Stochastic Quantization for the fractional Edwards Measure

0.15 MB·

by  Wolfgang Bock

STOCHASTIC QUANTIZATION FOR THE FRACTIONAL
EDWARDS MEASURE.
6
1 WOLFGANGBOCK,TORBENFATTLER,ANDLUDWIGSTEIT
0
2
Abstract. Weprovetheexistenceofadiffusionprocesswhoseinvariantmea-

n sure is the fractional polymer or Edwards measure for fractional Brownian
a motionindimensiond∈NwithHurstparameterH ∈(0,1)fulfillingdH <1.
J The diffusion is constructed via Dirichlet form techniques in infinite dimen-

4 sional (Gaussian) analysis. Moreover, we show that the process is invariant
2 undertimetranslations.

]
h
p
- 1. Introduction
h
t For a given probability measure ν on a measureable space X the stochastic
a
m quantization of ν means the construction of a Markov process which has ν as an
invariant measure. Stochastic quantization has been studied first by Parisi and
[
Wu for applications in quantum field theory, which were extended to Euclidean

1 quantum fields.
v The two-dimensional polymer measure is informally given as
6
0 dµ =Z−1e−gLdµ ,
g 0
4
whereµ denotestheWienermeasure,Ltheself-intersectionlocaltimeofBrownian
6 0
0 motion and Z is a normalization constant. In the two-dimensional case stochastic
. quantizationfor this ”polymer measure” has been studied by Albeverio, Roeckner,
1
0 Hu and Zhou [1]. Here we follow their approach, using Dirichlet forms also in the
6 fractional case.
1 Intersectionlocaltimes L ofBrownianmotionhavebeen studiedfor a longtime
:
v and by many authors, see e.g. [1],[2], [5]-[6], [10], [15], [17], [19] and [25]-[30], the
i intersections of Brownian motion paths have been studied even since the Forties
X
[18]. One can consider intersections of sample paths with themselves or e.g. with
r
a other, independent Brownian motions e.g. [29], one can study simple [6] or n-fold
intersections e.g. [7], [19] and one can ask all of these questions for linear, planar,
spatial or - in general - d-dimensional Brownian motion: self-intersections become
increasingly scarce as the dimension d increases.
Asomewhatinformalbutverysuggestivedefinitionofself-intersectionlocaltime
of a Gaussian process Y is in terms of an integral over Dirac’s - or Donsker’s - δ-
function
L(Y)≡ d2tδ(Y(t )−Y(t )),
2 1
Z
where for now Y =B is a Brownianmotion, intended to sum up the contributions
from each pair of ”times” t ,t for which the process Y is at the same point. In
1 2
Edwards’ modeling of long polymer molecules by Brownian motion paths, L is
Date:January26,2016.
1
2 WOLFGANGBOCK,TORBENFATTLER,ANDLUDWIGSTREIT
used to model the ”excluded volume” effect: different parts of the molecule should
not be located at the same point in space. As another application, Symanzik [25]
introduced L as a tool in constructive quantum field theory.
A rigorousdefinition, suchas e.g. througha sequence of Gaussiansapproximat-
ing the δ-function, will lead to increasingly singular objects and will necessitate
various”renormalizations”asthedimensiondincreases. Ford>1the expectation
will diverge in the limit and must be subtracted [17], [26], as a side effect such a
localtimewillthennomorebepositive. Ford>3variousfurtherrenormalizations
have been proposed [27] that will make L into a well-defined generalized function
of Brownian motion. For d = 3 a multiplicative renormalization gives rise to an
independent Brownian motion as the weak limit of regularized and subtracted ap-
proximationstoL[30];anotherrenormalizationhasbeenconstructedbyWestwater
to make the Gibbs factor e−g·L of the polymer model well-defined [28].
In this article we first introduce the setting along the lines of white noise or
Gaussiananalysis,usingthe fractionalwhite noisemeasure. Thiscanbe compared
to the approach in [22]. Moreover the results from [24] concerning the gradient
are extended to this setting. We show in the framework of Dirichlet forms, that
thereexistsaMarkovprocesswhichhasthefractionalpolymermeasureasinvariant
measure. The proof is based on the results of [13] and [14], which show that the
self-intersection local time in the case Hd < 1 is Meyer-Watanabe differentiable.
