Zero noise limits using local times

0.12 MB·English

by  Dario Trevisan

Zero noise limits using local times
3
∗
1 Dario Trevisan
0
2 January 29, 2013

n
a
J

8 Abstract
2
Weconsiderawell-knownfamilyofSDEswithirregulardriftsandthe

] correspondent zero noise limits. Using (mollified) local times, we show
R which trajectories are selected. The approach is completely probabilistic
P and relies on elementary stochastic calculus only.
.
h
t 1 Introduction and results
a
m
[ For fixed γ ∈[0,1), let us consider the following ODE, in integral form:

1 t
v x(t)= sgn(x(s))|x(s)|γds t≥0
0 Z0
1
5 wheresgn(x)=I{x>0}−I{x<0}. Itis wellknownthatthere areinfinitely many
6 solutions and they are all of the form ±H (t−t ), with
γ 0
.
1
0 Hγ(s)=[(1−γ)s+]1−1γ,
3
1 for some t ≥0.
0
v: Givenε>0,letusconsiderasmallrandomperturbationoftheODEabove:
i
X dXε =sgn(Xε)|Xε|γdt+εdW
t t t t (1)
r (cid:26) Xε =0.
a 0
WeakexistenceanduniquenessinlawarethenguaranteedbyGirsanovtheorem
and Novikov condition: indeed, the random varibale K T |W |2γ is exponen-
0 s
tially integrable for any T,K > 0. Therefore one can conRsider a weak solution
Xε defined on some space Ω,A,P,(F) ,(W ) and let µε =µε be
0≤t≤T t 0≤t≤T γ
(cid:16) (cid:17)
the law of Xε, which is a probability measure on the Borel sets of C [0,T],
0
equivalent to the Wiener measure.
∗Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126, Pisa, Italy . Email:
[email protected].
1
It is also known that the family of measures µε is tight: indeed, in
γ 0<ε≤1
the case γ =0, it follows from the estimate, with(cid:0)C =(cid:1) 1,
|Xε−Xε|≤C|t−s|+ε|W −W |.
t s t s
Ingeneral,estimatingthedriftterm|x|γ ≤1+|x|,oneobtainthatwitharbitrary
high probability Xε is uniformly bounded in the interval [0,T], and so the
t
estimateaboveholds,forsomeC >0. Bytightness,therefore,onecanconsider
some sequence εn → 0 such that µεn weakly converges to some probability
measure µ (depending on γ).
To characterize µ is the prototype of zero noise problems, which appear
in many contexts: for brevity, here we refer the extended overview presented
in Chapter 1, Section 5 in [8]. Among the results already there, we remark
that the work [1] discusses a zero noise limit for some linear PDEs of transport
type related to the family of ODEs introduced above. For more recent devel-
opments, not included in [8], we mention the forthcoming article [6], where a
two-dimensional zero noise problem with discontinuous drift is solved; [4] and
[5],wherezeronoiseproblemsforperturbedODEswithrespectivelycontinuous
and measurable drifts are discussed.
However, in the context of perturbed one-dimensional SDEs, the most gen-
eral results are still those in [2], which rely on explicit estimates for exit times,
obtained by solving related PDEs. The results obtained there show that, in
the particular case introduced above, µ is concentratedonly on the trajectories
±H , which leave immediately the origin.
γ
The aim of this short paper is to provide an entirely probabilistic proofof a
concentration result for µ, in this special case as strong as that obtainable by
applying the methods in [2], but relying only on applications of Itoˆ(-Tanaka)
formula and elementary estimates for stochastic integrals. In fact, local times
andTanakaformulaappearonly inthe proofforthe specialcaseγ =0,but the
generalcase is a technical development of the simple idea exploited there, after
a suitable mollification procedure.
Beforestatingthemainresults,weremarkthefamilyofexamplesintroduced
