;g/ ŒZ h  is constant in g and its

restriction to X is L X -harmonic, where L X denotes the infinitesimal generator of

the sub-diffusion .xt /t 0 . Since Inv..xt /t 0 / is supposed to be trivial, we deduce that

the function .x; g/ 7! E.x;g/ ŒZ h  is constant. In the sequel, we will denote by .h/

the value of this constant. Note that h 7! .h/ is a bounded measurable function

since h 7! Z h is. The resulting function

is precisely the one appearing in the

statement of Theorem 1. By construction, .h/ is the common value, for all starting

points .x; g/ 2 X G, of E.x;g/ ŒZ h , i.e. the expectation of Z “conditioned” by the

event that ! G exit in h instead of g1 .!/. The second step of the proof consists in

considering a “smooth version” of the map h 7! Z h , that will allow us to deal with

non-countable union of negligeable sets, which is necessary if we want to mimic

the above approach replacing h by g1 .!/. So let us introduce an approximate unity

. n /n 0 on G, fix g 2 G, n 2 N and consider the “conditioned and regularized”

version Z, namely:

Z

Z h .!/ n .gh 1 / .dh/:

Z g;n .!/ WD

G

The exact same reasoning as above shows that Z g;n is bounded and measurable

with respect to Inv..xt ; gt /t 0 / so that the function .x; g/ 7! E.x;g/ ŒZ g;n  is constant.

Hence, for all g 2 G, n 2 N and .x; g/ 2 X G, there exists a set ˝ g;n;.x;g/

˝

such that P.x;g/ .˝ g;n;.x;g/ / D 1 and such that for all paths ! in ˝ g;n;.x;g/ , we have:

Z g;n .!/ D lim E.xt .!/;gt .!// ŒZ g;n  D E.x0 .!/;g0 .!// ŒZ g;n  D E.x;g/ ŒZ g;n :

t!1

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J. Angst and C. Tardif

Let D be a countable dense set in G and consider the intersection

\

˝ .x;g/ WD

˝ g;n;.x;g/ :

g2D;n2N

We have naturally P.x;g/ .˝ .x;g/ / D 1 and for ! 2 ˝ .x;g/ :

8g 2 D; n 2 N;

Z g;n .!/ D E.x;g/ ŒZ g;n :

Since the above expressions are continuous in g, we deduce that the last inequality

is true for all g 2 G. In other words, we have shown that for all g 2 G and for all !

in ˝ .x;g/ :

Z

Z g;n .!/ D E.x;g/ ŒZ g;n  D

.h/ n .gh 1 / .dh/:

G

In particular, taking g D gQ 1 .!/, we obtain that for all ! 2 ˝ .x;g/ and for all n 2 N:

Z gQ1 .!/;n .!/ D

Z

.h/ n .Qg1 .!/h 1 / .dh/:

(5)

G

Recall that the Haar measure

Z

gQ 1 .!/;n

is right invariant so that

Z

.!/ D

Z

1

Z.h:!// n .h 1 / .dh/;

Z .!/ n .g1 .!/h / .dh/ D

h

G

G

and

Z

Z

.h/ n .Qg1 .!/h 1 / .dh/ D

G

.hQg1 .!// n .h 1 / .dh/:

G

Thus, Eq. (5) is equivalent to

Z

Z

1

.hQg1 .!// n .h 1 / .dh/:

Z.h:!/ n .h / .dh/ D

G

G

Taking the integral in ! with respect to P.x;g/ on ˝ .x;g/ , we deduce that for all n 2 N:

Z

Z

1

E.x;g/ ŒZ.h:!/ n .h / .dh/ D

G

E.x;g/ Œ .hQg1 / n .h 1 / .dh/;

G

which, from Lemma 2 yields

Z

H.x; hg/ n .h 1 / .dh/ D

G

Z

E.x;hg/ Œ .Qg1 / n .h 1 / .dh/:

G

Dévissage of a Poisson Boundary Under Equivariance and Regularity Conditions

211

From Hypothesis 4, bounded L -harmonic functions are continuous, hence we can

let n go to infinity in the above expressions to get the desired result, namely:

H.x; g/ D E.x;g/ Œ .g1 /:

t

u

3.2 Starting from a Non-trivial Poisson Boundary

Let us now consider the general case when Inv..xt /t 0 / is not trivial but generated

by a random variable `1 with values in a separable measure space .S; G /. We will

prove the following result:

Theorem 2 Suppose that the full diffusion .xt ; gt /t 0 satisfies Hypotheses 1–4.

Then, for all starting points .x; g/ 2 X G, the two sigma fields

Inv..xt ; gt /t 0 / and

.`1 ; g1 /

coincide up to P.x;g/ -negligeable sets. Equivalently, if H is a bounded L -harmonic

function, there exists a bounded measurable function on S G such that H can

be written as H.x; g/ D E.x;g/ Œ .`1 ; g1 / for all .x; g/ 2 X G.

