# REFLECTION PRINCIPLES FOR GENERAL WIENER FUNCTION SPACES

• Pierce, Ian ;
• Skoug, David
• Published : 2013.05.01
• 46 6

#### Abstract

It is well-known that the ordinary single-parameter Wiener space exhibits a reflection principle. In this paper we establish a reflection principle for a generalized one-parameter Wiener space and apply it to the integration of a class of functionals on this space. We also discuss several notions of a reflection principle for the two-parameter Wiener space, and explore whether these actually hold.

#### Keywords

generalized Wiener space;Yeh-Wiener space;generalized Brownian motion;distribution of supremum

#### References

1. E. Berkson and T. A. Gillespie, Absolutely continuous functions of two variables and well-bounded operators, J. London Math. Soc. (2) 30 (1984), no. 2, 305-321.
2. P. Billingsley, Convergence of probability measures, Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons Inc., New York, 1999.
3. R. H. Cameron, A Simpson rule for the numerical evaluation of Wiener's integrals in function space, Duke Math. J. 18 (1951), 111-130. https://doi.org/10.1215/S0012-7094-51-01810-8
4. Robert H. Cameron and David A. Storvick, Two related integrals over spaces of contin- uous functions, Pacific J. Math. 55 (1974), 19-37. https://doi.org/10.2140/pjm.1974.55.19
5. S. J. Chang, J. G. Choi, and D. Skoug, Integration by parts formulas involving generalized Fourier-Feynman transforms on function space, Trans. Amer. Math. Soc. 355 (2003), no. 7, 2925-2948. https://doi.org/10.1090/S0002-9947-03-03256-2
6. S. J. Chang and D. M. Chung, Conditional function space integrals with applications, Rocky Mountain J. Math. 26 (1996), no. 1, 37-62. https://doi.org/10.1216/rmjm/1181072102
7. S. J. Chang and H. S. Chung, Generalized Fourier-Wiener function space transforms, J. Korean Math. Soc. 46 (2009), no. 2, 327-345. https://doi.org/10.4134/JKMS.2009.46.2.327
8. S. J. Chang, H. S. Chung, and D. Skoug, Integral transforms of functionals in $L^2(C_{a,b}\left[0,T \right])$, J. Fourier Anal. Appl. 15 (2009), no. 4, 441-462. https://doi.org/10.1007/s00041-009-9076-y
9. S. J. Chang, Convolution products, integral transforms and inverse integral transforms of functionals in $L^2(C_0\left[0,T \right])$, Integral Transforms Spec. Funct. 21 (2010), no. 1-2, 143-151. https://doi.org/10.1080/10652460903063382
10. S. J. Chang and D. Skoug, Generalized Fourier-Feynman transforms and a first varia- tion on function space, Integral Transforms Spec. Funct. 14 (2003), no. 5, 375-393.
11. J. A. Clarkson and R. C. Adams, On definitions of bounded variation for functions of two variables, Trans. Amer. Math. Soc. 35 (1933), no. 4, 824-854. https://doi.org/10.1090/S0002-9947-1933-1501718-2
12. D. Freedman, Brownian Motion and Diffusion, Holden-Day, San Francisco, 1971.
13. V. Goodman, Distribution estimates for functionals of the two-parameter Wiener process, Ann. Probability 4 (1976), no. 6, 977-982. https://doi.org/10.1214/aop/1176995940
14. I. Karatzas and S. E. Shreve, Brownian motion and stochastic calculus, Graduate Texts in Mathematics 113, Springer-Verlag, New York, 1991.
15. L. Mattner, Complex differentiation under the integral, Nieuw Arch.Wiskd. (5) 2 (2001), no. 1, 32-35.
16. S. R. Paranjape and C. Park, Distribution of the supremum of the two-parameter Yeh-Wiener process on the boundary, J. Appl. Probability 10 (1973), 875-880. https://doi.org/10.2307/3212390
17. I. Pierce and D. Skoug, Comparing the distribution of various suprema on two-parameter Wiener space. To appear in Proc. Amer. Math. Soc.
18. D. Skoug, Converses to measurability theorems for Yeh-Wiener space, Proc. Amer. Math. Soc. 57 (1976), no. 2, 304-310. https://doi.org/10.1090/S0002-9939-1976-0422563-9
19. D. Skoug, Feynman integrals involving quadratic potentials, stochastic integration formulas, and bounded variation for functions of several variables, Rend. Circ. Mat. Palermo (2) Suppl. No. 17 (1987), 331-347.
20. J. Yeh, Approximate evaluation of a class of Wiener integrals, Proc. Amer. Math. Soc. 23 (1969), 513-517.
21. J. Yeh, Wiener measure in a space of functions of two variables, Trans. Amer. Math. Soc. 95 (1960), 433-450. https://doi.org/10.1090/S0002-9947-1960-0125433-1
22. J. Yeh, Cameron-Martin translation theorems in the Wiener space of functions of two variables, Trans. Amer. Math. Soc. 107 (1963), 409-420. https://doi.org/10.1090/S0002-9947-1963-0189138-6
23. J. Yeh, Stochastic Processes and the Wiener Integral, Pure and Applied Mathematics, 13, Marcel Dekker Inc., New York, 1973.
24. G. J. Zimmerman, Some sample function properties of the two-parameter Gaussian process, Ann. Math. Statist. 43 (1972), 1235-1246. https://doi.org/10.1214/aoms/1177692475

#### Cited by

1. THE TRANSLATION THEOREM ON THE GENERALIZED ANALOGUE OF WIENER SPACE AND ITS APPLICATIONS vol.26, pp.4, 2013, https://doi.org/10.14403/jcms.2013.26.4.735