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Self Avoiding Fractional Brownian Motion - the Edwards Model

Self-avoiding chain molecules Ludwig Streit

BiBoS, Univ. Bielefeld and CCM, Univ. da Madeira

Taipei, August 12, 2010

• L. Streit (Institute)

The fBm Edwards Model Taipei, August 12, 2010 1 / 26

Polymer Chains

Real linear polymer chains under liquid medium as recorded using an atomic force microscope.

http://commons.wikimedia.org/wiki/File:Single_Polymer_Chains_AFM.jpg

• L. Streit (Institute)

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"Brownian" Polymers?

Strategy: random paths x(s) 0 s l

• L. Streit (Institute)

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"Brownian" Polymers?

Strategy: random paths x(s) 0 s l Focus: avoid crossings, want x(s) 6= x(t) if s 6= t

• L. Streit (Institute)

The fBm Edwards Model Taipei, August 12, 2010 3 / 26

"Brownian" Polymers?

Strategy: random paths x(s) 0 s l Focus: avoid crossings, want x(s) 6= x(t) if s 6= t Brownian motion has many! How to suppress them?

• L. Streit (Institute)

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The discrete case

Background "self-avoiding random walks", intensively studied in discrete mathematics, often used to model polymers. Madras, N.; Slade, G. The Self-Avoiding Walk. Birkhäuser (1996)

• C. Vanderzande: Lattice Models of Polymers. Cambridge U. Press

(1998) .

• L. Streit (Institute)

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Self-intersection Local Time

To control self-intersections of paths consider L =

Z l

0 ds

Z l

0 dtδ (x(s) x(t) )

Widely studied for Brownian motion paths. For the white noise setting and many references see e.g.

• M. de Faria, T. Hida, L. Streit, H. Watanabe: "Intersection Local

Times as Generalized White Noise Functionals" Acta Appl. Math. 46, 351-362 (1997)

• L. Streit (Institute)

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The Edwards Strategy

• S. F. Edwards, The statistical mechanics of polymers with excluded
• volume. Proc.Roy. Soc. 85, 613-624 (1965).

Weakly self-avoiding paths via a "Gibbs factor" to suppress self-intersections: G = 1 Z exp

• g

Z l

0 ds

Z l

0 dtδ (x(s) x(t) )

• with

Z = E

• exp
• g

Z l

0 ds

Z l

0 dtδ (x(s) x(t) )

• .
• L. Streit (Institute)

The fBm Edwards Model Taipei, August 12, 2010 6 / 26

The Edwards Strategy

• S. F. Edwards, The statistical mechanics of polymers with excluded
• volume. Proc.Roy. Soc. 85, 613-624 (1965).

Weakly self-avoiding paths via a "Gibbs factor" to suppress self-intersections: G = 1 Z exp

• g

Z l

0 ds

Z l

0 dtδ (x(s) x(t) )

• with

Z = E

• exp
• g

Z l

0 ds

Z l

0 dtδ (x(s) x(t) )

• .

Problem: Z =?

• L. Streit (Institute)

The fBm Edwards Model Taipei, August 12, 2010 6 / 26

A closer look at self-intersection local times

How to make sense of L =

Z l

0 ds

Z l

0 dtδ (B(s) B(t) )

• L. Streit (Institute)

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A closer look at self-intersection local times

How to make sense of L =

Z l

0 ds

Z l

0 dtδ (B(s) B(t) )

Use delta sequences: δε(x) := 1 (2πε)d/2 e jxj2

2ε ,

ε > 0, Lε :=

Z T

dt

Z t

0 ds δε(B(t) B(s)).

(1)

• L. Streit (Institute)

The fBm Edwards Model Taipei, August 12, 2010 7 / 26

Removing the regularization

The main problem is the removal of the approximation, that is, ε & 0.

• L. Streit (Institute)

The fBm Edwards Model Taipei, August 12, 2010 8 / 26

Removing the regularization

The main problem is the removal of the approximation, that is, ε & 0. OK for d = 1.

