Exponential integrability and application to stochastic quantization

Applied Mathematics and Optimization

Volumn 37, Issue 3, 1998, Pages 295-353

  Hu, Y ^a,b   Kallianpur, G ^a  

^a University of North Carolina at Chapel Hill   (United States)

^b UNIVERSITY OF KANSAS   (United States)

Author keywords

Cameron Martin Girsanov theorem; Exponential integrability; Infinite dimensional stochastic differential equation; Multiple integral; P( )2 field; Stochastic quantization

Indexed keywords

DIFFERENTIAL EQUATIONS; INTEGRATION; OPTIMIZATION; RANDOM PROCESSES; THEOREM PROVING;

CAMERON-MARTIN-GIRSANOV THEOREM; STOCHASTIC QUANTIZATION;

INTEGRAL EQUATIONS;

EID: 0032072330     PISSN: 00954616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002459900078     Document Type: Article

References (52)

1. S. Albeverio and R. Høegh-Krohn, Dirichlet forms and diffusion processes on rigged Hilbert spaces, Z. Wahrsch. Verw. Gebiete 40 (1977), 1-58.
2. S. Albeverio and R. Høegh-Krohn, Hunt processes and analytic potential theory on rigged Hilbert spaces, Ann. Inst. H. Poincaré Sect. B 13 (1977), 269-291.
3. S. Albeverio and M. Röckner, Classical Dirichlet forms on topological vector spaces, construction of an associated diffusion process, Probab. Theory Relat. Fields 83 (1989), 405-435.
4. S. Albeverio and M. Röckner, Classical Dirichlet forms on topological vector spaces, Closability and a Cameron-Martin formula, J. Funct. Anal. 88 (1990), 395-436.
5. S. Albeverio and M. Röckner, Stochastic differential equations in infinite dimensions, Solutions via Dirichlet forms, Probab. Theory Relat. Fields 89 (1991), 347-386.
6. S. Albeverio, M. Röckner, and T.S. Zhang, Girsanov transform for symmetric diffusions with infinite-dimensional state space, Ann. Probab. 21 (1993), 961-978.
7. V.S. Borkar, R.T. Chari, and S.K. Mitter, Stochastic quantization of field theory in finite volume and infinite volume, J. Funct. Anal. 81 (1988), 184-206.
8. H.J. Brascamp and E.H. Lieb, Best constants in Young's inequality, its converse, and its generalization to more than three functions, Adv. in Math. 20 (1976), 151-173.
9. R.A. Carmona and D. Nualart, Traces of random variables on Wiener space and the Onsage-Machlup functionals, J. Funct. Anal. 107 (1992), 402-438.
10. S. Chaturvedi, A.K. Kapoor, and V. Shrinivasan, Stochastic Quantization Scheme of Parisi and Wu, Bibliopolis, 1990.
11. E.B. Davies, Quantum Theory of Open Systems, Academic Press, London, 1976.
12. E.B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989.
13. Dalecky and S.V. Fomin, Measures and Differential Equations in Infinite-Dimensional Space, Kluwer, Dordrecht, 1991.
14. P.H. Damgaard and H. Hüffel, Stochastic Quantization, World Scientific, Singapore, 1988.
15. C. Dellacherie, P.A. Meyer, and B. Maisonneuve, Probabilités et Potentiel, vol. 5, 1992.
16. N. Dunford and J.T. Schwartz, Linear Operators, Part II, Interscience, New York, 1963.
17. G. Da Prato and J. Zabczk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.
18. W. Faris and G. Jona-Lasinio, Large fluctuations for a nonlinear heat equation with noise, J. Phys. A 15 (1982), 3025-3055.
19. G.B. Folland, Real Analysis, Modern Techniques and Their Applications, Wiley, New York, 1984.
20. M. Fukushima, Dirichlet Forms and Markov Processes, North-Holland, Amsterdam, 1980, 1994.
21. M. Fukushima, On a stochastic calculus related to Dirichlet forms and distorted Brownian motion, Phys. Rep. 77 (1981), 255-262.
22. M. Fukushima, Energy forms and diffusion process, in Mathematics and Physics, Lecture Notes on Recent Results, ed. L. Streit, World Scientific, 1984.
23. D. Gatarek and B. Gotdys, Existence, uniqueness and ergodicity for stochastic quantization equation, Studia Math. 119 (1996), 179-193.
24. J. Glimm and A. Jaffe, Quantum Physics, a Functional Integral Point of View, Springer-Verlag, New York, 1981, 1987.