Proof The proof is very similar to the one of Theorem 1, but it requires an extra

argument to ensure the measurability of the function . So let H be a bounded L harmonic function and Z W ˝ ! R the associated bounded random variable which

is measurable with respect to Inv..xt ; gt /t 0 /. For g; h 2 G and n 2 N, we consider

the random variables

Z

Z h .!/ WD Z.h:Qg11 :!/;

Z g;n .!/ WD

Z h .!/ n .gh 1 / .dh/:

G

As in the proof of Theorem 1, the element h being fixed, the variable Z h is

bounded and Inv..xt ; gt /t 0 /-measurable, so that the function .x; g/ 7! E.x;g/ ŒZ h 

is bounded and L -harmonic. From Lemma 2, this function is constant in g and its

restriction to X is thus L X -harmonic. Hence, there exists a bounded measurable

function h W S ! R such that

8.x; g/ 2 X

G; E.x;g/ ŒZ h  D E.x;g/ Œ

h .`1 /:

(6)

By Hypothesis 1, the random variable `1 admits a density k with respect to the

reference probability measure on .S; G /, so that the last equation can be written

Z

8.x; g/ 2 X

G; E.x;g/ ŒZ h  D

h .`/k.x; `/

.d`/:

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J. Angst and C. Tardif

The difficulty here is that, a priori, the function .h; `/ 7! h .`/ is not measurable in

both variables. To deal with this difficulty, note that for any A 2 G , we have also:

Z

Ex;g Œ1`1 2A Z h  D Ex;g Œ1`1 2A

h .`1 /

1A .`/

D

h .`/k.x; `/

.d`/:

(7)

Let us fix x0 2 X and define

Qh .d`/ WD

h .`/

E.x0 ;g/ ŒZ h 

k.x0 ; `/ .d`/:

For each h, the measure Qh is absolutely continuous with respect to k.x0 ; `/ .d`/

and, by Eq. (7), the one parameter family .Qh /h is a measurable family of probability

measures. Recall that by Hypothesis 1, the measurable space .S; G / is separable,

thus Theorem 58 p. 57 of [6] applies and there exists a measurable map X W S G !

R such that X.:; h/ is a density of Qh with respect to k.x0 ; `/ .d`/, i.e. for all h 2 G

X.`; h/ D

h .`/

E.x0 ;g/ ŒZ h 

almost all `:

for

The map h 7! E.x0 ;g/ ŒZ h  being measurable, the function

Q .`; h/ WD X.`; h/E.x0 ;g/ ŒZ h 

is also measurable and for all .x; g/ 2 X G, we have E.x;g/ ŒZ h  D E.x;g/ Œ Q .`1 ; h/:

For all g 2 G, n 2 N and .x; g/ 2 X G, we thus have:

Z

E.x;g/ ŒZ h  n .gh 1 / .dh/ D

E.x;g/ ŒZ g;n  D

G

Z

E.x;g/ Œ Q .`1 ; h/ n .gh 1 / .dh/

G

ÄZ

Q .`1 ; h/ n .gh 1 / .dh/ :

D E.x;g/

G

Hence, .x; g/ 2 X

G being fixed, we obtain that P.x;g/ -almost surely

Z g;n D lim E.xt ;gt / ŒZ g;n 

t!C1

ÄZ

Q .`1 ; h/ n .gh 1 / .dh/ D

D lim E.xt ;gt /

t!C1

G

Z

Q .`1 ; h/ n .gh 1 / .dh/:

G

In other words, .g; n; .x; g// being fixed, there exists a set ˝ g;n;.x;g/

measure i.e. P.x;g/ .˝ g;n;.x;g/ / D 1 such that for all ! 2 ˝ g;n;.x;g/

Z

Z g;n .!/ D

G

Q .`1 .!/; h/ n .gh 1 / .dh/:

˝ of full

Dévissage of a Poisson Boundary Under Equivariance and Regularity Conditions

If D a countable dense set in G, we get that for all ! 2 ˝ .x;g/ WD

Z

8g 2 D; n 2 N;

213

T

g2D;n2N

˝ g;n;.x;g/ :

Q .`1 .!/; h/ n .gh 1 / .dh/:

Z g;n .!/ D

G

The above expressions being continuous in g, we can take g D gQ 1 .!/ to get

8! 2 ˝ .x;g/ ; 8n 2 N;

Z gQ1 .!/;n .!/ D

Z

Z

Q .`1 .!/; h/ n .Qg1 .!/h 1 / .dh/

G

Q .`1 .!/; hQg1 .!// n .h 1 / .dh/:

D

G

Taking the expectation under P.x;g/ , the left hand side gives :

E.x;g/ ŒZ

gQ 1 ;n

Z ÄZ

D

˝

Z h .!/ n .Qg1 .!/h 1 / .dh/ P.x;g/ .d!/

G

Z ÄZ

Z hQg1 .!/ .!/ n .h 1 / .dh/ P.x;g/ .d!/

D

˝

G

Z ÄZ

Z.h:!/ n .h 1 / .dh/ P.x;g/ .d!/

D

˝

G

Z ÄZ

Z.!/ n .h 1 / .dh/ P.x;hg/ .d!/

D

˝

Z

G

E.x;hg/ ŒZ n .h 1 / .dh/

D

G

and the right hand side

ÄZ

Q .`1 ; hQg1 / n .h 1 / .dh/ D

E.x;g/

Z

G

E.x;g/ Q .`1 ; hQg1 /

n .h

1

E.x;hg/ Q .`1 ; gQ 1 /

n .h

1

/ .dh/

G

Z

D

/ .dh/:

G

Since L -harmonic functions are continuous, letting n go to infinity, we deduce

E.x;g/ ŒZ D E.x;g/ Œ Q .`1 ; gQ 1 /:

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