Abstract
We consider various models of polymer conformations using paths of Gaussian processes such as Brownian motion. In each case, the calculation of the law of the moment of inertia of a random polymer structure (which is equivalent to the calculation of the partition function) is reduced to the problem of finding the law of a certain quadratic functional of a Gaussian process. We present a new method for computing the Laplace transforms of these quadratic functionals which exploit their special form via the Ray-Knight Theorem and which does not involve the classical method of eigenvalue expansions. We apply the method to several simple examples, then show how the same techniques can be applied to more complicated cases with the aid of a little excursion theory.
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Chan, T.: Indefinite quadratic functionals of Gaussian processes and least-action paths. Ann. Inst. H. Poincaré27, 2 (1991)
Chiang, T.S., Chow, Y., Lee, Y.J.: An abstract Wiener space approach to certain functional integrals. Preprint (1991)
Donati-Martin, C., Yor, M.: Fubini's theorem for double Wiener integrals and the variance of the Brownian path. Ann. Inst. H. Poincaré27, 2 (1991)
Duplantier, B.: Areas of planar Brownian curves. J. Phys. A22, 3033–3048 (1989)
Dynkin, E.B.: Markov processes and random fields. Bull. AMS3, 975–999 (1980)
Dynkin, E.B.: Markov processes, random fields and Dirichlet spaces. Phys. Reports77, 239–247 (1981)
Fixman, M.: Radius of gyration of polymer chains. J. Chem. Phys.36, 306–318 (1962)
Gaveau, B.: Principe de moindre action, propagation de chaleur et estimées sous-elliptiques sur certains groupes nilpotents. Acta Math.139, 95–153 (1977)
Helfer, A.D., Zhao, Z.: Gaussian integration on Wiener spaces. J. Appl. Prob.29, 46–55 (1992)
Krée, P.: A remark on Paul Lévy's stochastic area formula. In: J. Barroso (ed.) Aspects of Mathematics and its Applications. Elsevier (1986)
Lévy, P.: Wiener's random function and other Laplacian random functions. In: J. Neyman (ed.) Proceedings of the Second Berkeley Symposium in Mathematical Statistics and Probability 171–187 (1951)
McAonghusa, P., Pulé, J.V.: An extension of Lévy's stochastic area formula. Stochastics & Stochastics Reports26, 247–255 (1989)
McGill, P.: A direct proof of the Ray-Knight Theorem. In: Séminaire de Probabilités XV, LMN850, Berlin, Heidelberg, New York: Springer, pp. 206–209 (1981)
Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Berlin, Heidelberg, New York: Springer 1991
Rogers, L.C.G.: Itô excursion theory via resolvents. Z. Wahrscheinlichkeitstheorie verw. Gebiete63, 237–255 (1983)
Rogers, L.C.G.: A guided tour through excursions.Bull. LMS 21, 305–341 (1985)
Rogers, L.C.G., Williams, D.: Diffusions, Markov Processes and Martingales. Volume 2: Itô Calculus, Wiley (1987)
Rogers, L.C.G., Shi, Z.: Quadratic functionals of Gaussian processes, optimal control and the ‘Colditz’ example. Stochastics41, 201–218 (1992)
Williams, D.: Decomposing the Brownian path. Bull. Am. Math. Soc.76, 871–873 (1970)
Williams, D.: Some basic theorems on harnesses. In: D.G. Kendall, E.F. Harding (eds) Stochastic Analysis. New York: Wiley, pp. 349–366 (1973)
Williams, D.: Path decomposition and continuity of local time for one-dimensional diffusions, I. Proc. LMS28, 738–768 (1974)
Yor, M.: On stochastic areas and averages of planar Brownian motion. J. Phys. A22, 3049–3057 (1989)
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Communicated by M. Aizenman
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Chan, T., Dean, D.S., Jansons, K.M. et al. On polymer conformations in elongational flows. Commun.Math. Phys. 160, 239–257 (1994). https://doi.org/10.1007/BF02103275
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DOI: https://doi.org/10.1007/BF02103275