Through chaos decomposition we improve the Varadhan estimate for the rate of
convergence of the centered approximate self-intersection local time of planar
Brownian motion.Comment: 5 page

Bock, Wolfgang

da Silva, José Luis

Oliveira, Maria João

Streit, Ludwig

English

arXiv

Through chaos decomposition we improve the Varadhan estimate for the rate of
convergence of the centered approximate self-intersection local time of planar
Brownian motion.Financial support of FCT through the research project PTDC/MAT-STA/1284/2012
is gratefully acknowledged. W. B. and J. L. S. also would like to thank for financial
support of IGK and DFG through SFB-701 (University of Bielefeld),
respectively

Silva, José Luís da

The Repositório Aberto

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14 Polymer Measure: Varadhan’sRenormalization Revisited
Wolfgang Bock
CMAF, University of Lisbon, P 1649-003 Lisbon, Portugal
bock@campus.ul.pt
Maria Joa˜o Oliveira
Universidade Aberta, P 1269-001 Lisbon, Portugal
CMAF, University of Lisbon, P 1649-003 Lisbon, Portugal
oliveira@cii.fc.ul.pt
Jose´ Lu´ıs da Silva
CCEE, University of Madeira, P 9000-390 Funchal, Portugal
CCM, University of Madeira, P 9000-390 Funchal, Portugal
luis@uma.pt
Ludwig Streit
Forschungszentrum BiBoS, Bielefeld University, D 33501 Bielefeld, Germany
CCM, University of Madeira, P 9000-390 Funchal, Portugal
streit@physik.uni-bielefeld.de
Abstract
Through chaos decomposition we improve the Varadhan estimate
for the rate of convergence of the centered approximate self-intersection
local time of planar Brownian motion.
Keywords: Edwards model, self-intersection local time, Varadhan renor-
malization, white noise analysis
Mathematics Subject Classifications (2010): 28C20, 41A25, 60H40,
60J55, 60J65, 82D60
1
1 Introduction
The Edwards model [2] for self-repelling or ”weakly self-avoiding” d-dimensional
Brownian motion, with applications in polymer physics and quantum field
theory, is informally given by a Gibbs factor
G =
1
Z
exp
(
−g
∫ T
0
ds
∫ t
0
dtδ (B(s)− B(t) )
)
with g > 0 and
Z = E
(
exp
(
−g
∫ T
0
ds
∫ t
0
dtδ (B(s)− B(t) )
))
.
Using
δε(x) :=
1
(2piε)d/2
e−
|x|2
2ε , ε > 0,
one defines an approximate self-intersection local time by
Lε :=
∫ T
0
dt
∫ t
0
ds δε(B(t)− B(s)).
For d ≥ 2
lim
εց0
E(Lε) =∞.
For the planar case d = 2, Varadhan [3] has shown that centering
Lε,c := Lε − E(Lε) and Lc := lim
εց0
Lε,c
is sufficient to make the Gibbs factor G = Z−1 exp (−gLc) well defined. An
estimate for the rate of convergence
‖Lc − Lε,c‖
2
2 ≤ const.ε
α
for all α < 1/2, is in Varadhan’s words, ”the most difficult step of all and
