DOI QR코드

DOI QR Code

A CHANGE OF SCALE FORMULA FOR CONDITIONAL WIENER INTEGRALS ON CLASSICAL WIENER SPACE

  • 발행 : 2007.07.30

초록

Let $X_k(x)=({\int}^T_o{\alpha}_1(s)dx(s),...,{\int}^T_o{\alpha}_k(s)dx(s))\;and\;X_{\tau}(x)=(x(t_1),...,x(t_k))$ on the classical Wiener space, where ${{\alpha}_1,...,{\alpha}_k}$ is an orthonormal subset of $L_2$ [0, T] and ${\tau}:0 is a partition of [0, T]. In this paper, we establish a change of scale formula for conditional Wiener integrals $E[G_{\gamma}|X_k]$ of functions on classical Wiener space having the form $$G_{\gamma}(x)=F(x){\Psi}({\int}^T_ov_1(s)dx(s),...,{\int}^T_o\;v_{\gamma}(s)dx(s))$$, for $F{\in}S\;and\;{\Psi}={\psi}+{\phi}({\psi}{\in}L_p(\mathbb{R}^{\gamma}),\;{\phi}{\in}\hat{M}(\mathbb{R}^{\gamma}))$, which need not be bounded or continuous. Here S is a Banach algebra on classical Wiener space and $\hat{M}(\mathbb{R}^{\gamma})$ is the space of Fourier transforms of measures of bounded variation over $\mathbb{R}^{\gamma}$. As results of the formula, we derive a change of scale formula for the conditional Wiener integrals $E[G_{\gamma}|X_{\tau}]\;and\;E[F|X_{\tau}]$. Finally, we show that the analytic Feynman integral of F can be expressed as a limit of a change of scale transformation of the conditional Wiener integral of F using an inversion formula which changes the conditional Wiener integral of F to an ordinary Wiener integral of F, and then we obtain another type of change of scale formula for Wiener integrals of F.

키워드

참고문헌

  1. R. H. Cameron, The translation pathology of Wiener space, Duke Math. J. 21 (1954), 623-627 https://doi.org/10.1215/S0012-7094-54-02165-1
  2. R. H. Cameron and W. T. Martin, The behavior of measure and measurability under change of scale in Wiener space, Bull. Amer. Math. Soc. 53 (1947), 130-137 https://doi.org/10.1090/S0002-9904-1947-08762-0
  3. R. H. Cameron and D. A. Storvick, Some Banach algebras of analytic Feynman integrable functionals, Lecture Notes in Math. 798, Springer-Verlag, New York (1980), 18-67
  4. R. H. Cameron and D. A. Storvick, Relationships between the Wiener integral and the analytic Feynman integral, Rend. Circ. Mat. Palermo (2) Suppl. 17 (1987), 117-133
  5. R. H. Cameron and D. A. Storvick, Change of scale formulas for Wiener integral, Rend. Circ. Mat. Palermo (2) Suppl. 17 (1987), 105-115
  6. K. S. Chang, G. W. Johnson, and D. L. Skoug, Functions in the Fresnel class, Proc. Amer. Math. Soc. 100 (1987), no. 2, 309-318 https://doi.org/10.2307/2045963
  7. D. M. Chung and D. L. Skoug, Conditional analytic Feynman integrals and a related Schrodinger integral equation, SIAM J. Math. Anal. 20 (1989), no. 4, 950-965 https://doi.org/10.1137/0520064
  8. C. Park and D. L. Skoug, Conditional Wiener integrals II, Pacific J. Math. 167 (1995), no. 2, 293-312 https://doi.org/10.2140/pjm.1995.167.293
  9. C. Park and D. L. Skoug, A simple formula for conditional Wiener integrals with applications, Pacific J. Math. 135 (1988), no. 2, 381-394 https://doi.org/10.2140/pjm.1988.135.381
  10. I. Yoo and D. L. Skoug, A change of scale formula for Wiener integrals on abstract Wiener spaces, Internat. J. Math. Math. Sci. 17 (1994), no. 2, 239-247 https://doi.org/10.1155/S0161171294000359
  11. I. Yoo and D. L. Skoug, A change of scale formula for Wiener integrals on abstract Wiener spaces II, J. Korean Math. Soc. 31 (1994), no. 1, 115-129
  12. I. Yoo, T. S. Song, B. S. Kim, and K. S. Chang, A change of scale formula for Wiener integrals of unbounded functions, Rocky Mountain J. Math. 34 (2004), no. 1, 371-389 https://doi.org/10.1216/rmjm/1181069911
  13. I. Yoo and G. J. Yoon, Change of scale formulas for Yeh-Wiener integrals, Commun. Korean Math. Soc. 6 (1991), no. 1, 19-26

피인용 문헌

  1. Integral Transforms on a Function Space with Change of Scales Using Multivariate Normal Distributions vol.2016, 2016, https://doi.org/10.1155/2016/9235960
  2. Analogues of conditional Wiener integrals and their change of scale transformations on a function space vol.359, pp.2, 2009, https://doi.org/10.1016/j.jmaa.2009.05.023
  3. SCALE TRANSFORMATIONS FOR PRESENT POSITION-INDEPENDENT CONDITIONAL EXPECTATIONS vol.53, pp.3, 2016, https://doi.org/10.4134/JKMS.j150285
  4. A CHANGE OF SCALE FORMULA FOR GENERALIZED WIENER INTEGRALS II vol.26, pp.1, 2013, https://doi.org/10.14403/jcms.2013.26.1.111