The closability of the gradient Dirichlet form is then shown by an integration by
parts argument. The irreducibility follows as in the Brownian case, see [1].
2. Framework
For d ∈ N and Hurst parameter H ∈ (0,1) fractional Brownian motion in
dimension d is a Rd-valued centered Gaussian process BH with covariance
t t≥0
1 (cid:0) (cid:1)
cov (t,s):=E BHBH = t2H +s2H −|t−s|2H , s,t∈[0,∞).
H t s 2
(cid:2) (cid:3) (cid:0) (cid:1)
For s ∈ (0,∞) let Θ := and set Θ ,Θ := cov (t,s) for s,t ∈ [0,∞).
s 1[0,s) s t H H
Moreover,let X :=span Θ s>0 . Hence x,y ∈X are of the form
s (cid:0) (cid:1)
(cid:8) (cid:12) n (cid:9) m
(cid:12)
x= α Θ , y = β Θ
i si j tj
i=1 j=1
X X
with n,m∈N and
n m
x,y := α β Θ ,Θ
H i j si tj H
i=1j=1
(cid:0) (cid:1) XX (cid:0) (cid:1)
definesaninnerproductonX. Takingtheabstractcompletionoftheinnerproduct
space X,(·,·) we obtain a Hilbert space H,h·,·i , extending (·,·) to H.
H H H
Moreover, H,h·,·i has a countable orthonormal basis η . For k ∈N let
(cid:0) (cid:1) H (cid:0) (cid:1) k k∈N
λ ∈R such that
k
(cid:0) (cid:1) (cid:0) (cid:1)
∞
1
1<λ <λ <...<λ <λ <... and <∞.
1 2 k k+1 λ2
k=1 k
X
STOCHASTIC QUANTIZATION 3
Next we consider
∞
H∋f 7→Af := λ f,η η ∈H
k k k
H
k=1
X (cid:10) (cid:11)
and define for p∈N
H := f ∈H kApfk <∞ and N := H ,
p H p
p∈N
(cid:8) (cid:12) (cid:9) \
(cid:12)
wherek·k denotesthe inducednormonH. ThenN isacountablyHilbertspace,
H
which is Fr´echet and nuclear, compare e.g. [23]. Its topological dual is given by
N′ := H .
−p
p∈N
[
Thus we obtain the Gel’fand triple
N ⊂H⊂N′.
We denote complexifications by a subscript C.
Now by the Bochner-Minlos-Sazanov theorem, see e.g. [3] or [11], we define a
Gaussian measure µ on N′ by
H
1
exp(ihω,ξi ) dµ (ω):=exp − kξk .
ZN′ H H (cid:18) 2 H(cid:19)
Remark 2.1. Note that the measure has full support, i.e. every open set has pos-
itive measure. This can be seen by [9], Thm. 6 or the fact that the measure is
quasi translation invariant w.r.t. shifts in direction of the subspace N dense in N′,
compare e.g. [12], chapter 4B.
We obtain the probability space (N′,C ,µ ). Here C := σ(Cξ1,...,ξn ) denotes
σ H σ F1,...,Fn
the σ-algebra of cylinder sets
Cξ1,...,ξn = ω ∈N′ hξ ,ωi∈F ,...,hξ ,ωi∈F ,
F1,...,Fn 1 1 n n
n (cid:12)(cid:12) ξi ∈N,Fj ∈B(R),j =1,...,n, n∈N , (2.1)
o
where B(R) denotes the σ-algebra of Borel sets in R.
Note that since N is a nuclear countably Hilbert space we have, see e.g. [12]:
C (N′)=B (N′)=B (N′),
σ w s
where B (N′) (resp. B (N′)) is the Borel σ-algebra generated by the weak (resp.
w s
strong) topology.
We define by
N
P := p∈L2(N′;µH) p= hω⊗n,f⊗ni, f ∈NC
( )
(cid:12) nX=0
(cid:12)
the space of smooth polynomials. (cid:12)
We intend to construct the stochastic quantization Markov process via a local
Dirichlet form as e.g. in [8]. To this end we define differential operators as follows:
4 WOLFGANGBOCK,TORBENFATTLER,ANDLUDWIGSTREIT
Definition 2.2. Let p∈P and (ηk)k∈N ⊂N a CONS of H. Setting
N
p(ω+λη )−p(ω)
D p(ω)= lim k = nhη ⊗ω⊗n−1,f⊗ni,
ηk λ→0 λ k
n=1
X
we define
∇p:=(D p)∞ .