requires considerable estimation”. In this note we shall use a multiple Wiener
integral or chaos expansion for an alternate and comparatively straightfor-
ward argument, extending the estimate to all α < 1.
2
2 Fock space representation of the local time
The Ito-Segal-Wiener isomorphism relates the L2 space of planar Brownian
motion with the Fock space
F =
(
∞⊕
n=0
SymL2(Rn, n!dnx)
)⊗2
.
We shall use the multi-index notation
n = (n1, n2) n = n1 + n2, n! = n1!n2!
The Fock space norm is then
‖F‖22 =
∑
ni≥0
n! ‖F
n
‖2
2
.
For Lε,c the kernel functions Fn were computed explicitly in [1]. For the
planar case the result is
Proposition 1 [1]: For d = 2 the kernel functions F
n
of Lε,c(T ) and Lc(T )
have their support on [0, T ]n and are, with ε > 0, and ε = 0 respectively, for
n > 1
F2n,ε(u1, . . . , u2n) =
1
2pi
(
−
1
2
)n
1
n(n− 1)n!
×
(
1
(T + ε)n−1
−
1
(v + ε)n−1
−
1
(T − u+ ε)n−1
+
1
(v − u+ ε)n−1
)
,
where v := max1≤k≤2n uk ≤ T and u := min1≤k≤2n uk ≥ 0. For n = 1
F2,ε(u1, u2) = −
1
4pi
(ln(v + ε) + ln(T − u+ ε)− ln(v − u+ ε)− ln(T + ε)) .
All kernel functions F
n
with odd ni are zero.
2.1 The rate of convergence
Theorem 2 Given T > 0. Then for any α < 1 there is a constant CT,α > 0
such that for all ε > 0
‖Lε,c(T )− Lc(T )‖
2
2 ≤ CT,αε
α.
3
Proof: From Proposition 1
‖F2n,0 − F2n,ε‖
2
2
=
(
n(n− 1)2pi2nn!
)−2 ∫ T
0
d2nukK
2
ε (u, v, T )
where for n > 1
Kε(u, v, T ) =
(
T−n+1 − (T + ε)−n+1
)
−
(
v−n+1 − (v + ε)−n+1
)
−
(
(T − u)−n+1 − (T − u+ ε)−n+1
)
+
(
(v − u)−n+1 − (v − u+ ε)−n+1
)
.
Since Kε(u, v, T ) does not depend on 2n − 2 of the uk-variables, we may
integrate them out:
‖F2n,0 − F2n,ε‖
2
2
=
(
n(n−1)2pi2nn!
)−2
2n(2n−1)
∫ T
0
dv
∫ v
0
du(v−u)2n−2K2ε (u, v, T ) .
Of the four terms in Kε, the last one is dominant so that
‖F2n,0 − F2n,ε‖
2
2
≤ 16
2n(2n− 1)(
n(n− 1)2pi2nn!
)2
∫ T
0
dv
∫ v
0
du
(
1−
(
v − u
v − u+ ε
)n−1)2
= 16
2n(2n− 1)(
n(n− 1)2pi2nn!
)2
∫ T
0
dτ
∫ T−τ
0
du
(
1−
(
τ
τ + ε
)n−1)2
≤ 16
2n(2n− 1)T(
n(n− 1)2pi2nn!
)2
∫ T
0
dτ
(
1−
(
τ
τ + ε
)n−1)2
= 16
2n(2n− 1)T(
n(n− 1)2pi2nn!
)2
∫ T
0
dτ
(
τn−1(n− 1)
∫ ε
0
dx
(x+ τ)n
)2
(1)
By Ho¨lder’s inequality∫ ε
0
dx
(x+ τ)n
≤ ε1/q
(∫ ∞
0
dx
(x+ τ)np
)1/p
= ε1/q
(
1
np− 1
τ 1−np
)1/p
if 1
q
+ 1
p
= 1. Insertion of this estimate into (1) produces
∑
n:n>1
(2n)! ‖F2n,0 − F2n,ε‖
2
2
≤
4p
(2− p)pi2
T
2
p ε
2
q
∑
n:n>1
(2n)!
2n(2n− 1)
(n2nn!)2
1
(pn− 1)
2
p
which is convergent if 2
p
> 1, i.e., q > 2. For the n = 1 term an ε
2
q estimate
is likewise obtained via Ho¨lder’s inequality.
Hence, for any α < 1,
‖Lc(T )− Lε,c(T )‖
2
2 =
∑
n:n≥1
(2n)! ‖F2n,0 − F2n,ε‖
2
2
≤ C(T, α)εα, ∀ε > 0.

