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) The fBm Edwards Model Taipei, August 12, 2010 2 / 26
"Brownian" Polymers? Strategy: random paths x ( s ) 0 � s � l 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 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) The fBm Edwards Model Taipei, August 12, 2010 3 / 26
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) The fBm Edwards Model Taipei, August 12, 2010 4 / 26
Self-intersection Local Time To control self-intersections of paths consider Z l Z l L = 0 dt δ ( x ( s ) � x ( t ) ) 0 ds 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) The fBm Edwards Model Taipei, August 12, 2010 5 / 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: � � Z l Z l G = 1 Z exp � g 0 ds 0 dt δ ( x ( s ) � x ( t ) ) with � � �� Z l Z l Z = E exp � g 0 ds 0 dt δ ( 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: � � Z l Z l G = 1 Z exp � g 0 ds 0 dt δ ( x ( s ) � x ( t ) ) with � � �� Z l Z l Z = E exp � g 0 ds 0 dt δ ( 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 Z l Z l L = 0 ds 0 dt δ ( B ( s ) � B ( t ) ) L. Streit (Institute) The fBm Edwards Model Taipei, August 12, 2010 7 / 26
A closer look at self-intersection local times How to make sense of Z l Z l L = 0 ds 0 dt δ ( B ( s ) � B ( t ) ) Use delta sequences: ( 2 πε ) d / 2 e � j x j 2 1 2 ε , δ ε ( x ) : = ε > 0 , Z T Z t L ε : = dt 0 ds δ ε ( B ( t ) � B ( s )) . (1) 0 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 ε & 0 E ( L ε ) = ∞ lim 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 ε & 0 E ( L ε ) = ∞ lim For d = 2 have 2 π ln 1 l E ( L ε ) = ε + o ( ε ) and L 2 convergence after centering: L ε � E ( L ε ) ! L c . (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 ε & 0 E ( L ε ) = ∞ lim For d = 2 have 2 π ln 1 l E ( L ε ) = ε + o ( ε ) and L 2 convergence after centering: L ε � E ( L ε ) ! L c . (2) as ε tends to zero. For d � 3 a further multiplicative renormalization r ( ε ) is required r ( ε ) ( L ε � E ( L ε )) . (3) L. Streit (Institute) The fBm Edwards Model Taipei, August 12, 2010 8 / 26
The case d=2 Now consider exp ( � gL c ) L. Streit (Institute) The fBm Edwards Model Taipei, August 12, 2010 9 / 26
The case d=2 Now consider exp ( � gL c ) Problem: L c is unbounded below, and when L c ! � ∞ . exp ( � gL c ) ! ∞ L. Streit (Institute) The fBm Edwards Model Taipei, August 12, 2010 9 / 26
The case d=2 Now consider exp ( � gL c ) Problem: L c is unbounded below, and when L c ! � ∞ . exp ( � gL c ) ! ∞ 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 - o a c(n ,c) t1 ts h{ xr L-r z f 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 Show logarithmic lower bound 1 ε > � c ln 1 L c ε Show that the rate of convergence is 2 � ( L c ε � L c ) 2 � � A ε 1 / 2 E Now estimate 3 prob ( L c � � K ) . L. Streit (Institute) The fBm Edwards Model Taipei, August 12, 2010 10 / 26
prob ( L c � � K ) prob ( L c � L c ε + L c = ε � � K ) prob ( L c � L c ε � � K � L c = ε ) � � ε � � K + c ln 1 L c � L c � prob ε Now use the Chebychev inequality to get � � A ε 1 / 2 ε � � K + c ln 1 L c � L c prob � � � 2 ε K � c ln 1 ε Choosing 2 c ln 1 = K ε � � � K ε = exp 2 c the probability for large negative values � � K � prob ( L c � � K ) � 4 A exp 4 c K 2 is exponentially small. L. Streit (Institute) The fBm Edwards Model Taipei, August 12, 2010 11 / 26
As a result E ( exp ( � gL c )) < ∞ 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) The fBm Edwards Model Taipei, August 12, 2010 12 / 26
Fractional Brownian Motion Fractional Brownian motion fBm on R d , d � 1, with "Hurst parameter" H 2 ( 0 , 1 ) is a d -dimensional centered Gaussian process B H = f B H t : t � 0 g with covariance function � t 2 H + s 2 H � j t � s j 2 H � s ) = δ ij E ( B H t B H i , j = 1 , . . . , d , s , t � 0 . , 2 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 R d , d � 1, with "Hurst parameter" H 2 ( 0 , 1 ) is a d -dimensional centered Gaussian process B H = f B H t : t � 0 g with covariance function � t 2 H + s 2 H � j t � s j 2 H � s ) = δ ij E ( B H t B H i , j = 1 , . . . , d , s , t � 0 . , 2 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
Recommend
More recommend
