Table Of ContentAPPLICATION OF STOCHASTIC FLOWS TO THE INTERFACE
SDE ON A STAR GRAPH
6
Hatem Hajri (1) and Marc Arnaudon (2)
1
0
2
Abstract. Using a perturbation method of stochastic flows of kernels, we prove
r
a some results on a particular coupling of solutions to the interface SDE on a star
M graph, recently introduced in [8]. This coupling consists in two solutions which
areindependent giventhe driving Brownianmotion. As aconsequence,we deduce
4 that if the star graph contains 3 or more rays, the argument of the solution at a
2
fixed time is independent of the driving Brownian motion.
]
R
P 1. Introduction and main results
.
h Walsh Brownian motion [1] has acquired a particular interest since it was proved
t
a by Tsirelson that it can not be a strong solution to any SDE driven by a standard
m
Brownian motion, although it satisfies the martingale representation property with
[
respect to some Brownian motion [16]. Subsequently, Freidlin and Sheu [4] proved
2 that a Walsh Brownian motion X satisfy the following SDE
v
6 1
2 (1) df(Xt) = f′(Xt)dWt + f′′(Xt)dt
2
3
7
where W is the Brownian motion given by the martingale part of X and f runs over
0 | |
. an appropriate domain of functions with an appropriate definition of its derivative.
1
Freidlin and Sheu formula has led to the study of some variants of (1) in [5, 9, 8]
0
6 based on stochastic flows as well as to the development of a general framework of
1
stochastic calculus on graphs [11, 17].
:
v The purpose of the present paper is to show by the study of a simple example
i
X some particularities of stochastic differential equations on graphs and some possibly
r unexpected results about their solutions. The example studied here is the interface
a
SDE introduced in [8] and defined on a star graph G consisting of N half lines
(E ) sharingthesameorigin. ThisSDEisdrivenbyanN dimensionalBrownian
i 1≤i≤N
motion W = (W1, ,WN) and its solution X is a Walsh Brownian motion on G.
···
(1)Laboratoire de l’Intégration du Matériau au Système, Bordeaux. Email: Hatem.Hajri@ims-
bordeaux.fr
(2)Institut de Mathématiques de Bordeaux, Bordeaux. Email: [email protected]
bordeaux1.fr
1
While it moves inside E , X follows Wi so that the origin can be seen as an interface
i
at the intersection of the half lines. For N = 2, the interface SDE is identified with
dX = 1 dW1 +1 dW2
t {Xt>0} t {Xt≤0} t
which has a unique strong solution [15, 12] while for N 3, solutions of the interface
≥
SDE are only weak by the general result of [16].
The main result proved in [8] was the existence of a stochastic flow of mappings,
unique in law and a Wiener stochastic flow [13], unique up to a modification, which
solve the interface SDE. The problem of finding the flows of kernels which “inter-
polate” between these two particular flows was left open in [8]. The answer to this
question needs a complete understanding of weak solutions of this equation.
The main purpose of the present paper is to bring some light on the previous
problem by establishing some results on weak solutions of the interface SDE in the
case N 3. These results are very different from the case N = 2. Our proofs
≥
strongly rely on Tsirelson perturbation method [16] and on the existence of the
Wiener stochastic flow constructed in [8]. The present paper provides, in particular,
a direct application of stochastic flows to the study of weak solutions (see [7] for
another recent application by the same authors).
1.1. Notations.
This paragraph contains the main notations and definitions which will be used
throughout the paper.
Let (G,d) be a metric star graph with a finite set of rays (E ) and origin
i 1≤i≤N
denoted by 0. This means that (G,d) is a metric space, E E = 0 for all i = j
i j
∩ { } 6
and for each i, there is an isometry e : [0, [ E . We assume that d is the geodesic
i i
∞ →
distance on G in the sense that d(x,y) = d(x,0)+d(0,y) if x and y do not belong
to the same E .
i
For any subset A of G, we will use the notation A∗ for A 0 . Also, we define
\ { }
the function ε : G∗ 1, ,N by ε(x) = i if x E∗.