Acknowledgments
Financial support of FCT through the research project PTDC/MAT-STA/1284/2012
is gratefully acknowledged. W. B. and J. L. S. also would like to thank for fi-
nancial support of IGK and DFG through SFB-701 (University of Bielefeld),
respectively.
References
[1] M. de Faria, T. Hida, L. Streit and H. Watanabe: Intersection local
times as generalized white noise functionals, Acta Appl. Math. 46, 351-
362 (1997).
[2] S. F. Edwards: The statistical mechanics of polymers with excluded vol-
ume. Proc. Roy. Soc. 85, 613-624 (1965).
[3] S. R. S. Varadhan: Appendix to ”Euclidian quantum field theory” by
K. Szymanzik, in: R. Jost ed., Local Quantum Theory. Academic Press,
New York, 1969.
5


Reviews in Mathematical Physics

Polymer measure: varadhan's renormalization revisited

Repositório Aberto da Universidade Aberta

arXiv:1405.7150v1  [math-ph]  28 May 2014 Polymer Measure: Varadhan’sRenormalization RevisitedWolfgang BockCMAF, University of Lisbon, P 1649-003 Lisbon, Portugalbock@campus.ul.ptMaria Joa˜o OliveiraUniversidade Aberta, P 1269-001 Lisbon, PortugalCMAF, University of Lisbon, P 1649-003 Lisbon, Portugaloliveira@cii.fc.ul.ptJose´ Lu´ıs da SilvaCCEE, University of Madeira, P 9000-390 Funchal, PortugalCCM, University of Madeira, P 9000-390 Funchal, Portugalluis@uma.ptLudwig StreitForschungszentrum BiBoS, Bielefeld University, D 33501 Bielefeld, GermanyCCM, University of Madeira, P 9000-390 Funchal, Portugalstreit@physik.uni-bielefeld.deAbstractThrough chaos decomposition we improve the Varadhan estimatefor the rate of convergence of the centered approximate self-intersectionlocal time of planar Brownian motion.Keywords: Edwards model, self-intersection local time, Varadhan renor-malization, white noise analysisMathematics Subject Classifications (2010): 28C20, 41A25, 60H40,60J55, 60J65, 82D6011 IntroductionThe Edwards model [2] for self-repelling or ”weakly self-avoiding” d-dimensionalBrownian motion, with applications in polymer physics and quantum fieldtheory, is informally given by a Gibbs factorG =1Zexp(−g∫ T0ds∫ t0dtδ (B(s)− B(t) ))with g > 0 andZ = E(exp(−g∫ T0ds∫ t0dtδ (B(s)− B(t) ))).Usingδε(x) :=1(2piε)d/2e−|x|22ε , ε > 0,one defines an approximate self-intersection local time byLε :=∫ T0dt∫ t0ds δε(B(t)− B(s)).For d ≥ 2limεց0E(Lε) =∞.For the planar case d = 2, Varadhan [3] has shown that centeringLε,c := Lε − E(Lε) and Lc := limεց0Lε,cis sufficient to make the Gibbs factor G = Z−1 exp (−gLc) well defined. Anestimate for the rate of convergence‖Lc − Lε,c‖22 ≤ const.