• L. Streit (Institute)

The fBm Edwards Model Taipei, August 12, 2010 8 / 26

Removing the regularization

The main problem is the removal of the approximation, that is, ε & 0. OK for d = 1. For d 2 have lim

ε&0 E(Lε) = ∞

• L. Streit (Institute)

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Removing the regularization

The main problem is the removal of the approximation, that is, ε & 0. OK for d = 1. For d 2 have lim

ε&0 E(Lε) = ∞

For d = 2 have E(Lε) = l 2π ln 1 ε + o (ε) and L2 convergence after centering: Lε E(Lε) ! Lc. (2) as ε tends to zero.

• L. Streit (Institute)

The fBm Edwards Model Taipei, August 12, 2010 8 / 26

Removing the regularization

The main problem is the removal of the approximation, that is, ε & 0. OK for d = 1. For d 2 have lim

ε&0 E(Lε) = ∞

For d = 2 have E(Lε) = l 2π ln 1 ε + o (ε) and L2 convergence after centering: Lε E(Lε) ! Lc. (2) as ε tends to zero. For d 3 a further multiplicative renormalization r(ε) is required r(ε) (Lε E(Lε)) . (3)

• L. Streit (Institute)

The fBm Edwards Model Taipei, August 12, 2010 8 / 26

The case d=2

Now consider exp (gLc)

• L. Streit (Institute)

The fBm Edwards Model Taipei, August 12, 2010 9 / 26

The case d=2

Now consider exp (gLc) Problem: Lc is unbounded below, and exp (gLc) ! ∞ when Lc ! ∞.

• L. Streit (Institute)

The fBm Edwards Model Taipei, August 12, 2010 9 / 26

The case d=2

Now consider exp (gLc) Problem: Lc is unbounded below, and exp (gLc) ! ∞ when Lc ! ∞. Need to show that large values occur only with a very small probability such that the expectation is nevertheless …nite.

• L. Streit (Institute)

The fBm Edwards Model Taipei, August 12, 2010 9 / 26

,A

• a

c(n ,c)

t1

ts

h{ xr L-r f

z

ZH 0Fl

?1 ;-a Ll \H ?\J

iZ'?'

7

\.,J (-

cN;

Hii

rro>" Z':

a

\J

r.

ördi

rl-{ V|.-

v H=

\

r:1

• =rr{F

^

\J \I

z

a ri^ fd Fn

Fa \/

c'.j

L./ rr F< tr9 ?\) r 7

|-- t9..

r f-l

0c

tJ2

;

an

d=2: Varadhan’s Method

1

Show logarithmic lower bound Lc

ε > c ln 1

ε

2

Show that the rate of convergence is E (Lc

ε Lc)2 Aε1/2

3

Now estimate prob (Lc K) .

• L. Streit (Institute)

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prob (Lc K) = prob (Lc Lc

ε + Lc ε K)

= prob (Lc Lc

ε K Lc ε )

• prob
• Lc Lc

ε K + c ln 1

ε

• Now use the Chebychev inequality to get

prob

• Lc Lc

ε K + c ln 1

ε

• A ε1/2
• K c ln 1

ε

2 Choosing K = 2c ln 1 ε ε = exp

• K

2c

• the probability for large negative values

prob (Lc K) 4A exp K

4c

• K 2

is exponentially small.

• L. Streit (Institute)

The fBm Edwards Model Taipei, August 12, 2010 11 / 26

As a result E (exp(gLc)) < ∞ if g > 0 is su¢ciently small. By a scaling argument this can be extended to all g > 0.

• S. R. S. Varadhan: Appendix to "Euclidean quantum …eld theory" by
• K. Symanzik, in: R. Jost, ed., Local Quantum Theory, Academic

Press, New York, p. 285 (1970)

• E. Nelson in Analysis in Function Space, W. T. Martin and I. Segal,
• eds. , Cambridge, 1964, p.87.
• B. Simon: The P(ϕ)2 (Euclidean) Quantum Field Theory. Princeton

University Press.