Let C2(G∗) denot→e t{he ·s·et· of a}ll continuous func∈tionis f : G R such that for all
b →
i [1,N], f e is C2 on ]0, [ with bounded first and second derivatives both with
i
∈ ◦ ∞
finitelimitsat0+. Forx = e (r) G∗,set f′(x) = (f e )′(r)andf′′(x) = (f e )′′(r).
i i i
∈ ◦ ◦
Let p , ,p (0,1) such that N p = 1 and define
1 ··· N ∈ i=1 i
P N
= f C2(G∗) : p (f e )′(0+) = 0 .
D ∈ b i ◦ i
( )
i=1
X
For f C2(G∗), we will take the convention f′(0) = N p (f e )′(0+) and
∈ b i=1 i ◦ i
f′′(0) = N p (f e )′′(0+)sothat canbewrittenas = f C2(G∗) : f′(0) = 0 .
i=1 i ◦ i D D P{ ∈ b }
We are now in position to recall the following
P 2
Definition 1.1. A solution of the interface SDE (I) on G with initial condition X =
0
x is a pair (X,W) of processes defined on a filtered probability space (Ω, ,( ) ,P)
t t
A F
such that
(i) W = (W1,...,WN) is a standard ( )-Brownian motion in RN;
t
F
(ii) X is an ( )-adapted continuous process on G;
t
F
(iii) For all f ,
∈ D
N t 1 t
(2) f(X ) = f(x)+ f′(X )1 dWi + f′′(X )ds
t s {Xs∈Ei} s 2 s
i=1 Z0 Z0
X
It has been proved in [8] (Theorem 2.3) that for all x G, (I) admits a solution
∈
(X,W) with X = x and moreover the law of (X,W) is unique. We will denote this
0
law by Q . Theorem 2.3 in [8] also states that X is a Walsh Brownian motion on G
x
and X is σ(W)-measurable if and only if N 2.
≤
Let us explain the meaning of the previous equation. Given a Walsh Brownian
motion X started from 0, we will denote from now on the martingale part of X by
| |
BX. Comparing (2) with Freidlin-Sheu formula [4] (see also [10]), shows that
N t
(3) BX = 1 dWi
t {Xs∈Ei} s
i=1 Z0
X
Thus, while it moves inside E , X follows the Brownian motion Wi.
i
Let us now introduce the following
Definition 1.2. We say that (X,Y,W) is a coupling of solutions to (I) if (X,W) and
(Y,W) satisfy Definition 1.1 on the same filtered probability space (Ω, ,( ) ,P).
t t
A F
A trivial coupling of solutions to (I) is given by (X,X,W) where (X,W) solves
(I). This is also the law unique coupling of solutions to (I) if N 2 as σ(X) σ(W)
≤ ⊂
in this case. Let us now introduce another interesting coupling.
Definition 1.3. A coupling (X,Y,W) of solutions to (I) is called the Wiener cou-
pling if X and Y are independent given W.
The existence of the Wiener coupling is easy to check. For this note there exists a
lawuniquetriplet(X,Y,W)suchthat(X,W)and(Y,W)aredistributedrespectively
as Q and Q and moreover X and Y are independent given W. It remains to check
x y
that W is a standard ( )-Brownian motion in RN where = σ(X ,Y ,W ,u t).
t t u u u
F F ≤
This holds from the conditional independence between X and Y given W and the
fact that W is a Brownian motion with respect to the natural filtrations of (X,W)
and (Y,W). The reason for choosing the name Wiener for this coupling will be
justified in Section 2.2 in connection with stochastic flows of kernels.
3
1.2. Main results.
Given a Walsh Brownian motion X on G, we define the process X by
N e−1(X )
X = 1 1 e i t
t {Xt6=0} {ε(Xt)=i} × i Np
i=1 (cid:18) i (cid:19)
X
Note that X = X if p = 1 for all 1 i N. Following the terminology used
i N ≤ ≤
in [2], the process X is a spidermartingale (“martingale-araignée”). In fact, for all
1 i N, define
≤ ≤
i i
(4) X = X if X E and X = 0 if not
t | t| t ∈ i t
Note that Xi = fi(X ), where fi(x) = |x| 1 . Applying Freidlin-Sheu formula
t t Npi {x∈Ei}
for X and the function fi shows that
1 t 1
(5) Xi = 1 dBX + L ( X )
t Np {Xs∈Ei} s N t | |
i Z0
where L ( X )isthe localat zero ofthe reflected Brownian motion X . Inparticular,
t
| | | |
i j
X X is a martingale for all i,j [1,N]. Proposition 5 in [2] shows that X is a
t − t ∈
spidermartingale.
