Table Of ContentSelf attracting diffusions on a sphere and
5 application to a periodic case
∗
1
0
2 Carl-Erik Gauthier †
p
e September 7, 2015
S
4
] Abstract
R
P This paper proves almost-sure convergence for the self-attracting dif-
. fusion on the unit sphere
h
t t
ma dX(t)=σdWt(X(t))−aZ ∇SnVXs(Xt)dsdt, X(0)=x∈Sn
0
[ where σ>0, a<0, Vy(x)=hx,yi is the usual scalar product in Rn, and
2 (Wt(.))t>0 isaBrownianmotiononSn. Fromthisfollowsthealmost-sure
v convergence of thereal-valued self-attracting diffusion
7
t
82 dϑt =σdWt+aZ0 sin(ϑt−ϑs)dsdt,
4
0 where (Wt)t>0 is a real Brownian motion.
.
1 keywords: reinforced process, self-interacting diffusions, asymptotic pseudo-
0 trajectories, rate of convergence.
5
MSC 60K35,60G17, 60J60
1
:
v
i 1 Introduction
X
r In this paper, we are interested in the asymptotic behaviour of the solutions to
a
the stochastic differential equation (SDE)
t
dX(t)=σdWt(X(t))−a ∇SnVXs(Xt)dsdt, X(0)=x∈Sn (1)
Z0
with σ > 0, a R, (W (.)) is a Brownian motion on the n-dimensional Eu-
t t
clidean unit sph∈ere, Sn is the gradient on Sn and Vy(x) = x,y where .,.
stands for the canoni∇cal scalar product on Rn+1. h i h i
∗ThisworkwassupportedbytheSwissNationalFoundation GrantFN200020 149871/1.
†Institut de Math´ematiques, Universit´e de Neuchˆatel, Rue E´mile Argand 11, 2000
Neuchˆatel, Switzerland. Email: [email protected]
1
Themotivationforinvestigating(1)liesinthestudy ofthe longtimebehaviour
of the real-valued SDE:
t
dϑ =σdW +a sin(ϑ ϑ )dsdt, ϑ =0, (2)
t t t s 0
−
Z0
where (W ) is a real Brownian motion.
t t
Identify ϑ to (cos(ϑ ),sin(ϑ )) S1 and define d(x,y) as the square of the
t t t
∈
euclideandistancebetween(cos(x),sin(x))and(cos(y),sin(y));sothatd(x,y)=
2 2cos(x y). Deriving d with respect to x gives ∂ d(x,y)=2sin(x y).
x
− − −
Therefore, depending on the sign of a, asin(ϑ y) points forward/outward
t
−
(cos(y),sin(y)); that is (cos(ϑ ),sin(ϑ )) is attracted by (cos(y),sin(y)) if a<0
t t
and repelled from (cos(y),sin(y)) if a > 0. Hence, intuitively, (cos(ϑ ),sin(ϑ ))
t t
should turn around the circle if a > 0 and it should converge to some point if
a<0.
Concerning the case a > 0, it is proved in the preprint [3] of the author in
collaborationwith M.Bena¨ım that
Theorem 1 The law of
(cos(ϑ ),sin(ϑ ))
t t
converges to the uniform law on the circle.
Thispaperintendstoprovethattheintuitionconcerningtheattractivecase
(whena<0)istrue. Thefirstmainresultofthis paperisthe followingresult.
Theorem 2 If a<0, thereexists arandom variable X suchthat ϑ X =
t
O(t 1/2logγ/2(t)), with γ >1. ∞ | − ∞|
−
In 1995, M.Cranston and Y. Le Jan proved a convergence result in [4] in the
cases where asin(x) is replaced by ax (linear case) or a sgn(x), with a < 0
×
(constant case); this last case being extended in all dimension by O.Raimond
in [11] in 1997. A few years later, S.Herrmann and B.Roynette weakened the
condition of the profile function f around 0 and were still able to get almost
sure convergence(see [6]) for the solutionto the stochastic differentialequation
t
dϑ =σdW + f(ϑ ϑ )dsdt. (3)
t t t s
−
Z0
Rateofconvergenceweregivenin[7]byS.HerrmannandM.Scheutzow. Forthe
linear case, the optimal rate is proven and turns out to be O(t 1/2√t).
−
However, a common fundamental property of these three papers lies in the
fact that the associated profile function f is monotone.
