i. Bonic R. and Frampton J., Differentiable functions on certain Banach spaces, Bull. Amer. Math. Soc. 71(1965),

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1 References i. Bonic R. and Frampton J., Differentiable functions on certain Banach spaces, Bull. Amer. Math. Soc. 71(1965), Cameron R. H. and Graves R., Additive functionals on a space of continuous funcitons, Trans. Amer. Math. Soc. 70 (1951), Cameron R. H. and Martin W. T., Transformation of Wiener integrals under translations, Ann. Math. 45(1944), , Transformation of Wiener integrals under a general class of linear transformations, Trans. Amer. Math. Soc. 58(1945), , The transformation of Wiener integrals by nonlinear transformations, Trans. Amer. Math. Soc. 66(1949), Donsker M. D. and Lions J. L., Volterra variational equations, boundary value problems and function space integrals, Acta Math. 108(1962), Donsker M. D., On function space integrals, in "Analysis in function space edited by W.T. Martin and I.E. Segal (1963), , Lecture notes in "Integration in function space", Courant Institute, New York University (1971). 9. Dudley R.M., Feldman J. and LeCam L., On semi-norms and probabilities, and abstract Wiener spaces., Ann. Math 93

2 219 (1971), i0. Feldman J., Equivalence and perpendicularity of Gaussian processes, Pacific J. Math. 8(1958), ii., A short proof of the Levy continuity theorem in Hilbert space, Israel J. Math. 3 (1965), Fernique M.X., Int~grabilit~ des Vecteurs Gaussiens, Academie des Sciences, Paris, Comptes Rendus, 270 S~ries A (1970), Gel'fand I.M. and Vilenkin N.Ya., Generalized functions, vol. 4, English translation, Academic Press, New York(1964). 14. Gihman I.I. and Skorokhod A.V., Densities of probability measures in function space, Russian Math. Surveys (Engl. transl.) 21 (1966), Goodman V., A divergence theorem for Hilbert space, Trans. Amer. Math. Soc. 164 (1972), Gross L., Measurable functions on Hilbert space, Trans. Amer. Math. Soc. 105(1962), , Harmonic analysis on Hilbert space, Memoirs Amer. Math. Soc. no. 46(1963). 18., Abstract Wiener spaces, Proc. 5th. Berkeley Sym. Math. Stat. Prob. 2(1965), , Potential theory on Hilbert space, J. Func. Anal. 1(1967), Hajek J., A property of J-divergences of marginal probability distributions, Czech. Math. J. 8(1958), , On a property of normal distributions of an

3 220 arbitrary stochastic process, Czech. Math. J. 8(1958), Helms L.L., Mean convergence of martingales, Trans. Amer. Math. Soc. 87(1958), Ito K., The topological support of Gauss measures on Hilbert space, Nagoya Math. J. 38(1970), , Lecture notes in "Stochastic integrals", Cornell University (1972). 25. Kac M., On distributions of certain Wiener integrals, Trans. Amer. Math. Soc. 65(1949), Kakutani $., On equivalence of infinite product measures, Ann. Math. 49(1948), Kallianpur G., Abstract Wiener processes and their reproducing kernel Hilbert spaces, Z. Wahrscheinlichkeitstheorie 17(1971), Kato T., Perturbation theory for linear oeprators, Springer- Verlag, Berlin and New York (1966). 29. Kuelbs J., Gaussian measures on a Banach space J. Func. Anal. 5(1970), Kuo H.-H., Integration theory on infinite-dimensional manifolds, Trans. Amer. Math. Soc. 159(1971), , Stochastic integrals in abstract Wiener space, Pacific J. Math. 41(1972), , Diffusion and Brownian motion on infinis dimensional manifolds, Trans. Amer. Math. Soc. 169(1972),