Our main result in this paper is the following
Theorem 1.4. Suppose N 3. Let (X,Y,W) be the Wiener coupling of solutions
≥
to (I) with X = Y = 0. Then
0 0
(i) d(X ,Y ) N−2( X + Y ) is a martingale. In particular,
t t − N | t| | t|
N 2 2t
E[d(X ,Y )] = 2 −
t t
N π
r
(ii) Call gX and gY the last zeroes before t of X and Y, then for all t > 0,
t t
P(gX = gY) = 0 and P(X = Y ) = 0.
t t t t
(iii) ε(X ) and ε(Y ) are independent for all t > 0.
t t
The claim (ii) says that common zeros of X and Y are rare. It has been proved
in [8], that couplings (X,Y) to (I) have the same law before coalescence and that
coalescence occurs in a finite time with probability one. The strong Markov property
shows then that the set of common zeros of X and Y is infinite.
This theorem yields the following important
Corollary 1.5. Suppose N 3. Let (X,W) be a solution of (I) with X = 0. Then
0
≥
for each t > 0, ε(X ) is independent of W.
t
4
This corollary seems to us quite remarkable. In fact, admitting Tsirelson theorem
[16] and using (3), it can be deduced that ε(X ) is not σ(W)-measurable (actually
t
neither ε(X ) nor X are σ(W)-measurable). However, Corollary (1.5) gives a much
t t
| |
stronger result than this non-measurability. Comparing this with the case N = 2 in
which ǫ(X ) is σ(W)-measurable, shows that stochastic differential equations on star
t
graphs with N 3 rays involve interesting “phase transitions”.
Corollary 1.5≥is easy to deduce from Theorem 1.4. For this, define C = P(ǫ(X ) =
t t
i W). Since X and Y are independent given W and (X,W), (Y,W) have the same
|
law,
P(ε(X ) = i)2 = P(ε(X ) = i,ε(Y ) = i) = E[C2]
t t t t
Thus E[C ] = E[C2]12 and so there exists a constant c such that C = c a.s. Taking
t t t t t
the expectation shows that c = p .
t i
Let us now explain our arguments to prove Theorem 1.4. In Section 2.1, we prove
that for any coupling (X,Y,W) of solutions to (I), L (D) = 0 where D = d(X ,Y ).
t t t t
Next, weconsidertheWienerstochasticflowofkernelsK constructedin[8]whichisa
strong solution to the flows of kernels version of (2). Inspired by Tsirelson arguments
[16], we then consider a perturbation Kr of K,r 1. The semigroup
→
Qr(f g)(x,y) = E[K f(x)Kr g(y)]
t ⊗ 0,t 0,t
on G2 is Feller. Interesting results about the Markov process (Xr,Yr) associated to
Qr are known [2, 3, 16]. This process also converges in law as r 1 to the Wiener
→
coupling (X,Y) described above. The passage to the limit r 1 allows to deduce
→
the properties of (X,Y) mentioned in Theorem 1.4.
2. Proofs
2.1. The local time of the distance.
The subject of this paragraph is to prove the following
Proposition 2.1. Let (X,Y,W) be a coupling of two solutions to (I) with X =
0
Y = 0 and let D = d(X ,Y ). Then L (D) = 0.
0 t t t t
Proof. We follow the proof of Proposition 4.5 in [8] and first prove that a.s.
da t d D
(6) La(D) = 1 h is <
t a {Ds>0} D ∞
Z]0,+∞] Z0 s
5
By (5),
N
X = Xi = M1 +L ( X )
| t| t t t | |
i=1
X
N
Y = Yi = M2 +L ( Y )
| t| t t t | |
i=1
X
with
N 1 t N 1 t
M1 = 1 dBX, M2 = 1 dBY
t Np {Xs∈Ei} s t Np {Ys∈Ei} s
i=1 i Z0 i=1 i Z0
X X
In particular,
N 1 t N 1 t
M1 = 1 ds, M2 = 1 ds
h it (Np )2 {Xs∈Ei} h it (Np )2 {Ys∈Ei}
i=1 i Z0 i=1 i Z0
X X
and
N 1 t
M1,M2 = 1 ds.