We pointoutthatselfinteractingdiffusioninvolvingperiodicityhas already
receivedsomeattentionin2002byM.Bena¨ım,M.LedouxandO.Raimond([1]),
t
but in the normalized case; that is, when sin(X X )ds is replaced by
0 t − s
1 tsin(X X )ds. The interpretation is therefore different. While the drift
t 0 t− s R
term of (2) can be “seen” as a summation over [0,t] of the interaction between
R
2
the current position X and its position at time s and thus an accumulation
t
of the interacting force , their drift is then the average of the interacting force.
The asymptotic behaviour is then given by the following Theorem.
Theorem 3 (Theorem 1.1, [1], Bena¨ım, Ledoux, Raimond) Let (ϑ ) be a
t t>0
solution to the SDE
c t
dϑ =dW + sin(ϑ ϑ )dsdt,
t t t s
t −
Z0
with initial condition ϑ = 0 and c R. Set X = ϑ mod 2π S1 = R/2πZ
0 t t
∈ ∈
and defined the normalized occupation measure
1 t
µ = δ ds.
t t Xs
Z0
1. If c > 1, then µ converges almost surely (for the topology of weak*
t
converg−ence) tow{ard}the normalized Lebesgue measure on S1 [0,2π],
∼
λ(dx)= dx.
2π
2. If c < 1, then there exists a constant β(c) and a random variable ς
− ∈
[0,2π[ such that µ converges almost surely toward the measure
t
{ }
exp(β(c)cos(x ς))
µ (dx)= − λ(dx).
c,ς
exp(β(c)cos(y))λ(dy)
S1
R
An intermediate framework between those considered in Theorem 2 and The-
orem 3 is to add a time-dependent weight g(t) to the normalized case that
increases to infinity when time increases, but “not too fast”. In that case,
O.Raimond proved the following Theorem
Theorem 4 (Theorem 3.1, [12], Raimond) Let (ϑ ) be the solution to the
t t>0
SDE
g(t) t
dϑ =dW sin(ϑ ϑ )dsdt, (4)
t t t s
− t −
Z0
whereg isanincreasingfunctionsuchthatlim g(t)= ,thereexistspositives
t
c,t such that for t>t , g(t)6alog(t) and g→(t) =O(∞t γ), with γ ]0,1]. Set
0 0 ′ −
| | ∈
X =ϑ mod 2π.
t t
Then, there exists a random variable X in S1 such that almost surely,
µ = 1 tδ ds converges weakly towards δ ∞.
t t 0 Xs X∞
InterpreRting ϑ as an angle also provide the link between (1) and (2). Indeed,
t
for x,y Sn, one has
∈
x y 2 =2 2 x,y =2 2cos(D(x,y)), (5)
k − k − h i −
where D(.,.) is the geodesicdistance onSn. Therefore,Theorem2 followsfrom
the more general Theorem
3
Theorem 5 If a<0, there exists a random variable X Sn such that
∞ ∈
X(t) X =O(t−1/2logγ/2(t)),
k − ∞k
with γ >1 and . is the standard Euclidean norm in Rn+1.
kk
We emphasize that Theorems 1, 3 and4 areparticular casesfrom moregeneral
results proved in the respective papers (Theorem 5 in [3], Theorem 4.5 in [1]
and Theorem 3.1 in [12]).
1.1 Reformulation of the problem
From now on, we assume that a < 0 and that n is fixed. Since the values of σ
anda donotplayanyparticularrole,weassumewithoutlossofgeneralitythat
σ =1 and a= 1. Thus (1) becomes
−
t
dX(t)=σdWt(X(t))+ ∇SnVXs(Xt)dsdt, X(0)=x∈Sn (6)
Z0
where V (x)= x,y =:V(y,x). Since V satisfies Hypothesis 1.3 and 1.4 in [1],
y
h i
then(6)admitsauniquestrongsolutionbyProposition2.5in[1]. Weemphasize
that equation (2) admits a unique strong solutionbecause the function sin(.) is
Lipschitz continuous (see for example Proposition 1 in [6]). We shall begin by
clarifying the two quantities that appear in (6). Firstly, by Example 3.3.2 in
[8], we have
dW (X(t))=dB(t) X X , dB , (7)
t t t t
− h ◦ i
where stands for the Stratonovitch integraland (B(t)) is a (n+1) dimen-
t>0
◦
sional Brownian motion.
