Abstract
In this paper, we give a nonstandard construction of the free Euclidean field via S-white noise. This provides a flat integral realization of the free Euclidean field measure, which extends N. J. Cutland's flat integral representation of Wiener measure. Moreover, we show how a Cameron-Martin type formula for translations of the free field measure and a Schilder type large deviation principle for the scalar free field measure can be deduced from our nonstandard construction.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Albeverio, S.: Nonstandard analysis in mathematical physics, in: N. J. Cutland (ed.), Nonstandard Analysis and its Applications, LMS Student Text 10, Cambridge Univ. Press, 1988, pp. 182–220.
Albeverio, S., Fenstad, J. E., Høegh-Krohn, R., and Lindstrøm, T.: Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Academic Press, New York, 1986.
Albeverio, S. and Høegh-Krohn, R.: Martingale convergence and the exponential interaction in 347–1, in: L. Streit (ed.), Quantum Fields-Algebras, Processes, Springer-Verlag, Berlin, 1980, pp. 331–353.
Albeverio, S. and Wu, J. L.: Nonstandard flat integral representation of the free Euclidean field and a large deviation bound for the exponential interaction, in: N. Cutland, V. Neves, A. J. Franco de Oliveira and J. Sousa Pinto (eds), Developments in Nonstandard Mathematics (Proceedings of the CIMNS 94 Conference), Longman, Harlow.
Anderson, R. M.: A nonstandard representation for Brownian motion and Itô integration, Israel J. Math. 25 (1976), 15–46.
Cutland, N. J.: Nonstandard measure theory and its applications, Bull. London Math. Soc. 15 (1983), 529–589.
Cutland, N. J.: Infinitesimals in action, J. London Math. Soc. 35 (1987), 202–216.
Cutland, N. J.: The Brownian bridge as a flat integral, Math. Proc. Cambridge Phil. Soc. 106 (1989), 343–354.
Cutland, N. J.: An action functional for Lévy Brownian motion, Acta Appl. Math. 18 (1990), 261–281.
Cutland, N. J.: On large deviations in Hilbert space, Proc. Edinburgh Math. Soc. 34 (1991), 487–495.
Cutland, N. J.: Nonstandard representation of Gaussian measures, in: T. Hida, H.-H. Kuo, J. Potthoff and L. Streit (eds), White Noise-Mathematics and Application, World Scientific, Singapore, 1990, pp. 73–92.
Deuschel, J. D. and Stroock, D. W.: Large Deviations, Academic Press, San Diego, 1989.
Gelfand, I. M. and Vilenkin, N. Ya.: Generalized Functions, Vol. 4. Applications of Harmonic Analysis, Academic Press, New York, 1964.
Glimm, J. and Jaffe, A.. Quantum Physics: A Functional Integral Point of View (2nd edn), Springer-Verlag, New York, 1987.
Guerra, F., Rosen, L., and Simon, B.: The 348–1 Euclidean quantum field theory as classical statistical mechanics, Ann. Math. 101 (1975), 111–259.
Guerra, F., Rosen, L., and Simon, B.: Boundary conditions for the 348–2 Euclidean field theory, Ann. Inst. Henri Poincaré A 25 (1976), 231–334.
Hida, T., Kuo, H.-H., Potthoff, J., and Streit, L.: White Noise: An Infinite Dimensional Calculus, Kluwer Acad. Publ., Dordrecht, 1993.
Itô, K.: Foundation of Stochastic Differential Equations in Infinite Dimensional Spaces, CBMS 47, SIAM, Philadelphia, 1984.
Jona-Lasinio, G. and Mitter, P. K.: Large deviation estimates in the stochastic quantization of 348–3, Comm. Math. Phys. 130 (1990), 111–121.
Lindstrøm, T.: Hyperfinite stochastic integration I, Math. Scand. 46 (1980), 265–292.
Lindstrøm, T.: An invitation to nonstandard analysis, in: N. J. Cutland (ed.), Nonstandard Analysis and its Applications, LMS Student Text 10, Cambridge Univ. Press, 1988, pp. 1–105.
Mück, S.: Large deviations w.r.t. quasi-every starting point for symmetric right processes on general state spaces, Probab. Theory Related Fields 99 (1994), 527–548.
Nelson, E.: The free Markov field, J. Funct. Anal. 12 (1973), 211–227.
Reed, M. and Rosen, L.: Support properties of the free measure for Boson fields, Comm. Math. Phys. 36 (1974), 123–132.
Reed, M. and Simon, B.: Methods of Mathematical Physics I: Functional Analysis (revised and enlarged edition), Academic Press, New York, 1980; Fourier Analysis, Self-Adjointness, Academic Press, New York, 1975.
Simon, B.: The 348–4 Euclidean (Quantum) Field Theory, Princeton University Press, Princeton, 1974.
Stoll, A.: A nonstandard construction of Lévy Brownian motion, Probab. Theory Related Fields 71 (1986), 321–334.
Stroock, D. W.: An Introduction to the Theory of Large Deviations, Springer-Verlag, New York, 1984.
Yan, J.-A.: Generalizations of Gross' and Minlos' theorems, in: J. Azéma, P. A. Meyer, and M. Yor (eds), Séminaire de Probabilités, XXII. Lect. Notes in Math. 1372, Springer-Verlag, Berlin, 1989, pp. 395–404.
Author information
Authors and Affiliations
Additional information
SFB 237 Essen-Bochum-Düseldorf; BiBoS-Research Centre; CERFIM, Locarno, Switzerland.
Rights and permissions
About this article
Cite this article
Albeverio, S., Wu, JL. A mathematical flat integral realization and a large deviation result for the free Euclidean field. Acta Appl Math 45, 317–348 (1996). https://doi.org/10.1007/BF00047027
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00047027