ηk k=1
Remark2.3. Notethatthis definesD and∇onadensesubspaceofL2(N′;µ ).
ηk H
For p∈P we have
∞ ∞ N N
(D p)2 = mnhω⊗m−1⊗η ,f⊗mihω⊗n−1⊗η ,f⊗ni
ηk k k
k=1 k=1m=1n=1
X X X X
N N
= m nhω⊗m+n+2,(f,f) f⊗n+m−2i.
H
m=1 n=1
X X
Furthermoreforu∈HtheadjointD∗ =h·,ui −D onadensesubspace,e.g.poly-
u H u
nomials in L2(N′;µ ), see e.g. [24].
H
3. Results
In the following we will just write L for L(BH), where BH, H ∈ (0, 1), is a
d
d-dimensional fractional Brownian motion with Hurst parameter H.
Moreover we denote by ν the measure ν =e−gLµ .
g g H
Theorem 3.1. The bilinear form
E (u,v):=E(e−gL∇u·∇v), u,v ∈P,
νg
isadenselydefined, closable, symmetricpre-Dirichlet form andgives risetoalocal,
quasi-regular Dirichlet form (E ,D(E )). Here E denotes expectation w.r.t. µ .
νg νg H
Theorem3.2. ThereexistsadiffusionprocessM=(Ω,F,(Ft)t≥0,(Xt)t≥0,(Pω)ω∈N′)
which is associated with (E ,D(E )).
νg νg
Theorem3.3. Thereexistsaconstantc >0(seeLemma4.1below), suchthatfor
0
all g <c the form (E ,D(E )) is irreducible (i.e. u∈D(E ) with (E (u,u)=0
0 νg νg νg νg
implies u is a constant), equivalently the associated diffusion is invariant under
time translations.
4. Proofs
Proof of Thm. 3.1: Note that the polynomials are dense in L2(N′;e−gLµ ,
H)
compare the proof of Prop. 10.3. [12], p.371 and Prop. 2.3.2 from [23]. We show
closability using an integration by parts criterion. Let (η ) be a CONS. We can
k k
write
∞
E (u,v)= E e−gLD u·D v .
νg ηk ηk
k=1
X (cid:0) (cid:1)
Itis enoughto show,see e.g.[20], thateachterminthe sumis closable. Indeed we
have for the components of the gradient:
E(D u·(vexp(−gL)))=E(uD∗ (v·exp(−gL)))
ηk ηk
=E(u(h·,η ivexp(−gL)−D v−vgD L)exp(−gL)). (4.1)
k ηk ηk
STOCHASTIC QUANTIZATION 5
We intendto showthat the lastexpressionis finite foru,v ∈P. This is evidentfor
the first two terms. For the last one we have
|E(v(DηkL)exp(−gL))|≤kvkL2(N′;µH)·kDηkLkL2(N′;µH),
which is finite due to the Meyer-Watanabe differentiability of L, see e.g. [13]. For
u,v ∈ P the expression (4.1) is well-defined, i.e. the adjoint of the gradient is
densely defined. Hence the form is closable, see e.g. [16].
The rest follows by [20] Section IV.b., Thm. 3.5.
(cid:4)
Proof of Thm. 3.2 The proof is an immediate result of [20] Section IV.b., Thm.
3.5. (cid:4)
For the proof of Theorem 3.3 we will follow the lines of [1]. In order to do so, we
need a further result from [14].
Lemma 4.1 ([14], Thm. 1). Suppose that Hd<1. Then the self-intersection local
time L fulfills
E(eLp)<∞,
for any p < 1 . Moreover there exists a constant c > 0 such that for all c < c
Hd 0 0
one has
1
E(ecLHd)<∞,
In particular, E(ecL) is then finite. This we will use in the following proof.