Secondly, for a function F :Rn+1 R, we have
→
∇Sn(F|Sn)(x)=∇Rn+1F(x)−hx,∇Rn+1F(x)ix; x∈Sn. (8)
Hence equation (6) rewrites
t
dX(t)=dB(t) X(t) X(t), dB(t) + (X X ,X X )dsdt, (9)
s t s t
− h ◦ i −h i
Z0
Following the same idea as in [3], we set U := tX(s)ds Rn+1 in order to
get the SDE on Sn Rn+1: t 0 ∈
× R
dX(t)=dB(t) X(t) X(t), dB(t) + n+1U (t)[e X (t)X(t)]dt
− h ◦ i j=1 j j − j
( dU(t)=X(t)dt P
(10)
withinitialcondition(X(0),U(0))=(x,0)ande , ,e standforthecanon-
1 n+1
ical basis of Rn+1. ···
A first property is
4
Lemma 6 d( U(t) 2)=2 U(t),X(t) dt, where . stands for the standard Eu-
clidean norm ikn Rn+k1. h i kk
Proof. It is a direct consequence of Itoˆ’s Formula.
The paper is organised as follow. In Section 2, we present the detailed
strategy used for proving Theorem 5 whereas the more technical proofs are
presented in Section 3.
2 Guideline of the proof of Theorem 2
Set R = U(t) and define V Sn and Θ [ 1,1] as follows:
t t t
k k ∈ ∈ −
U(t)/R if R >0
t t
V(t)= (11)
( X(t) otherwise
and
Θ = V ,X(t) (12)
t t
h i
t
Since ( V(s) Θ X(s),dB(s)) is a realvaluedmartingale,thenby the
0h − s t>0
Martingale representationTheorem by a Brownianmotion (see Theorem5.3 in
R
[5]), there exists a real valued Brownian motion (W ) such that
t t
t t
V(s) Θ X(s),dB(s) = 1 Θ2dW . (13)
h − s i − s s
Z0 Z0
p
Lemma 7 ((Θ ,R )) is solution to
t t t>
dY = 1 Y2dW +[(r + 1)(1 Y2) nY ]dt
t − t t t rt − t − 2 t (14)
( dr =Ypdt
t t
whenever R >0.
t
Proof. From Lemma 6, we deduce
dR =Θ dt. (15)
t t
Applying Itoˆ’s Formulae to U(t),X(t) gives
h i
t t
U(t),X(t) = t+ ( U(s) 2 U(s),X(s) 2)ds+ U(s) X(s),U(s) X(s), dB .
s
h i k k −h i h −h i ◦ i
Z0 Z0
t n t
= (1 X(s),U(s) ds+ ( U(s) 2 U(s),X(s) 2)ds (16)
− 2h i k k −h i
Z0 Z0
t
+ U(s) X(s),U(s) X(s),dB
s
h −h i i
Z0
5
Since U(t),X(t) =R Θ , we have by Itoˆ’s Formulae
t t
h i
1
dΘ = (d( U(t),X(t)) Θ dR ). (17)
t t t
R h −
t
Combining (13), (15) and (16), we obtain
1 n Θ2
dΘ = ( Θ +R (1 Θ2))dt tdt+ V(t) Θ X(t),dB(t)
t R − 2 t t − t − R h − t i
t t
1 n
= [(R + )(1 Θ2) Θ ]dt+ 1 Θ2dW (18)
t R − t − 2 t − t t
t
q
A first important result, whose proof is postponed to Section 3, is
Lemma 8 We have liminf Rt >1 almost-surely.
t→∞ √t
From this Lemma, we prove in Section 3
Lemma 9 Wehave that (Θt)t>0 and (Rtt)t>0 converge almost surelyto1. Fur-
thermore the rate of convergence is O(t 1logγ(t)), with γ >1.
−
From (15) and the definition of U(t), it follows from Itoˆ’s Formulae
1
dV = (X(t) Θ V )dt. (19)
t t t
R −
t
Lemma 10 V converges almost surely.
t
Proof. Since
X(t) Θ V = 1 Θ2, (20)
k − t tk − t
it follows from (19) that q
1
X(t) ΘtVt =O(t−3/2logγ/2(t)), (21)
R k − k
t
which is an integrable quantity.
We can now prove the main result.
Proof of Theorem 2.
By Lemmas 9 and 10, there exists a random variable X Sn such that
∞ ∈
lim Θ V =X . The rate of convergencefollows from the triangle inequal-
t t t
→∞ ∞
ity, (20), (21) and Lemma 9.