h it (Np )2 {Xs∈Ei, Ys∈Ei}
i=1 i Z0
X
Proposition 7 [2] tells us that
t
D 1 (dM1 +dM2)
t − {ε(Xs)6=ε(Ys)} s s
Z0
t
1 sgn(M1 M2)(dM1 dM2)
− {ε(Xs)=ε(Ys)} s − s s − s
Z0
is a continuous increasing process. Consequently,
N
1
d D = 1 (1 +1 )ds C1 ds
h is (Np )2 {ε(Xs)6=ε(Ys)} {Xs∈Ei} {Ys∈Ei} ≤ {ε(Xs)6=ε(Ys)}
i
i=1
X
whereC isapositiveconstant. NotethereexistsC′ > 0suchthatD C′( X + Y )
s s s
≥ | | | |
for all s such that ε(X ) = ε(Y ). Thus, to get (6), it is sufficient to prove
s s
6
t ds
1 1 <
{Xs6=0,Ys6=0} {ǫ(Xs)6=ǫ(Ys)} X + Y ∞
Z0 | |s | s|
Let us prove for example that
t 1
(1) = X + Y 1{Xs∈E1∗,Ys∈/E1}ds < ∞
Z0 | s| | s|6
Define f(z) = z if z E and f(z) = z if not and set x = f(X ),y = f(Y ).
1 t t t t
| | ∈ −| |
Clearly
1 1 sgn(x ) sgn(y )
X + Y 1{Xs∈E1∗,Ys∈/E1} = 2| xs −y s |1{ys<0<xs}.
s s s s
| | | | | − |
As in [8], let (f ) C1(R) such that f sgn pointwise and (f ) is uniformly
n n n n n
⊂ →
bounded in total variation. Defining zu = (1 u)x +uy , we have
s − s s
t f (x ) f (y ) ds
n s n s
(1) liminf 1 | − |
≤ n Z0 {ys<0<ys} |xs −ys| 2
t 1 ds
liminf 1 f′(zu) du
≤ n {ys<0<ys} n s 2
Z0 Z0
(cid:12) (cid:12)
Writing Freidlin-Sheu formula for the function f a(cid:12)pplied t(cid:12)o X and Y shows that on
y < 0 < x ,
s s
{ }
d 1
zu = u2 +(1 u)2
dsh is − ≥ 2
Thus
1 t
(1) liminf 1 f′(zu) d zu du
≤ n {ys<0<xs} n s h is
Z0 Z0
1 (cid:12) (cid:12)
liminf f′(a) La((cid:12)zu)dadu(cid:12)
≤ n Z0 ZR n t
(cid:12) (cid:12)
So a sufficient condition for (1) to be fin(cid:12)ite is(cid:12)
sup E La(zu) <
a∈R,u∈[0,1] t ∞
(cid:2) (cid:3)
By Tanaka’s formula
t
E La(zu) = E zu a E zu a E sgn(zu a)dzu
t t − − 0 − − s − s
(cid:20)Z0 (cid:21)
(cid:2) (cid:3) (cid:2)(cid:12) (cid:12)(cid:3) (cid:2)(cid:12) t (cid:12)(cid:3)
E[(cid:12)zu zu(cid:12) ] E (cid:12) sgn(cid:12)(zu a)dzu
≤ t − 0 − s − s
(cid:20)Z0 (cid:21)
(cid:12) (cid:12)
Since x and y are two skew(cid:12)Browni(cid:12)an motions, it is easily seen that sup E[ zu
u∈[0,1] t −
zu ] < . The same argument shows that
0 ∞ (cid:12)
(cid:12)
(cid:12) t
(cid:12) E sgn(zu a)dzu
s − s
(cid:20)Z0 (cid:21)
is uniformly bounded with respect to (u,a) and this shows that (1) is finite. Finally
La(D)da is finite a.s. Since lim La(D) = L0(D), we deduce L0(D) = 0. (cid:3)