• L. Streit (Institute)

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Fractional Brownian Motion

Fractional Brownian motion fBm on Rd, d 1, with "Hurst parameter" H 2 (0, 1) is a d-dimensional centered Gaussian process BH = fBH

t : t 0g with covariance function

E(BH

t BH s ) = δij

2

• t2H + s2H jt sj2H

, i, j = 1, . . . , d, s, t 0. H = 1/2 is ordinary Brownian motion.

• F. Biagini, Y. Hu, B. Oksendal: Stochastic Calculus for Fractional

Brownian Motion and Applications. Springer, Berlin, 2007..

• Y. Hu and D. Nualart. Renormalized self-intersection local time for

fractional Brownian motion. Ann. Probab. 33:948–983, (2005).

• M. J. Oliveira, J. L. Silva, and L. Streit. Intersection local times of

independent fractional Brownian motions as generalized white noise

• functionals. arXiv:math.PR/1001.0513 Preprint, 2010.
• L. Streit (Institute)

The fBm Edwards Model Taipei, August 12, 2010 13 / 26

Fractional Brownian Motion

Fractional Brownian motion fBm on Rd, d 1, with "Hurst parameter" H 2 (0, 1) is a d-dimensional centered Gaussian process BH = fBH

t : t 0g with covariance function

E(BH

t BH s ) = δij

2

• t2H + s2H jt sj2H

, i, j = 1, . . . , d, s, t 0. H = 1/2 is ordinary Brownian motion. For larger resp. smaller H the paths are smoother resp. curlier than those of BM, and continuous.

• F. Biagini, Y. Hu, B. Oksendal: Stochastic Calculus for Fractional

Brownian Motion and Applications. Springer, Berlin, 2007..

• Y. Hu and D. Nualart. Renormalized self-intersection local time for

fractional Brownian motion. Ann. Probab. 33:948–983, (2005).

• M. J. Oliveira, J. L. Silva, and L. Streit. Intersection local times of

independent fractional Brownian motions as generalized white noise

• functionals. arXiv:math.PR/1001.0513 Preprint, 2010.
• L. Streit (Institute)

The fBm Edwards Model Taipei, August 12, 2010 13 / 26

Applications of fBm

Fractional Brownian motion (fBm) appears naturally in the modeling of many situations, for example, when describing

1

The widths of consecutive annual rings of a tree,

2

Local temperature as a function of time,

3

Water level of a river as a function of time (Hurst 1951)

4

Solar activity as a function of time,

5

values of the log returns of a stock,

6

polymer models

7

Financial turbulence (Mendes, Oliveira 2004)

8

The prices of electricity in a liberated electricity market H>1/2 in examples 1-5 while 7 and 8 have H<1/2

• L. Streit (Institute)

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Properties of fBm

Self-similarity: BH(at) ahBH(t)

• L. Streit (Institute)

The fBm Edwards Model Taipei, August 12, 2010 15 / 26

Properties of fBm

Self-similarity: BH(at) ahBH(t) Continuous paths

• L. Streit (Institute)

The fBm Edwards Model Taipei, August 12, 2010 15 / 26

Properties of fBm

Self-similarity: BH(at) ahBH(t) Continuous paths Stationary increments

• L. Streit (Institute)

The fBm Edwards Model Taipei, August 12, 2010 15 / 26

Properties of fBm

Self-similarity: BH(at) ahBH(t) Continuous paths Stationary increments Increments are uncorrelated for H = 1/2, correlated ("persistent" paths) for H > 1/2, anticorrelated for H < 1/2

• L. Streit (Institute)

The fBm Edwards Model Taipei, August 12, 2010 15 / 26

Fractional BM in Terms of WN

Find functions K (t, s) such that the Gaussian, centered random variables

Z

K (t, s) ω(s)ds have the same covariance as BH

t . A canonical, adapted solution, with BH

depending only on ω(s) with 0 < s < t is BH

t =

Z t

0 K (t, s) ω(s)ds

where K (t, s) = Θt(s) if H = 1/2 K (t, s) = cHs1/2H

Z t

s (u s)H3/2 uH1/2du if H 6= 1/2

with cH =

• H (2H 1))

β (2 2H, H 1/2)

• ,

β (x, y) Γ (x) Γ (y) Γ (x + y) .