Proof of Thm. 3.3: For ϕ∈L2(N′,e−gLµ ), and 1 + 1 =1,p,q>1 we have
H p q
g g
2 2
|ϕ|p dµ (ω)= |ϕ|p exp − L exp L dµ (ω)
H p p H
Z Z (cid:16) (cid:17) (cid:16) (cid:17)
1 1
p q q
≤ |ϕ|2exp(−gL)dµ (ω) · exp gL dµ (ω) ,
H p H
(cid:18)Z (cid:19) (cid:18)Z (cid:16) (cid:17) (cid:19)
by Ho¨lder inequality. The last term is finite if qg < c as in Lemma 4.1. In this
p 0
case we obtain
|ϕ|p2 dµH(ω)≤CkϕkLp22(H;e−gLdµH). (4.2)
Z
Hence for any polynomial v ∈P, by setting ϕ=∇v, we have
2
2 p
k∇vk2 := k∇vkpdµ (ω) ≤CE (v,v). (4.3)
H,p2 H H νg
(cid:18)Z (cid:19)
So we see already that E (v,v)=0 implies v =const. a.e. for any v ∈P.
νg
Consider now u ∈ D(E ) such that E (u,u) = 0. Then by closability we
νg νg
find a sequence (v ) in P such that v → u in D(E ). Then by (4.3) we see
n n n νg
that k∇v k → 0. The convergence in L2(N′;e−gLdµ ) implies convergence in
n H,p2 H
Lp2(N′,dµ ), due to (4.2). The same holds for the gradients by (4.3).
H
Altogether,we havea polynomialsequencewhichapproximatesuin D . Here
1,p2
D denotes the corresponding Malliavin–Sobolev space, see e.g. [21] Sect. 1.2.
1,p2
Thus by [21] Prop. 1.5.5. we obtain u ∈ D , if 2 > 1, which is the case if we
1,p2 p
choose p∈(1,2).
6 WOLFGANGBOCK,TORBENFATTLER,ANDLUDWIGSTREIT
Moreover,inthiscasewehavethat∇u=0a.e.. Thenby[21]Prop.1.2.5,since
∇u=0 and u∈D , we find that u=E(u), which is a constant.
1,p2
The properties hold µ -a.e. and thus e−gLµ -a.e. by absolute continuity of the
H H
measure.
Thus we have shown the assertion. (cid:4)
Acknowledgement: WetrulythankM.Ro¨cknerforhelpfuldiscussions. Further-
more we thank M. Grothaus and M. J. Oliveira for helpful comments. Financial
supportby CRC 701and the mathematics departmentof the University ofKaiser-
slautern for researchvisits at Bielefeld university are gratefully acknowledged.
References
[1] Albeverio, S., Hu,Y.-Z.,R¨ockner, M., Zhou, X. Y.: Stochastic Quantization of the Two-
DimensionalPolymerMeasure.Appl.Math.Optim.40,341-354.(1996).
[2] Bass, R. F., Khoshnevisan, D., Intersection Local Times and Tanaka Formulas. Ann. Inst.
H. Poincar´e, 29,(1993), 419-451.
[3] Berezansky,Yu.M.,Kondratiev,Yu.G.,SpectralMethods inInfinite-DimensionalAnalysis.
NaukovaDumka,Kiev,1988(inRussian).Englishtranslation,KluwerAcademicPublishers,
Dordrecht,1995.
[4] deFaria,M.,Hida,T.,Streit,L.,Watanabe,H.,Intersectionlocaltimesasgeneralizedwhite
noisefunctionals.ActaAppl.Math.,46:351–362, 1997.
[5] de Faria, M., Drumond, C., Streit, L., The renormalization of self-intersection local times.
I. The chaos expansion. Infin. Dimens. Anal. Quantum Probab. Relat. Top.,3(2):223–236,
2000.
[6] Dvoretzky,A.,Erd¨os,P.,Kakutani,S.DoublepointsofpathsofBrownianmotioninn-space.
ActaSci. math. Szeged,12,(1950), 75-81.
[7] Dvoretzky, A., Erd¨os, P., Kakutani, S., Taylor, S. J., Triplepoints of the Brownian motion
in3-space.Proc. Cambridge Philos. Soc. 53(1957), 856–862.
[8] Fukushima,M.,Oshima,Y.,Takeda,M.: DirichletFormsandSymmetricMarkovProcesses.
deGruyter,Berlin,(1994)
[9] Kondratiev,Y.G.,Streit,L.andWesterkamp,W.,AnoteonpositivedistributionsinGauss-
iananalysis,Ukr.Mat.J.,Vol.47,No.5,(1996)
[10] He,S. W.,Yang, W. Q.,Yao,R.Q.,Wang, J.G., Local timesof self-intersectionformulti-
dimensionalBrownianmotion.Nagoya Math. J.138(1995), 51–67.