aboveiswellstudiedinthe literatureandmuchmorecanalreadybesaidabout
the limit probability µ. In [9] and [10], large deviations estimates are proved,
by computing explicitly the density of Xε and expanding it in terms of eigen-
t
functions of a Sch¨rodingeroperator (there are currently many efforts to extend
the classical Wentzell-Freidlin large deviations theory in the case of irregular
coefficients: see e.g.[3] and the monograph[7]). Another approachis presented
in [11], where a general setting for small noise problems is introduced, using
Malliavin calculus both to prove strong existence and compactness of families
of strong solutions for the SDE. We remark that computations involving Itoˆ-
Tanaka formulas and local times appear also in these works, but they are not
used to investigate the concentration properties of µ. In the proof of Proposi-
tion 3 in [9], local times appear when manipulating the expression provided by
Girsanovtheorem, while in [11] they appear in Example 2.11,in the expression
for the Malliavin derivative of a solution Xε.
2
Theconcentrationresultforµthatwearegoingtoproveisindeedacorollary
of the following theorem.
Theorem 1.1. Given T > 0, there exist positive numbers t¯,h,α, depending
only on γ,T,ε, infinitesimal as ε→0 (and the other parameters are fixed) such
that, given any weak solution (Xε) of (1), with probability greater than
t 0≤t≤T
1−α, it holds
• either for any t¯≤t≤T, |Xε−Hγ(t)|≤h
t
• or for any t¯≤t≤T, |Xε+Hγ(t)|≤h.
t
The main feature of this result, together with its proof, is that it providesa
rigorous deduction of the following intuition, which is not evident at all in the
classical approach in [2]: as ε→0, the trajectory Xε(ω) is forced by the noise
to follow closely one of the two extremal trajectories ±Hγ, and this selection
happens in a small time interval [0,t¯]. Moreover, the quantities t¯,h,α can be
computed explicitly.
We deduce immediately the following existence and characterization result
for the zero noise limit probability µ .
γ
Corollary 1.2. The weak limit µ=lim µε exists and is given by
ε→0
1 1
µ= δ + δ . (2)
Hγ −Hγ
2 2
Proof. Since (µǫ) is tight, it is enoughto consider a convergentsubsequence
ǫ>0
µǫn and prove that its limit is given by the expression above.
For fixed t,η >0, it holds
limµǫ{ω ∈C [0,T] : ||ω(t)|−H (t)|>η}≤α,
0 γ
ε→0
whenever ε is small enough so that t¯≤ T and h ≤ η, where t¯,h,α are those
provided by the theorem above.
By lower semicontinuity of weak convergence of measures on open sets, it
holds therefore
µ(ω ∈C [0,T] : ||ω(t)|−H (t)|>η)=0,
0 γ
which entails that µ is a probability measure concentrated at most on ±H ,
γ
being t,η arbitrary.
The simmetry of the problem allows us to conclude that µ is given by (2),
since every µǫ is invariant under the transformation ω 7→−ω.
2 Proof of Theorem 1.1
2.1 Case γ = 0
Given ε>0 and a weak solution Xǫ, we write Itoˆ-Tanaka formula for the local
time at 0, with respect to the semimartingale Xε (Theorem 1.2, Chapter VI in
3
[12]), i.e.
t t
|Xε|= [sgn(Xε)]2ds+ε sgn(Xε)dW +L0[Xε] ,
t Z s Z s s t
0 0
for any t ≥ 0. Since sgn(x)2 = I and the local time process L0[Xε] is
{x6=0} t
non negative, we obtain
t t
|Xε|≥ I ds+ε sgn(Xε)dW .
t Z {Xsε6=0} Z s s
0 0
As already remarked,by Girsanov theorem, the law of (Xε) is equivalent
0≤t≤T
to the Wiener measure and therefore
Xε(ω)6=0 P⊗L-a.e. (ω,t)∈Ω⊗[0,T],
t
where L is the Lebesgue measure on the interval: indeed the same holds true
foraWienerprocessinplaceofXε. Itfollowsthat,almostsurely,for0≤t≤T,