• L. Streit (Institute)

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Sel…ntersection Local Time for fBm

Set Lε :=

Z l

0 dt

Z t

0 ds δε(BH(t) BH(s)).

(4) and study the limit ε & 0. Results for H < 3/4 can be found in

• Y. Hu and D. Nualart. Renormalized self-intersection local time for

fractional Brownian motion. Ann. Probab., 33:948–983, 2005.

• L. Streit (Institute)

The fBm Edwards Model Taipei, August 12, 2010 17 / 26

Properties of the local time (Hu&Nualart)

For Hd < 1 the approximated self-intersection local time Lε converges in L2: Lε ! L 0 Hence exp (gL) is well-de…ned. Varadhan’s method is based on logarithmic divergence, Hd = 1 would seem tractable!

• L. Streit (Institute)

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Properties of the local time (Hu&Nualart)

For Hd < 1 the approximated self-intersection local time Lε converges in L2: Lε ! L 0 Hence exp (gL) is well-de…ned. For 1 Hd < 3/2 the approximated and centered self-intersection local time Lc

ε converges in L2:

Lc

ε = Lε E (Lε) ! Lc

Varadhan’s method is based on logarithmic divergence, Hd = 1 would seem tractable!

• L. Streit (Institute)

The fBm Edwards Model Taipei, August 12, 2010 18 / 26

Properties of the local time (Hu&Nualart)

For Hd < 1 the approximated self-intersection local time Lε converges in L2: Lε ! L 0 Hence exp (gL) is well-de…ned. For 1 Hd < 3/2 the approximated and centered self-intersection local time Lc

ε converges in L2:

Lc

ε = Lε E (Lε) ! Lc

E(Lε) = 8 > < > : CH,dεd/2+1/(2H) + o(ε), if 1 < Hd < 3/2

l 2H(2π)d/2 ln(1/ε) + o(ε),

if Hd = 1 where CH,d is a positive constant which depends of H and d. Varadhan’s method is based on logarithmic divergence, Hd = 1 would seem tractable!

• L. Streit (Institute)

The fBm Edwards Model Taipei, August 12, 2010 18 / 26

The rate of convergence

Varadhan’s method will require E (Lc

ε Lc)2 Aεα

for some α > 0, i.e. we need to determine the rate of convergence for Lc

ε = Lε E (Lε) ! Lc.

• L. Streit (Institute)

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More Hu and Nualart results

E(L2

ε ) =

1 (2π)d

Z

T dτ

1 ((λ + ε)(ρ + ε) µ2)d/2 , where T := f(s, t, s0, t0) : 0 < s < t < T, 0 < s0 < t0 < Tg (5) and, for each τ = (s, t, s0, t0) 2 T , λ = λ(τ), ρ = ρ(τ), µ = µ(τ) are given by λ := (t s)2H, ρ := (t0 s0)2H, and µ := 1 2 h js t0j2H + js0 tj2H jt t0j2H js s0j2Hi .

• L. Streit (Institute)

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The crucial estimate

• M. Grothaus, M.J. Oliveira, L.S.: Self-avoiding fractional Brownian

motion - The Edwards model. arXiv:1007.3445v1 [math-ph] 20 Jul 2010

Theorem

Assume that dH = 1. Then there is a positive constant k such that E

• [(Lε E(Lε)) Lc]2

kε1/2 (6) for all ε > 0.