[11] Hida,T.,StationaryStochasticProcesses ,MathematicalNotesSeries,PrincetonUniversity
Press,Princeton1970.
[12] Hida,T.,Kuo,H.H.,Potthoff,J.andStreit,L.,WhiteNoise.Aninfinitedimensionalcalculus.
Kluwer,Dordrecht,516p.,(1993)
[13] Hu,Y.: Self-intersectionlocaltimeoffractional Brownianmotions—viachaos expansion. J.
Math.KyotoUniv.41,233–250(2001).
[14] Hu, Y., Nualart, D., Song, J., Integral representation of renormalized self-intersection local
times.J.Func.Ana.,Vol255,No.9,(2008)
[15] Imkeller,P.,P´erez-Abreu, V.,Vives,J.,Chaos expansions ofdoubleintersection localtimes
ofBrownianmotioninRd andrenormalization.Stochastic Process. Appl.56(1995), 1–34.
[16] Kato, T., Perturbation theory for linear operators, 2nd edition, SpringerVerlag, Berlin-
Heidelberg-NewYork,(1976).
[17] LeGall,J.F.: LeGall,J.F.,Surletempslocald’intersectiondumouvementbrownienplanet
lam´ethodederenormalisationdeVaradhan.S´em.Prob.XIX,LectureNotesinMathematics
1123,Springer,Berlin,1985,314–331.
[18] L´evy,P.: L´evy,P.,Lemouvement brownienplan.Amer. J. Math.62(1940), 487–550.
[19] Lyons, T. J., The critical dimension at which quasi-every Brownian motion is self-avoiding.
Adv. inAppl. Probab. (Spec. Suppl.1986), 87–99.
[20] Ma, Z.-M. and Ro¨ckner, M., An introduction to the theory of (non-symmetric) Dirichlet
forms,Springer,Berlin,(1992).
[21] Nualart, D., The Malliavincalculus and related topics., Second edition. Probability and its
Applications(NewYork).Springer-Verlag,Berlin,(2006).
STOCHASTIC QUANTIZATION 7
[22] Hu,Y.andØksendal,B.,FractionalWhiteNoiseCalculusandApplicationstoFinance.Infin.
Dimens. Anal. Quantum. Probab. Relat. Top.Vol.06,Nr.1(2003).
[23] Obata N., White Noise Calculus and Fock Spaces,volume 1577 of LNM, Springer Verlag,
Berlin,Heidelberg,NewYork,1994.
[24] Potthoff,J.: OndifferentialoperatorsinWhiteNoiseanalysis.ActaApplicandaeMathemat-
icae63,333-347(2000).
[25] Symanzik, K., Euclidean quantum field theory, in ”Local Quantum Theory” (R. Jost, ed.),
AcademicPress,NewYork,1969.
[26] Varadhan,S.R.S.,Appendixto”Euclideanquantumfieldtheory”byK.Symanzik,in”Local
QuantumTheory”(R.Jost.ed.),AcademicPress,NewYork,1969.
[27] Watanabe, H.,Thelocaltimeofself-intersectionsofBrownianmotionsasgeneralizedBrow-
nianfunctionals.Lett. Math. Phys.23(1991), 1–9.
[28] Westwater, J.,OnEdward’smodelforlongpolymerchains.Comm. Math. Phys.72(1980),
131–174.
[29] Wolpert,R.,Wienerpathintersectionandlocaltime.J. Funct. Anal.30(1978), 329–340.
[30] Yor,M.,Renormalisationetconvergenceenloipourlestempslocauxd’intersectiondumou-
vement brownien dans R3. S´eminaire de Probabilit´e, Lecture Notes in Mathematics 1123,
Springer,Berlin,1985,350–365.
TechnomathematicsGroup,University of Kaiserslautern
E-mail address: [email protected]
FunctionalAnalysisandStochastic AnalysisGroup,University of Kaiserslautern
E-mail address: [email protected]
BiBoS, Universita¨t Bielefeld, Germany; CCM, Unversidade da Madeira, Funchal,
PortugalandPhysics Deptartment,MSU-IIT,Iligan,Philipinnes
E-mail address: [email protected]