Previous |  Up |  Next


analogue of Wiener measure; Cameron-Martin translation theorem; conditional analytic Feynman $w_\varphi $-integral; conditional Wiener integral; Kac-Feynman formula; simple formula for conditional $w_\varphi $-integral
Let $C[0,T]$ denote the space of real-valued continuous functions on the interval $[0,T]$ with an analogue $w_\varphi $ of Wiener measure and for a partition $ 0=t_0< t_1< \cdots < t_n <t_{n+1}= T$ of $[0, T]$, let $X_n\: C[0,T]\to \mathbb R^{n+1}$ and $X_{n+1} \: C [0, T]\to \mathbb R^{n+2}$ be given by $X_n(x) = ( x(t_0), x(t_1), \cdots , x(t_n))$ and $X_{n+1} (x) = ( x(t_0), x(t_1), \cdots , x(t_{n+1}))$, respectively. \endgraf In this paper, using a simple formula for the conditional $w_\varphi $-integral of functions on $C[0, T]$ with the conditioning function $X_{n+1}$, we derive a simple formula for the conditional $w_\varphi $-integral of the functions with the conditioning function $X_n$. As applications of the formula with the function $X_n$, we evaluate the conditional $w_\varphi $-integral of the functions of the form $F_m(x) = \int _0^T (x(t))^m d t$ for $x\in C[0, T]$ and for any positive integer $m$. Moreover, with the conditioning $X_n$, we evaluate the conditional $w_\varphi $-integral of the functions in a Banach algebra $\mathcal S_{w_\varphi }$ which is an analogue of the Cameron and Storvick's Banach algebra $\mathcal S$. Finally, we derive the conditional analytic Feynman $w_\varphi $-integrals of the functions in $\mathcal S_{w_\varphi }$.
[1] Ash, R. B.: Real analysis and probability. Academic Press, New York-London (1972). MR 0435320
[2] Cameron, R. H., Martin, W. T.: Transformations of Wiener integrals under translations. Ann. Math. 45 (1944), 386-396. DOI 10.2307/1969276 | MR 0010346 | Zbl 0063.00696
[3] Cameron, R. H., Storvick, D. A.: Some Banach algebras of analytic Feynman integrable functionals. Lecture Notes in Math. 798, Springer, Berlin-New York (1980). DOI 10.1007/BFb0097256 | MR 0577446 | Zbl 0439.28007
[4] Chang, K. S., Chang, J. S.: Evaluation of some conditional Wiener integrals. Bull. Korean Math. Soc. 21 (1984), 99-106. MR 0768465 | Zbl 0576.28023
[5] Cho, D. H.: A simple formula for an analogue of conditional Wiener integrals and its applications. Trans. Amer. Math. Soc. 360 (2008), 3795-3811. DOI 10.1090/S0002-9947-08-04380-8 | MR 2386246 | Zbl 1151.28017
[6] Chung, D. M., Skoug, D.: Conditional analytic Feynman integrals and a related Schrödinger integral equation. SIAM J. Math. Anal. 20 (1989), 950-965. DOI 10.1137/0520064 | MR 1000731 | Zbl 0678.28007
[7] Im, M. K., Ryu, K. S.: An analogue of Wiener measure and its applications. J. Korean Math. Soc. 39 (2002), 801-819. DOI 10.4134/JKMS.2002.39.5.801 | MR 1920906 | Zbl 1017.28007
[8] Laha, R. G., Rohatgi, V. K.: Probability theory. John Wiley & Sons, New York-Chichester-Brisbane (1979). MR 0534143 | Zbl 0409.60001
[9] Park, C., Skoug, D.: A simple formula for conditional Wiener integrals with applications. Pacific J. Math. 135 (1988), 381-394. DOI 10.2140/pjm.1988.135.381 | MR 0968620 | Zbl 0655.28007
[10] Ryu, K. S., Im, M. K.: A measure-valued analogue of Wiener measure and the measure-valued Feynman-Kac formula. Trans. Amer. Math. Soc. 354 (2002), 4921-4951. DOI 10.1090/S0002-9947-02-03077-5 | MR 1926843 | Zbl 1017.28008
[11] Yeh, J.: Transformation of conditional Wiener integrals under translation and the Cameron-Martin translation theorem. Tôhoku Math. J. 30 (1978), 505-515. DOI 10.2748/tmj/1178229910 | MR 0516883 | Zbl 0409.28006
[12] Yeh, J.: Inversion of conditional Wiener integrals. Pacific J. Math. 59 (1975), 623-638. DOI 10.2140/pjm.1975.59.623 | MR 0390162 | Zbl 0365.60073
[13] Yeh, J.: Inversion of conditional expectations. Pacific J. Math. 52 (1974), 631-640. DOI 10.2140/pjm.1974.52.631 | MR 0365644 | Zbl 0323.60003
[14] Yeh, J.: Stochastic processes and the Wiener integral. Marcel Dekker, New York (1973). MR 0474528 | Zbl 0277.60018
Partner of
EuDML logo