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Main theorem

Theorem

The Edwards model is well de…ned for all H < 1/d, with G = 1 Z exp

• g

Z l

0 ds

Z l

0 dtδ

• BH(s) BH(t)
• L. Streit (Institute)

The fBm Edwards Model Taipei, August 12, 2010 22 / 26

Main theorem

Theorem

The Edwards model is well de…ned for all H < 1/d, with G = 1 Z exp

• g

Z l

0 ds

Z l

0 dtδ

• BH(s) BH(t)
• For H = 1/d,

G = lim

ε&0

1 Zε exp

• g

Z l

0 ds

Z l

0 dtδε

• BH(s) BH(t)
• ,

with Zε E

• exp
• g

Z l

0 ds

Z l

0 dtδε

• BH(s) BH(t)
• is well-de…ned.
• L. Streit (Institute)

The fBm Edwards Model Taipei, August 12, 2010 22 / 26

A glimpse of the proof

T \ fs < s0g = T1 [ T2 [ T3, where T1 := f(t, s, t0, s0) : 0 < s < s0 < t < t0 < Tg, T2 := f(t, s, t0, s0) : 0 < s < s0 < t0 < t < Tg, T3 := f(t, s, t0, s0) : 0 < s < t < s0 < t0 < Tg. We set D := d + 1. Subregion T1: Set a := s0 s, b := t s0, and c = t0 t for (t, s, t0, s0) 2 T1. Thus have λ(t, s, t0, s0) =: λ1(a, b, c) = (a + b)2H, ρ(t, s, t0, s0) = (b + c)2H µ(t, s, t0, s0) =: µ1(a, b, c) = 1 2 h (a + b + c)2H + b2H c2H a2Hi .

• L. Streit (Institute)

The fBm Edwards Model Taipei, August 12, 2010 23 / 26

On the region T1 one can bound E

• (Lc

ε Lc)2

• E00 Eε0

= d 2(2π)d

Z

T dτ ρ

Z ε

0 dx

1 (δ + xρ)d/2+1 1 ((λ + x)ρ)d by the …rst term only, and to estimate the latter we shall use the Schwartz inequality, yielding ρ

Z ε

0 dx

1 (δ1 + xρ )(D+1)/2 Aε1/2ρ1/2δD/2

1

. From Hu and Nualart have δ1 k(a + b)H(b + c)HaHcH k(abc)4H/3, and we deduce

Z

[0,T ]3 da db dc δD/2 1

k

Z

[0,T ]3 da db dc (abc)2DH/3 < ∞,

using DH < 3/2. In conclusion the term from integration over T1 is of

• rder ε1/2. Similar estimates prevail for T2, T3.
• L. Streit (Institute)

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Polymers and the Hurst Index

"....by suitable choice of the parameter H, the average con…gurational behavior of the chain can be made to correspond to its actual behavior in solvents of di¤erent quality, thereby eliminating the need to account in detail for the nature of the intermolecular potential appropriate to the given solvent. For instance, the choice H = 3/5 models polymers in good solvents, while the choice H = l/3 models polymers in compact or collapsed phases. This is not to say that this approach is faithful to the underlying chemical constitution of the chain, even in a coarse-grained sense, only that it is a useful calculational tool."

• P. Biswas, B. J. Cherayil: Dynamics of Fractional Brownian Walks. J.
• Phys. Chem. 99, 816-821 (1995).
• L. Streit (Institute)

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Westwater

For polymers in good solvents the models should have H > 1/2. Those would require a multiplicative renormalization r(ε) Lren = lim

ε&0 r(ε) (Lε E(Lε))

(9) like in the case of ordinary Brownian motion in three space d = 3. For the latter see e.g.

• J. Westwater: On Edwards’ model for polymer chains. Comm. Math.
• Phys. 72, 131-174 (1980)
• J. Westwater: On Edwards’ model for polymer chains. III. Borel
• summability. Comm. Math. Phys. 84, 459 470 (1982)
• E. Bolthausen: On the Construction of the three dimensional polymer
• measure. Prob. Theory. Rel. Fields 97, 81-101 (1993).
• L. Streit (Institute)

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