t
I ds=t and therefore
0 {Xsε6=0}
R
t
|Xε|≥t−ε sgn(Xε)dW . (3)
t (cid:12)Z s s(cid:12)
(cid:12) 0 (cid:12)
(cid:12) (cid:12)
On the other hand, for any t≥0, dir(cid:12)ectly from (1) wr(cid:12)itten in integralform, we
deduce that
|Xε|≤t+ε|W |. (4)
t t
The estimates (3) and (4) above imply that, given η >0, the event
sup |Xε|−t >η
(cid:26) t (cid:27)
0≤t≤T(cid:12) (cid:12)
(cid:12) (cid:12)
is contained in the union
t η η
sup sgn(Xε)dW > ∪ sup |W |> .
(cid:26) Z s s ε(cid:27) (cid:26) t ε(cid:27)
0≤t≤T(cid:12) 0 (cid:12) 0≤t≤T
(cid:12) (cid:12)
On the other hand, Doob’s inequality and Itoˆ’s isometry assure that
t η η T
P sup sgn(Xε)dW > +P sup |W |> ≤2ε2 .
(cid:18) Z s s ε(cid:19) (cid:18) t ε(cid:19) η2
0≤t≤T(cid:12) 0 (cid:12) 0≤t≤T
(cid:12) (cid:12)
In order to compute t¯,h,α as required by the theorem, we fix any a, with
0 < a < 1 and put η = εa above, so that, with probability greater than 1−α,
where α=2ε2(1−a)T, it holds
sup ||Xε|−t|≤εa.
t
0≤t≤T
Then, we puth=εa andt¯=2h sothat, inthe eventabove,it holdsfor any
t¯≤t≤T,
|Xε|≥t−h≥t¯−h=h>0.
t
and in particular Xε does not change sign. Therefore,
t
4
• either for any t¯≤t≤T, |Xε|=Xε and |Xε−t|≤h,
t t t
• or for any t¯≤t≤T, |Xε|=−Xε and |Xε+t|≤h.
t t t
2.2 Case γ ∈ (0,1)
The main difficulty in this case is due to the fact that the drift term is in-
finitesimal in zero. Indeed, if we repeat the same passages as above, we obtain
that
t t
|Xε|= |Xε|γds+ε sgn(Xε)dW +L0[Xε] . (5)
t Z s Z s s t
0 0
If we simply drop the local time we cannot conclude that |Xε| grows enough
t
and the solution leaves the origin. We are going to see that the local time
termindeedcontainsexactlythe informationthatweneedtoconclude thatthe
solution with high probability moves away from zero. This, combined with the
fact that the drift drags the solution away from zero in a finite time, will lead
to the conclusion.
To extract easily this information from the local time term, we mollify the
map x7→|x| and define
1 δ
x7→|x| = ρ(y/δ)|x−y|dy,
δ δ Z
−δ
where ρ(x)∈C∞(R) is non negative, supported in [−1,1] with ρ(x)≥3/4 for
x∈[−1,1] and 1 ρ(x)dx=1.
2 2 −1
The positiveRmap defined in this way is smooth and for any x∈R, it holds
||x|−|x| |≤δ, |x|′ ≤1 and |x|′′ ≥0.
δ δ δ
Moreover, the assumption ρ(x) ≥ 3/4 for x ∈ [−1,1] entails that, when |x| ≥
2 2
δ/2, it holds |x|′ sgn(x) ≥ 1/2, while for |x| ≥ δ, it holds |x|′ sgn(x) ≥ 1.
δ δ
Finally, for |x|≤δ/2, |x|′′ ≥3/4δ.
δ
We apply Itoˆ formula to |Xε| so that, for t≥0, it holds
t δ
t t 1 t
|Xε| =|0| + |Xε|′ sgn(Xε)|Xε|γ+ε |Xε|′ dW + ε2 |Xε|′′ds. (6)
t δ δ Z s δ s s Z s δ s 2 Z s δ
0 0 0
Let us state a key estimate in form of a lemma.
Lemma 2.1. Fix a > 2γ/(1+γ). Then there are positive numbers t¯, δ, de-
pending only on γ,a,ε, infinitesimal as ε → 0 (and the other parameters are
fixed) such that, a.s. on the event
t
sup ε |Xε|′ dW ≤εa , (7)
(cid:26) (cid:12)Z s δ s(cid:12) (cid:27)
0≤t≤T (cid:12) 0 (cid:12)
(cid:12) (cid:12)
for every t, with t¯≤t≤T, it hol(cid:12)ds |Xε| ≥2δ(cid:12)+εa and |Xε|≥δ.
t δ t
5
Proof. Using the estimates for |·| and its derivatives, from (6) we obtain that
δ
t δγ 3ε2 δγ t
|Xε| ≥ I +I − ds+ε |Xε|′ dW .
t δ Z (cid:20) {|Xsε|≥δ/2}2γ+1 {|Xsε|<δ/2}(cid:18) 8δ 2γ(cid:19)(cid:21) Z s δ s
0 0
Weputδ =c ε2/(1+γ),wherec =2(γ−2)/(1+γ)isapositivenumberdepend-
1 1
ing only on γ, such that
3ε2 δγ δγ
− = .
(cid:18) 8δ 2γ(cid:19) 2γ+1
Thanks to this choice, for any t≥0, it holds
δγ t
|Xε| ≥ t−ε |Xε|′ dW .
t δ 2γ+1 (cid:12)Z s δ s(cid:12)