3 Proofs of Lemma 8 and Lemma 9
3.1 Proof of Lemma 8.
t
Set M = 2 U(s) X(s),U(s) X(s),dB . Then its quadratic variationis
t − 0h −h i si
R t
M = 4 U(s) X(s),U(s) X(s) 2ds.
t
h i k −h i k
Z0
t
= 4 ( U(s) 2 U(s),X(s) 2)ds. (22)
k k −h i
Z0
6
So, from (16) and Lemma 6, we obtain
1 1
exp( 2( X(t),U(t) + R2))=exp( 2t)exp(M M ), (23)
− h i 4 t − t− 2h ti
Since
t t t3
M 64 U(s) 2ds68 U(0) 2+s2ds=8( + U(0) 2t), (24)
h it k k k k 3 k k
Z0 Z0
M satisfies the Novikov Condition (see [10], Chapter V, section D, page 198).
t
Therefore
1
(t):=exp(M M )
EM t− 2h it
is a martingale having 1 as expectation.
Thus
1
E(exp( 2( X(t),U(t) + R2))=exp( 2t). (25)
− h i 4 t −
Consequently,
1 1
exp( 2( X(t),U(t) + R2))=exp( 2(R Θ + R2))
− h i 4 t − t t 4 t
converges almost surely to 0 by the Markov inequality and the Borel-Cantelli
Lemma. Furthermore, for all 0< c <2, there exists a random variable T such
that almost surely
1
exp( 2(R Θ + R2))6exp( ct), t>T. (26)
− t t 4 t − ∀
Hence, for t>T,
1 c
R Θ + R2 > t.
t t 4 t 2
Therefore, by choosing c=1, we obtain
R > 1+√t for t>T. (27)
t
−
This concludes the proof.
3.2 Proof of Lemma 9.
BeforestartingtheproofofLemma9,letusrecalltheDefinitionofanasymptotic
pseudotrajectory introduced by Bena¨ım and Hirsch in [2].
Definition 11 Let (M,d) be a metric space and Φ a semiflow; that is
Φ:R M M :(t,x) Φ(t,x)=Φ (x)
+ t
× → 7→
is a continuous map such that
Φ =Id and Φ =Φ Φ
0 t+s t s
◦
7
for all s,t R .
+
A continuo∈us function X :R M is an asymptotic pseudotrajectory for Φ if
+
→
lim sup d(X(t+h),Φ (X(t)))=0 (28)
h
t→∞06h6T
for any T > 0. In words, it means that for each fixed T > 0, the curve X :
[0,T] M :h X(t+h)shadows the Φ-trajectory over the interval [0,T]with
→ 7→
arbitrary accuracy for sufficiently large t.
If X is a continuous random process, then X is an almost-surely asymptotic
pseudotrajectory for Φ if (28) holds almost-surely.
Theorem 12 (Theorem 1.2 in[2]) Supposethat X([0, ))has compact closure
∞
in M and set L(X) = X([t, )). Let A be an attractor for Φ with basin
t>0 ∞
W. If X W for some sequence t , then L(X) A.
tk ∈ T k →∞ ⊂
The following result gives a sufficient condition between a SDE on R and the
related ODE when the diffusion term vanishes.
Theorem 13 (Proposition 4.6in[2])Letg :R RbeaLipschitz functionand
σ : R R R a continuous function. Assum→e there exists a non-increasing
+
function×ε:R→ R such that σ2(t,x)6ε(t) for all (t,x) and such that
+ +
→
k >0, ∞exp( k/ε(t))dt< 1. (29)
∀ − ∞
Z0
Then, all solution of
dx =g(x )dt+σ(t,x )dB
t t t t
is with probability 1 an asymptotic pseudotrajectory for the flow induced by
the ODE X˙(t) = g(X(t)). Furthermore, for all T > 0, there exist constant
C,C(T)>0, such that for all β >0,
P( sup x Φ (x ) >β)6Cexp( (βC(T))2/ε(t)). (30)
t+h h t
| − | −
06h6T
Remark 14 The same result holds if (x ) is solution to the SDE
t t>0
dx =g(x )dt+σ(t,x )dB +δ(t)h(x )dt,
t t t t t
wherehisaboundedfunctionandδisanon-negativefunctionwithlim δ(t)=
t
→∞
0. In that case, for all T >0, there exist constant C,C(T,h)>0, such that for
all β >0,
P( sup x Φ (x ) >β)6Cexp( (β sup δ(s))2C(T,h)/ε(t)). (31)
t+h h t
| − | − −
06h6T s [t,t+T]
∈
1Forexampleε(t)=O(1/(log(t))α)withα>1.
8
Proof of Lemma 9:
The proof is divided into two parts.
