Invariant measures for intermittent transport

Michel Zinsmeister

Description

Niveau: Supérieur, Doctorat, Bac+8
Invariant measures for intermittent transport Athanasios BATAKIS and Michel ZINSMEISTER Abstract: We are interested in the existence and properties of limits of invariant measures for Brownian diffusions started at distance from the boundary of a given domain and stopped when they hit back this boundary, when goes to 0. 1 Introduction The motivation of the following work has its origin in experimental physics. Some long molecules are solvable in a liquid (for instance imogolite in water or DNA in lithium) and the molecules forming the liquid show an intermittent dynamics, alternating diffusion in the bulb and adsorption on the long molecules. For the physicist's point of view, it is very important to a knowledge have as precise as possible of the statistics of these brownian flights. In [GKL+06] a connection is established between the statistics of the long flight lengths and the geometry of the long molecules (more precisely their Minkowski dimension). This connection has been made rigorous in [BLZ11],[BZ10]. Nevertheless, the statistics of Brownian flights are depending on the distribution of the initial starting point. In all previous papers this distribution is taken uniform on the set ?? of the points at distance ? to the boundary. This choice, justified by experimental data, seemed mathematically unfounded. In fact, iteration of Brownian flights seems to have a limite steady state : uniform distribution is stationnary.

• fq ?


• let ?


• green function


• dimensions coincide


• brownian motion


• uniform capacity


• whitney dimension


• qy ?


• taken uniform

Sujets

• Ancona
• Green function
• Brownian motion

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 20
• Langue: English

Extrait

Invariant measures for intermittent transport
Athanasios BATAKIS and Michel ZINSMEISTER
Abstract: We are interested in the existence and properties of limits of invariant
measures for Brownian di usions started at distance from the boundary of a given
domain and stopped when they hit back this boundary, when goes to 0.
1 Introduction
The motivation of the following work has its origin in experimental physics. Some long
molecules are solvable in a liquid (for instance imogolite in water or DNA in lithium) and
the molecules forming the liquid show an intermittent dynamics, alternating di usion in
the bulb and adsorption on the long molecules. For the physicist’s point of view, it is very
important to a knowledge have as precise as possible of the statistics of these brownian
ights.
+In [GKL 06] a connection is established between the statistics of the long ight lengths
and the geometry of the long molecules (more precisely their Minkowski dimension). This
connection has been made rigorous in [BLZ11],[BZ10].
Nevertheless, the statistics of Brownian ights are depending on the distribution of the
initial starting point. In all previous papers this distribution is taken uniform on the set
of the points at distance " to the boundary. This choice, justi ed by experimental data,"
seemed mathematically unfounded. In fact, iteration of Brownian ights seems to have a
limite steady state : uniform distribution is stationnary.
The aim of this paper is to rigorously prove this statement (all de nitions of objects will
be reminded in the following section) .
dFor a Green domain
inR we can de ne (following [LS84], [BL96]) a random walk on
the centers of dyadique Whitney cubes in
with time-homogeneous transition probabilities
and (discrete) Green function equal a constant times the Green function of . The positive
harmonic functions associated to this Markov chain are the traces of positive harmonic
functions on the centers of Whitney cubes. The trajectories of the so-de ned random walk
are called discretized Brownian paths.
We choose any " > 0 and we consider the collectionS of all dyadique Whitney cubes"
intersecting =fx2
; dist(x;@ ) = "g. Let be a (discrete) probability measure on"
S and choose a cube Q with probability (Q)."
1QFor every discretized Brownian path started at the center of Q, we consider the last
0 Qcube Q = ofS visited by the path . The Markov chain being transient this exit time""
is well de ned and is a.s. nite. This de nes a function on the set of discrete probability
0 Qmeasures onS , that assigns to a new mesure () : (Q ) =E ." "
Theorem 1.1 For every " > 0, there exists a unique probability measure such that"
( ) = . Moreover, there exists a constant not depending on " such that for all Q2S" " "
1 1 1
:"
#S #S" "
Some mild hypothesis on the domain is needed to prove this theorem, and the last exit
time must be properly rede ned. To carry out the proofs, we will suitably discretize Brownian
motion, following [BL96] and [LS84] and apply an adapted version of the Perron-Frobenius
theorem to a nite Markov chain.
2 Backgound and Motivation
dIn the sequel,
will always denote a domain inR with compact boundary. The main tool
we need to use is the notion of Whitney cubes. We thus recall the
dProposition 2.1 (cf. [Gra08], p. 463) Given any non-empty open proper subset
of R ,
there exists a familyW of closed dyadic cubesfQg such thatj j
Q =
and the cubes Q ’s have disjoint interiorsj jj
p p
d‘(Q ) dist(Q ;@ ) 4 d‘(Q )j j j
if Q and Q touch then ‘(Q ) 4‘(Q )j k j k
d for a given Whitney cube Q there are at most 12 Whitney cubes Q ’s that touch Q .j k j
In this statement,‘(Q) stands for the side-length of the cubeQ and, for> 0, Q is the
cube of the same center and of sidelength ‘ (Q). Fork2Z, we denote byQ , the collectionk
kof Whitney cubesQ with‘(Q ) = 2 . We also recall the de nition of the Minkowski sausage:j j
for r> 0,
M =fx2
; dist(x;@ ) rgr
and
=fx2
; dist(x;@ ) = rgr
We then de ne S as the collection of Whitney cubes intersecting . Notice thatS isr r r
a nite set.
2
SDe nition 2.1 Let " > 0. We will call Brownian ight the random process F;t 0t
consisting in picking at random with equiprobability one of the dyadic Whitney cubes ofS"
and starting from the center of the cube a Brownian motiong killed once it reaches@
. Wet
denote by = infft ; F 2=
g the lifetime of this process.
t
De nition 2.2 The Minkowski dimension of K is
log (N )j2
d (K) = lim supM
jj!1
We can de ne similarly the Whitney dimension of @
as
log (W )j2
d =d (@ ) = lim sup ; (1)W W
jj!1
where W is the number of elements ofQ .j j
Under very mild conditions (see [Tri83], [Bis96], [JK82], [BLZ11]) these two dimensions
coincide. If the boundary of
has some self similarity we can moreover say that there is a
constant c> 0 such that
1 d dM M" #S c" ; (2)"
c
for all "R , where d =d (@ ).
M M
We also suppose that the domain
satis es so-called -regularity condition (see also
[JW88], [Anc86], [HK93]): there exists L> 0 such that for all x2 , if d = dist(x;@ ) <x
R then