εαfor all α < 1/2, is in Varadhan’s words, ”the most difficult step of all andrequires considerable estimation”. In this note we shall use a multiple Wienerintegral or chaos expansion for an alternate and comparatively straightfor-ward argument, extending the estimate to all α < 1.22 Fock space representation of the local timeThe Ito-Segal-Wiener isomorphism relates the L2 space of planar Brownianmotion with the Fock spaceF =(∞⊕n=0SymL2(Rn, n!dnx))⊗2.We shall use the multi-index notationn = (n1, n2) n = n1 + n2, n! = n1!n2!The Fock space norm is then‖F‖22 =∑ni≥0n! ‖Fn‖22.For Lε,c the kernel functions Fn were computed explicitly in [1]. For theplanar case the result isProposition 1 [1]: For d = 2 the kernel functions Fnof Lε,c(T ) and Lc(T )have their support on [0, T ]n and are, with ε > 0, and ε = 0 respectively, forn > 1F2n,ε(u1, . . . , u2n) =12pi(−12)n1n(n− 1)n!×(1(T + ε)n−1−1(v + ε)n−1−1(T − u+ ε)n−1+1(v − u+ ε)n−1),where v := max1≤k≤2n uk ≤ T and u := min1≤k≤2n uk ≥ 0. For n = 1F2,ε(u1, u2) = −14pi(ln(v + ε) + ln(T − u+ ε)− ln(v − u+ ε)− ln(T + ε)) .All kernel functions Fnwith odd ni are zero.2.1 The rate of convergenceTheorem 2 Given T > 0. Then for any α < 1 there is a constant CT,α > 0such that for all ε > 0‖Lε,c(T )− Lc(T )‖22 ≤ CT,αεα.3Proof: From Proposition 1‖F2n,0 − F2n,ε‖22=(n(n− 1)2pi2nn!)−2 ∫ T0d2nukK2ε (u, v, T )where for n > 1Kε(u, v, T ) =(T−n+1 − (T + ε)−n+1)−(v−n+1 − (v + ε)−n+1)−((T − u)−n+1 − (T − u+ ε)−n+1)+((v − u)−n+1 − (v − u+ ε)−n+1).Since Kε(u, v, T ) does not depend on 2n − 2 of the uk-variables, we mayintegrate them out:‖F2n,0 − F2n,ε‖22=(n(n−1)2pi2nn!)−22n(2n−1)∫ T0dv∫ v0du(v−u)2n−2K2ε (u, v, T ) .Of the four terms in Kε, the last one is dominant so that‖F2n,0 − F2n,ε‖22≤ 162n(2n− 1)(n(n− 1)2pi2nn!)2∫ T0dv∫ v0du(1−(v − uv − u+ ε)n−1)2= 162n(2n− 1)(n(n− 1)2pi2nn!)2∫ T0dτ∫ T−τ0du(1−(ττ + ε)n−1)2≤ 162n(2n− 1)T(n(n− 1)2pi2nn!)2∫ T0dτ(1−(ττ + ε)n−1)2= 162n(2n− 1)T(n(n− 1)2pi2nn!)2∫ T0dτ(τn−1(n− 1)∫ ε0dx(x+ τ)n)2(1)By Ho¨lder’s inequality∫ ε0dx(x+ τ)n≤ ε1/q(∫ ∞0dx(x+ τ)np)1/p= ε1/q(1np− 1τ 1−np)1/pif 1q+ 1p= 1. Insertion of this estimate into (1) produces∑n:n>1(2n)! ‖F2n,0 − F2n,ε‖22≤4p(2− p)pi2T2p ε2q∑n:n>1(2n)!2n(2n− 1)(n2nn!)21(pn− 1)2pwhich is convergent if 2p> 1, i.e., q > 2. For the n = 1 term an ε2q estimateis likewise obtained via Ho¨lder’s inequality.Hence, for any α < 1,‖Lc(T )− Lε,c(T )‖22 =∑n:n≥1(2n)! ‖F2n,0 − F2n,ε‖22≤ C(T, α)εα, ∀ε > 0.AcknowledgmentsFinancial support of FCT through the research project PTDC/MAT-STA/1284/2012is gratefully acknowledged. W. B. and J. L. S. also would like to thank for fi-nancial support of IGK and DFG through SFB-701 (University of Bielefeld),respectively.References[1] M. de Faria, T. Hida, L. Streit and H. Watanabe: Intersection localtimes as generalized white noise functionals, Acta Appl. Math. 46, 351-362 (1997).[2] S. F. Edwards: The statistical mechanics of polymers with excluded vol-ume. Proc. Roy. Soc. 85, 613-624 (1965).[3] S. R. S. Varadhan: Appendix to ”Euclidian quantum field theory” byK. Szymanzik, in: R. Jost ed., Local Quantum Theory. Academic Press,New York, 1969.5

 Through chaos decomposition, we improve the Varadhan estimate for the rate of convergence of the centered approximate self-intersection local time of planar Brownian motion. </jats:p

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