25. L. Gross, Abstract Wiener spaces, Proc. 5th Berkeley Symp. on Mathematics, Statistics, and Probability, vol. 2, 1965, pp. 31-42.
26. 2 Euclidean quantum field theory as classical statistical mechanics, Ann. of Math. 101 (1975), 111-259.
27. Y.Z. Hu and P.A. Meyer, Chaos de Wiener et intégrales de Feynman, Lecture Notes in Mathematics, vol. 1321, Springer-Verlag, Berlin, 1988, pp. 51-71.
28. Y.Z. Hu and RA. Meyer, Sur les intégrales multiples de Stratonovitch, Lecture Notes in Mathematics, vol. 1321, Springer-Verlag, Berlin, 1988, pp. 72-81.
29. Y.Z. Hu and P.A. Meyer, On the approximation of Stratonovitch multiple integrals, in Stochastic Processes, a Festschrift in Honor of G. Kallianpur, eds. S. Cambanis et al., Springer-Verlag, New York, 1993, pp. 141-147.
30. N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2nd edn., Elsevier/North-Holland, Amsterdam, 1989.
31. G. Jona-Lasinio and P.K. Mitter, On the stochastic quantization of field theory, Comm. Math. Phys. 101 (1985), 409-436.
32. G.W. Johnson and G. Kallianpur, Some remarks on Hu and Meyer's paper and infinite-dimensional calculus on finitely additive canonical Hilbert space, Theory Probab. Appl. 34 (1989), 742-752.
33. G.W. Johnson and G. Kallianpur, Multiple Wiener integrals on abstract Wiener spaces and lifting of p-linear forms, in White Noise Analysis, eds. T. Hida et al., World Scientific, Singapore, 1990, pp. 208-219.
34. G.W. Johnson and G. Kallianpur, Homogeneous chaos, p-forms, scaling, and the Feynman integral, Trans. Amer. Math. Soc. 340 (1993), 503-549.
35. G.W. Johnson and G. Kallianpur, The analytic Feynman integral of the natural extension of pth homogeneous chaos, Rend. Cire. Mat. Palermo, to appear.
36. G. Kallianpur, Trace, natural extension and Feynman distribution, in Gaussian Random Fields, eds. K. Itô and H. Hida, World Scientific, Singapore, 1991, pp. 14-28.
37. N. Kazamaki, The equivalence of two conditions on weighted norm inequalities for martingales, in Proc. Internat. Symp. on Stochastic Differential Equations, Kyoto, 1976, ed. K. Itô, Wiley, New York, 1978.
38. G. Kallianpur, I. Mitoma, and R.L. Wolpert, Diffusion equations in duals of nuclear spaces, Stochastics 20 (1990), 285-329.
39. G. Kallianpur and A.S. Ustunel, Distributions, Feynman integrals and measures on abstract Wiener spaces, in Stochastic Analysis and Related Topics, eds. H. Körezlioglu and A.S. Ustunel, pp. 237-284.
40. P.A. Meyer, Quantum Probability for Probabilists, Lecture Notes in Mathematics, vol. 1538, Springer-Verlag, Berlin, 1993.
41. M. Namiki, Stochastic Quantization, Springer-Verlag, Berlin, 1992.
42. E. Nelson, A quadratic interaction in two dimensions, in Mathematical Theory of Elementary Particles, eds. R. Goodman and I. Segal, M.I.T. Press, Cambridge, MA, 1966, pp. 69-73.
43. D. Nualart and E. Pardoux, Stochastic calculus with anticipating integrals, Probab. Theory Relat. Fields 78 (1988), 535-581.
44. D. Nualart and M. Zakai, Generalized stochastic integrals and the Malliavin calculus, Probab. Theory Relat. Fields 73 (1986), 255-280.
45. D.W. Robison, Elliptic Operators and Lie Groups, Clarendon Press, Oxford, 1991.
46. M. Reed and B. Simon, Methods of Modern Mathematical Physics, III: Scattering Theory, Academic Press, New York, 1972.
47. 2 Euclidean Field Theory, Princeton University Press, Princeton, NJ, 1974.
48. W. Stummer, The Novikov and entropy conditions of multi-dimensional diffusion processes with singular drift, Probab. Theory Relat. Fields 97 (1993), 515-542.
49. e Sér. 113 (1989), 463-474.
50. D.W. Stroock and S.R.S. Varadhan, Multidimensional Diffusion Processes, Springer-Verlag, Berlin, 1979.
51. T. Yamada, Sur une construction des solutions d'équation differentielle stochastique dans le cas non lipschitzien, Sem. Prob. LNM 649 (1978).
52. K. Yosida, Functional Analysis, sixth edn., Springer-Verlag, New York, 1980.