x! (@ ) L; (3)B(x;2d )\
x
xwhere ! is the distribution law of the hitting point of Brownian motion starting atB(x;2d )\
x
x and killed when reaching the boundary of B(x; 2d )\ . This is a very mild conditionx
2(satis ed, for instance, by all domains in R with non-trivial connected boundary) that
appears frequently in related literature in various forms (for instance \uniform capacity
condition" or Hardy inequality).
The following result has been proven in [BLZ11]:
Theorem 2.3 Let " < r < R . The probability that the hitting point of F is at distance

greater than r from the starting point x is comparable to
dM d 2#S rr
(4)
#S ""
Notice that we do not assume (2) for this theorem. If we do, we have
dM d 2 d (d 2)M#S r rr
(5)
#S " ""
Aknowledgment: The authors wish to thank Alano Ancona for helpful and enlightning
discussions on the discretization of Brownian motion.
3
??????????3 Discretization of Brownian Motion
We will modify the continuous di usion process into a discrete one, with the same potential
theory. In this section,
is a Green domain,B stands for Brownian motion in , is thet

exit time (for brownian motion) of , ie. the hitting time of @ .
dIf G denotes the Green function of the domain
R and Q is a cube in , recall that
there exist a constant C such that for all y2Q
‘(Q) C‘(Q)
log G(x ;y) log ;Q
jx yj jx yjQ Q
2C depending on
R and
1 1 1
G(x;y) ;
d 2 d 2 d 2jjx yjj ‘(Q) jjx yjjQ Q
dfor domains
R , d 3. Moreover,
‘(Q) 2‘(Q)
log G (x ;y) log ;Q Q
2jx yj jx yjQ Q
and p
1 C 1 d
G (x;y) ;Qd 2 d 2 d 2 d 2jjx yjj ‘(Q) jjx yjj ‘(Q)Q Q
for d 3, G being the Green function of the cube Q.Q
We denote byN be the collection of the centers of cubes inW and we consider the
complete graphG associated. Let x 2N be the center of a Whitney cube Q2W.Q
3.1 Planar domains
We consider separately planar domains not (only) because of the recurrence of brownian
2motion inR but in order to better explain the ideas of the proof.
Let F () =fy2
; G (x ;y)g. Clearly, F () is a compact connected set, suchQ Q Q Q
that x 2 F . Furthermore, by the preceeding observations and the de nition of WhitneyQ Q
cubes we can deduce that, for big enough, F =F ()Q and that there is a constantQ Q
c < 1 not depending onQ such thatc QF , wherecQ will denote the (contracted) cube0 0 Q
centered at x but of sidelength ‘(cQ) =c‘(Q).Q
The triplet (N; F; W), where F =fF ; Q2Wg, and W =fQ ; Q2Wg is a balancedQ
Lyons-Sullivan data, de ned in [BL96]. For convienience of the reader we remind hereby the
principal facts of this paper.
41. The collection F is recurrent for Brownian motion in , ie.
P (9t< ;B 2 F ) = 1 for all x2
:x
t Q
F
2. x 2F Q, for all Q2W,Q Q
0 03. F \Q =;, for all Q =Q 2W,Q
4. there exists a constant c such that for allQ2W, any positive harmonic function h in
Q and all z2F we haveQ
1
h(x )h(z)ch(x )Q Q
c
Following [BL96] we de ne a Markov chain X onN : for y2 F = F denote byQF
(y)2N the center of the unique cubeQ =Q 2W containingy. For a path in the spacey
of brownian paths starting at y2F , letS () be the exit time of fromQ . Recursively,0 y
we de ne the stopping times R and S in the following wayn n
R () = infft>S () ; (t)2Fgn n 1
S () = infft>R () ; (t)2= Q g.n n 1 (R ())n 1
xRecall that, if V is an open set and for any x2V we denote by ! the harmonic measureV
of V at x. By our hypothesis, there exist C such that for all Q2W and all y2F ,Q
1 xQ y xd! d! Cd! : QQ QC
Let now
((R ()))n
d! ((S ()))n1 Q((R ()))n () = 1n (R ())nCd! ((S ()))nQ((R ()))n
Using these stopping times Ballmann and Ledrappier consider the probability space
N N~ ~( = [0; 1] ;P =P
);y y
~ being the Lebesgue measure in [0; 1]. For (;)2 de n