Proof of the convergence:
First we prove that Θ converges almost surely to 1. Recall that
t
1 n
dΘ = 1 Θ2dW +[(R + )(1 Θ2) Θ ]dt. (32)
t − t t t R − t − 2 t
t
q
Defineα(t)=(23t)23 sothatα˙(t)=α−21(t). SetZt :=Θα(t) andMt =Wα(t).
Thus (M ) is a martingale with respect to the filtration = σ W 0 6 s 6
t t t s
α(t) , whose quadratic variation at time t is α(t)= t( Gα˙(s))2d{s. |
} 0
Thenbythe TheoremofMartingaleRepresentationby aBrownianmotion(see
R p
Theorem 5.3 in [5]), there exists a Brownian motion (B(α)) adapted to ( )
t t Gt t
such that
t
M = α˙(s)dB(α).
t s
Z0
p
Then
α(t) α(t) 1 n α(t)
Z = 1 Θ2dW + (R + )(1 Θ2)ds Θ d(s33)
t − s s s R − s − 2 s
Z0 Z0 s Z0
= t αp˙(s) 1 Z2dB(α) t Rα(s)+ Rα1(s)ds n tα˙(s)Z ds,
− s s − α(s) − 2 s
Z0 Z0 Z0
p p
For y [ 1,1], let (Yy) be the solution topthe SDE taking values in [ 1,1]
∈ − t t>0 −
dYy = α˙(t) 1 (Yy)2dB(α)+[1(1 (Yy)2) nα˙(t)Yy]dt
t − t t 2 − t − 2 t (34)
( Yy =yp p
0
Wedividetheproofoftheconvergenceintwosteps. Inthefirstone,weprovefor
ally [ 1,1],Yy convergesalmostsurelyto1;andthenprovethe convergence
∈ − t
of Z to 1 in the second one.
t
Step I: Let y [ 1,1]. In order to lighten the notation, we omit the su-
perscript y in Yy∈in−this step. We start by proving that Y is an asymptotic
t t
pseudotrajectory for the flow induced by the ODE
1
x˙ = (1 x2). (35)
2 −
In order to achieve it, we use Theorem 13. Since x (1 x2) is Lipschitz
7→ −
continuous on[ 1,1]and that Z [ 1,1]for all t>0, it remains to prove the
t
− ∈ −
hypothesis concerning the noise term.
Set
3
1
ε(t):=α˙(t)=( t)−3. (36)
2
We have to show that ε(t) satisfies (29). But this is immediate because for all
k >0
∞exp( kt1/3)dt< . (37)
− ∞
Z0
9
Since Y [ 1,1]for all t>0, it is clear that the condition in Remark 14 is
t
∈ −
satisfied. Consequently,byTheorem13,(Y ) isanasymptoticpseudotrajectory
t t
of (35).
Because 1 is an attractor for the flow induced by (35) with basin ] 1,1]
{ } −
and that almost-surely Y ] 1,1] infinitely often, then
t
∈ −
lim Y =1 a.s. (38)
t
t
→∞
Step II: Our goal is to prove
P(lim Z =1)=1. (39)
t
t
→∞
Define the stopping times τ =0,
0
R 1
τ =inf(t>σ α(t) = ), j >1 (40)
j j
| α(t) 2
and p
R 3
σ =inf(t>τ α(t) = ), j >1. (41)
j j 1
− | α(t) 4
By Lemma 8, we have p
P( τ = )=1 and P(σ < τ < )=1. (42)
j j j 1
{ ∞} ∞| − ∞
j>1
[
Let start by computing P(lim Z =1, τ = ). For s [σ ,τ ], we have
t t j j j
→∞ ∞ ∈
Rα(s)+ Rα1(s) > 1.
α(s) 2
p
So, by a comparisonresult (see Theorem 1.1, Chapter VI in [9]),
P(Z >YZσj , t>0)=1. (43)
(t+σj)∧τj (t+σj)∧τj ∀
As a consequence, we have
P(lim Z =1, τ = , σ < ) > P(lim YZσj =1, τ = , σ < )
t t j ∞ j ∞ t t+σj j ∞ j ∞
→∞ = P(τ→∞= , σ < ). (44)
j j
∞ ∞
where the last equality follows from Step I. Because σ = τ = , it
j j
{ ∞} ⊂ { ∞}
follows from (42)
P(lim Z =1, τ = )=P(lim Z =1, τ = )+P(τ = , σ < ).
t j t j 1 j j
t ∞ t − ∞ ∞ ∞
→∞ →∞ (45)
10