]0,+∞] t a a↓0 t
7
R
2.2. The Wiener stochastic flow of kernels solution of the interface SDE.
Stochastic flows of kernels play a crucial role in our proofs. These objects were
introduced in [13] and offer a robust method for the study of weak solutions as will
be demonstrated in this paper (see also [7]). For K, a stochastic flow of kernels on
G,
(7) Pnf(x , ,x ) = E f(y , ,y )K (x ,dy ) K (x ,dy )
t 1 ··· n 1 ··· n 0,t 1 1 ··· 0,t n n
(cid:20)ZG (cid:21)
definesaFellersemigrouponGn. Moreover (Pn) isacompatiblefamily(inasense
n≥1
explained in [13]) of Feller semigroups acting respectively on C (Gn) that uniquely
0
characterize the law of K. Conversely, it has been proved in [13] that to each family
of compatible Feller semigroups (Pn) is associated a (law unique) stochastic flow
n≥1
of kernels such that (7) holds for every n 1.
≥
Definition 2.2. (Real white noise) A family (W ) is called a real white noise if
s,t s≤t
there exists a Brownian motion on the real line (Wt)t∈R, that is (Wt)t≥0 and (W−t)t≥0
are two independent standard Brownian motions such that for all s t, W =
s,t
≤
W W (in particular, when t 0, W = W and W = W ).
t s t 0,t −t −t,0
− ≥ −
Fora familyofrandomvariablesZ = (Z ) , define Z = σ(Z ,s u v t)
s,t s≤t Fs,t u,v ≤ ≤ ≤
for all s t. The extension to flows of kernels of the interface SDE is deduced from
≤
Definition (1.1) as follows.
Definition 2.3. LetK bea stochasticflow of kernelsonG and = ( i,1 i N)
W W ≤ ≤
be a family of independent real white noises. We say that (K, ) solves (I) if for all
W
s t, f and x G, a.s.
≤ ∈ D ∈
N t 1 t
K f(x) = f(x)+ K (1 f′)(x)d i + K f′′(x)du.
s,t s,u Ei Wu 2 s,u
i=1 Zs Zs
X
When K = δ , we only say (ϕ,W) solves (I). We say K is a Wiener solution if for
ϕ
all s t, K W.
≤ Fs,t ⊂ Fs,t
Note that when K = δ , we find the interface SDE as previously defined.
ϕ
Assume (K, ) solves (I), then for all s t τ (x) = inf u s : x + i = 0
W ≤ ≤ s { ≥ | | Ws,u }
and x E , a.s K (x) = δ [8]. Since this holds for all x, we deduce that
∈ i s,t x+Wsi,t
W K for all s t. For this reason, we may only say K solves (I) as is a
Fs,t ⊂ Fs,t ≤ W
function of K.
It has been proved in [8], that given a real white noise, there exists a stochastic
W
flow of mappings ϕ such that (ϕ, ) solves (I). Filtering this flow with respect to
W
gives rise to a Wiener stochastic flow of kernels K (x) = E[δ W] (to be
W s,t ϕs,t(x)|Fs,t
more precise K (x) is defined along a dense set (s ,t ,x ) as a regular conditional
s,t i i i
{ }
8
expectation of ϕ (x ) given W and then extended to all values of (s,t,x) by a
si,ti i Fsi,ti
density argument, see Lemma 3.2 in [13]). The Wiener stochastic flow of kernels K
is unique up to modification in the sense that if K′ is another flow of kernels such
that (K′, ) solves (I) and K′ W for all s t, then for all x G and s t
W Fs,t ⊂ Fs,t ≤ ∈ ≤
a.s K (x) = K′ (x).
s,t s,t
By considering the Feller semigroups Qn(f g)(x,w) = E[K⊗nf(x)g(w+ )], one
t ⊗ 0,t Wt
can prove that stochastic flows of kernels satisfying Definition (2.3) arethe projective
limits of compatible weak solutions to the sticky equation satisfying Definition (1.1)
(see Proposition 2.1 in [6] for more details in a similar context).
Note that in the case N = 2, the Wiener flow and the flow of mappings coincide:
K = δ while K = δ if N 3 and other flows solving (I) may exist in this case.
ϕ ϕ
6 ≥
From now on, (K, ) will denote the Wiener stochastic flow which solves (I). Let
Q be the Feller semigWroup on G2 RN defined by
×
Q (f g h)(x,y,w) = E[K f(x)K g(y)h(w+ )].
t 0,t 0,t 0,t
⊗ ⊗ W
Denoteby(X,Y,W)theMarkovprocessassociatedto(Q ) andstartedfrom(x,y,0).
t t
Proposition 2.4. (X,Y,W) is the Wiener coupling solution of (I) with X = x and
0
Y = y.