(cid:12) 0 (cid:12)
(cid:12) (cid:12)
In the event (7), it entails that, for 0≤t(cid:12)≤T, (cid:12)
δγ
|Xε| ≥ t−εa.
t δ 2γ+1
Weputt¯=2γ+1(2δ+2εa)/δγ,whichisimmediatelyseentobeinfinitesimal
asε→0,sinceγ <1anda>2γ/(1+γ). Withthischoicethe inequalityabove
entails that, for any t¯≤t≤T,
|Xε| ≥2δ+ǫa,
t δ
which leads to the thesis, since |Xǫ|≥|Xǫ| −δ.
t t δ
In order to conclude the proof of Theorem 1.1, let us fix a∈]2γ/(1+γ),1[,
so that the lemma just proved provides some t¯, δ.
Applying Itoˆ’s formula to |Xε| , starting from t¯, we obtain
t δ
t t
|Xε| ≥|Xε| + |Xε|′ sgn(Xε)|Xε|γ −ε |Xε|′ dW ,
t δ t¯ δ Zt¯ s δ s s (cid:12)(cid:12)Zt¯ s δ s(cid:12)(cid:12)
(cid:12) (cid:12)
since |x|′′ ≥0. On the event (7), it holds therefore, fo(cid:12)r t¯≤t≤T, (cid:12)
δ
t
|Xε|≥δ+ |Xε|′ sgn(Xε)|Xε|γds,
t Zt¯ s δ s s
where we used the fact that ||x| −|x||≤δ and the estimate on Xǫ provided
δ t¯ δ
by the lemma. But the lemma shows that also |Xǫ| ≥ δ, so th(cid:12)at,(cid:12)as already
remarked, |Xǫ|′ sgn(Xǫ)≥1 and therefore for t¯≤tt≤T, (cid:12) (cid:12)
t δ t
t
|Xε|≥δ+ |Xε|γds.
t Zt¯ s
Lemma 2.2 below allows us to conclude that, in the event (7), a.s. it holds,
for t¯≤t≤T,
|Xε|≥Hγ(t−R(t¯,δ)).
t
6
whereR(t¯,δ)issome(explicit)quantitydependingalsoonγ,whichisinfinites-
imalasε→0,sothatther.h.s.aboveconvergesuniformlyint∈[0,T]toHγ(t)
as ǫ→0.
On the other hand, directly from (1), we obtain the estimate
t
|Xε|≤ |Xε|γds+ε|W |,
t Z s t
0
that, thanks to another application of Lemma 2.2, entails that on the event
sup ε|W |≤εa (8)
t
(cid:26) (cid:27)
0≤t≤T
it holds
|Xε|≤Hγ(t−R(0,εa)).
t
Again,asε→0,R(0,εa)→0andthereforether.h.s.aboveconvergesuniformly
in t∈[0,T] to Hγ(t).
ByapplyingDoob’sinequalityandItoˆ’sisometry,weseethattheintersection
ofthe events(7)and(8)hasprobabilitygreaterthan1−α,whereweintroduce
the infinitesimal α=2Tε2−a. On this intersection, it holds, for t¯≤t≤T,
||Xε|−H(t)|≤h,
t
where
h= max {|Hγ(t)−Hγ(t−R(t¯,δ))|,|Hγ(t)−Hγ(t−R(0,εa))|}
0≤t≤T
which is easily seen to be infinitesimal, as ε→0.
On the other hand, we already know from Lemma 2.1 above that |Xε| ≥
t
δ 6=0, for t¯≤t≤T andtherefore we canconclude as in the prooffor the γ =0
case. Indeed, in the intersection of the events (7) and (8), whose probability is
greater than 1−α, almost surely,
• either for any t¯≤t≤T, |Xε|=Xε and |Xε−Hγ(t)|≤h,
t t t
• or for any t¯≤t≤T, |Xε|=−Xε and |Xε+Hγ(t)|≤h,
t t t
andtheproofiscompleted. Itremainsonlythefollowingcomparisonlemma
that was used above.
Lemma 2.2. Given 0 ≤ t¯≤ T, δ > 0 and a non-negative continuous function
f on [t¯,T], such that for any t∈[t¯,T],
t
f(t)≥δ+ f(s)γds (respectively, ≤).
Zt¯
Then, for t∈[t¯,T],
f(t)≥Hγ(t−R(t¯,δ)) (respectively, ≤).
where R(t¯,δ)=t¯−δ1+γ/(1−γ).
7
Proof. The term R(t¯,δ) is defined in such a way that t 7→ Hγ(t−R) is the
solution of the ODE,
t
x(t)=δ+ x(s)γds, (t¯≤t≤T).
Zt¯
Moreover, Hγ(t−R) is continuously differentiable on [t¯,T].
Let R=R(t¯,δ) and set
t
D(t)=δ+ f(s)γds−Hγ(t−R),
Zt¯
which is continuously differentiable on [t¯,T], with D(t¯)=0 and derivative
D′(t)=f(t)γ −(Hγ)′(t−R)=f(t)γ −(Hγ(t−R))γ
by the remark above. Now, since x 7→ xγ is increasing for x > 0, from the
hypothesiswe obtainthatfor t∈[t¯,T],the conditionD(t)≥0 implies D′(t)≥
0 and so we conclude that D(t) ≥ 0 for t in this range. The other case is
similar.
Acknowledgements
The author is grateful to M. Maurelli for many discussions on the subject and
thanks F. Flandoli and M. Pratelli for their support.
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