4 , Stochastic integrals in abstract Wiener space :Regularity properties, Nagoya Math. J. 50(1973), , Integration by parts for abstract Wiener measures, Duke Math. J. 41(1974), Kuo H.-H. and Piech M.A., Stochastic integral and parabolic equation in abstract Wiener space, Bull. Amer. Math. Soc. 79(1973), Maruyama G., Notes on Wiener integrals, Kodai Math. Seminar Rep. 3(1950), Piech M.A., A fundamental solution of the parabolic equation on Hilbert space, J. Func. Anal. 3(1969), Prohorov Yu. V., Convergence for random processes and limit theorems in probability theory, Teor. Veroj. i Prim. 1(1956), , The method of characteristic functionals, Proc. 4th Berkeley Sym. Math. Stat. Prob. (1961), Sazonov V. V., A remark on characteristic functionals, Teor. Veroj. i Prim. 3(1958), Segal I.E., Tensor algebras over Hilbert spaces, Trans. Amer. Math. Soc. 81(1956), , Distributions in Hilbert space and canonical systems of operators, Trans. Amer. Math. Soc. 88(1958)~ Shepp L.A., Gaussian measures in function spaces, Pacific J. Math. 17(1966), Skorokhod A.V., Notes on Gaussian measures in a Banach space, Teor. Veroj. i Prim. 15(1970),

5 Sunouchi G., Harmonic analysis and Wiener integrals, Tohoku Math. J. 3(1951), Varadhan S.R.S., Stochastic processes, Courant Institute, New York University (1968). 47. Whitfield J.H.M.,Differentiable functions with bounded nonempty support on Banach spaces, Bull. Amer. Math. Soc. 72 (1965), Wiener N., The average value of a functional, Proc. London Math. Soc. 22(1922), , Differential space, J. Math. Phys. 58(1923), Selected Papers of Nobert Wiener, SIAM and MIT Press (1965). 51. Yeh J., Stochastic processes and the Wiener integral, Marcel Dekker, New York (1973).

6 INDEX abstract Wiener space 54,63,86,153 Brownian motion, 108,214 characteristic functional, 19 classical Wiener space, 63 compact operator, 7 conditional Banach space, 199 cone, 171 covariance function, 128 covariance operator, 15 cylinder function, 54 cylinder set, 36,54,63,92 cylinder set measure, 92 degenerator operator, 13 dichotomy theorem, i10,125,127 divergence, 213 divergence theorem, 208,213 Donsker, M.D., 51 Donsker's delta function, 50 Donsker's flat integral, 112 equivalence and orthogonality, 110,125,127 Feldman-Hajek's theorem, 118 Fernique, M.X., 159,216 Fr~chet differentiability, 145,168 Gauss measure, 54,74 Gaussian measure, 1,28,128,153 generalized Laplacian, 171 Goodman, V., 83,208 Green measure, 165,184 Gross, L., 63,109,151,216,217 Gross-Sazonov's theorem, 102 H-C 1 boundary, 209 H-C 1 partition, 208 H-C 1 surface, 208 H-differentiability, 145,168 Hellinger integral, 116 Helms' theorem, 130 Hilbert-Schmidt operator, 2,3 Hilbert-Schmidt type n-linear map, 103 intergration by parts, 149 Ito's integral, 108 Ito's lemma, 197,198,200 J-functional, 129 Kac's formula, 48 Kakutani's theorem, 116 Kallianpur, G., 157 Kuelbs, J., 153 Kuo, H.-H., 85

7 224 Laplacian, 168 L~vy's continuity theorem, 98,99,109 linear transformation of Wiener measure, 141 Lions, J.L., 51 martingale, 130 mean, 18 measurable semi-norm, 59 nonanticipating, 189 normal (outward), 212 normal distribution, 56,93 normal projection, 211 Piech, M. Ann, 187 polar decomposition, 8 positive definite functional, 19 potential, 165,184 potential theory, 165 Prohorov's metric, 97 Prohorov's theorem, 20,29,97 quasi-invariance, 151 Riemann-Wiener manifold, 214 simple nonanticipating process, 192 Skorokhod, A.V., 163,216 stochastic integral, 87,188 stochastic integral equatior 202 strong regularity, 171 surface measure, 212 test operator, 174 m-topology, 94,99,102,109 trace class operator, 2,9 trace class type bilinear map, 104,199 translation of Wiener measure, 113,146 uniformly~m-continuous near zero, 109 Varadhan, S.R.S., 110 weak convergence, 97 weak distribution, 92 Wiener, N., 43 Wiener measure, 36,74 Wiener process, 159,170,189 rotation invariance, 33 S-operator, 16 Sazonov, V.V., 102,109 Segal, I.E.,56,92 Shepp, L.A., 110,151