0
Proof. We follow the proof of Lemma 6.2 in [8]. Let = σ(X ,Y ,W ;s t).
t s s s
F ≤
Clearly X and Y are two ( )-Walsh Brownian motions. We first prove that (X,W)
t
F
(and similarly (Y,W)) solves (I). By Freidlin-Sheu’s formula, for all f ,
∈ D
t 1 t
(8) f(X ) = f(x)+ f′(X )dBX + f′′(X )ds.
t s s 2 s
Z0 Z0
Thus for (X,W) to solve (I), it will suffice to show BX = N t1 dWi.
t i=1 0 {Xs∈Ei} s
Define
P R
(9) = f : f,f′,f′′ C (G)
1 0
D { ∈ D ∈ }
By the condition f′,f′′ C (G), it is meant here that for all i [1,N], the limits
0
∈ ∈
lim f′(x) are equal and the same holds for f′′ (not to confuse this with the
x∈Ei,x6=0
conventions taken for f′(0),f′′(0)). As f , this yields lim f′(x) = 0 for all
∈ D x∈Ei,x=6 0
i. Denote by A the generator of Q (f g) = Q (f I g) and (A) its domain,
t t
then C2(RN) (A) and for all ⊗f and g⊗ C⊗2(RN), D
D1 ⊗ 0 ⊂ D ∈ D1 ∈ 0
e
N
1 1 ∂g
A(f g)(x,w) = f(x)∆g(w)+ f′′(x)g(w)+ (f′1 )(x) (w)
⊗ 2 2 Ei ∂wi
i=1
9 X
In particular, for all f and g C2(RN),
∈ D1 ∈ 0
t
(10) f(X )g(W ) A(f g)(X ,W )ds is a martingale.
t t s s
− ⊗
Z0
On another hand, using (8), it is possible to write Itô’s formula for f(X )g(W ).
t t
Combining this Itô’s formula with (10), it comes that
N t ∂g N t ∂g
(f′1 )(X ) (W )ds = (f′1 )(X ) (W )d BX,Wi
Ei s ∂wi s Ei s ∂wi s h is
i=1 Z0 i=1 Z0
X X
Since this holds for all f and g C2(RN), by an approximation argument, we
∈ D1 ∈ 0
deduce BX,Wi = t1 ds for all i.
h it 0 {Xs∈Ei}
Now it remains to prove that X and Y are independent given W. We will check
R
that
n n
E f (X )g (Y )h (W ) = E E[f (X ) W]E[g (Y ) W]h (W )
i ti i ti i ti i ti | i ti | i ti
" # " #
i=1 i=1
Y Y
for all measurable and bounded test functions (f ,g ,h ) . Since K is a measurable
i i i i
function of , we may assume K (and so ) is defined on the same space as X
W W
and Y and that W = . By an easy induction (see the proof of Proposition 4.1
t 0,t
W
in [6]),
n n
(11) E f (X )g (Y )h (W ) = E K f (x)K g (y)h (W )
i ti i ti i ti 0,ti i 0,ti i i ti
" # " #
i=1 i=1
Y Y
From(11),wealsodeduceK f (x) = E[f (X ) W ]andK g (y) = E[g (Y ) W ].
This completes the proof. 0,ti i i ti |F0,ti 0,ti i i ti |F0,(cid:3)ti
Since (Q ) is Feller, the proposition shows in particular that (X,Y) is a Feller
t t
process on G2.
2.3. Perturbation of the Wiener flow.
We now fix a measurable function F such that if is a real white noise on RN,
V
then K = F( ) solves (I). Let and ′ be two independent real white noises on
RN and for rV> 0, define the reaWl whiteWnoise on RN
r = r +√1 r2 ′
W W − W
Let K = F( ) and Kr = F( r) and note that (K,Kr) and (Kr,K) have the
W W
same law since this is also true for ( , r) and ( r, ). Define now
W W W W
Qr(f g h)(x,y,w) = E[K f(x)Kr g(y)h(w+ )]
t ⊗ ⊗ 0,t 0,t W0,t
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