CONDITIONAL TRANSFORM WITH RESPECT TO THE GAUSSIAN PROCESS INVOLVING THE CONDITIONAL CONVOLUTION PRODUCT AND THE FIRST VARIATION

Title & Authors
CONDITIONAL TRANSFORM WITH RESPECT TO THE GAUSSIAN PROCESS INVOLVING THE CONDITIONAL CONVOLUTION PRODUCT AND THE FIRST VARIATION
Chung, Hyun Soo; Lee, Il Yong; Chang, Seung Jun;

Abstract
In this paper, we define a conditional transform with respect to the Gaussian process, the conditional convolution product and the first variation of functionals via the Gaussian process. We then examine various relationships of the conditional transform with respect to the Gaussian process, the conditional convolution product and the first variation for functionals F in $\small{S_{\alpha}}$ [5, 8].
Keywords
Brownian motion process;Wiener integral;Gaussian process;conditional convolution product;simple formula;conditional transform with respect to Gaussian process;
Language
English
Cited by
1.
Series expansions of the transform with respect to the Gaussian process, Integral Transforms and Special Functions, 2015, 26, 4, 246
2.
Generalized conditional transform with respect to the Gaussian process on function space, Integral Transforms and Special Functions, 2015, 26, 12, 925
References
1.
R. H. Cameron, Some examples of Fourier-Wiener transforms of analytic functionals, Duke Math. J. 12 (1945), 485-488.

2.
R. H. Cameron and W. T. Martin, Fourier-Wiener transforms of functionals belonging to $L_2$ over the space C, Duke Math. J. 14 (1947), 99-107.

3.
S. J. Chang, H. S. Chung, and D. Skoug, Some basic relationships among transforms, convolution products, first variations and inverse transforms, Cent. Eur. J. Math. 11 (2013), no. 3, 538-551.

4.
J. G. Choi and S. J. Chang, A rotation on Wiener space with applications, ISRN Appl. Math. 2012 (2012), Art. ID 578174, 13 pages.

5.
H. S. Chung, J. G. Choi, and S. J. Chang, Conditional integral transforms with related topics on function space, Filomat 26 (2012), no. 6, 1151-1162.

6.
D. M. Chung, C. Park, and D. Skoug, Generalized Feynman integrals via conditional Feynman integrals, Michigan Math. J. 40 (1993), no. 2, 337-391.

7.
H. S. Chung, D. Skoug, and S. J. Chang, Relationships involving transform and convolutions via the translation theorem, Stoch. Anal. Appl. 32 (2014), no. 2, 348-363.

8.
H. S. Chung and V. K. Tuan, Generalized integral transforms and convolution products on function space, Integral Transforms Spec. Funct. 22 (2011), no. 8, 573-586.

9.
D. L. Cohn, Measure Theory, Birkhauser-Verlag, Boston, 1980.

10.
G. W. Johnson and D. L. Skoug, Scale-invariant measurability in Wiener space, Pacific J. Math. 83 (1979), no. 1, 157-176.

11.
B. J. Kim, Conditional integral transforms, conditional convolution products and first variations for some conditioning functions, Far East J. Math. Sci. 19 (2005), no. 3, 245-258.

12.
B. S. Kim, B. J. Kim, and D. Skoug, Conditional integral transforms, conditional convolution products and first variations, Panamer. Amer. Math. J. 14 (2004), no. 3, 27-47.

13.
B. S. Kim and D. Skoug, Integral transforms of functionals in $L_2(C_0$[0, T]), Rocky Mountain J. Math. 33 (2003), no. 4, 1379-1393.

14.
I. Y. Lee, H. S. Chung, and S. J. Chang, Relationships among the transform with respect to the Gaussian process, the ${\diamond}$-product and the first variation of functionals on function space, to submitted for publications.

15.
C. Park and D. Skoug, A simple formula for conditional Wiener integrals with applications, Pacific J. Math. 135 (1988), no. 2, 381-394.

16.
W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1966.