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Book Review

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Book Information:

Author: Paul Malliavin
Title: Stochastic analysis
Additional book information: Springer, 1997, 343+xi pp., ISBN 3-540-57024-1, $125.00$

Jean-Michel Bismut, Large deviations and the Malliavin calculus, Progress in Mathematics, vol. 45, Birkhäuser Boston, Inc., Boston, MA, 1984. MR 755001

Denis R. Bell, The Malliavin calculus, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 34, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987. MR 902583

-, ``Degenerate Stochastic Differential Equations and Hypoellipticity,'' (Pitman monographs and surveys in pure and applied mathematics; 79), Longman, Essex, England, 1995. CMP 98:01

Nicolas Bouleau and Francis Hirsch, Dirichlet forms and analysis on Wiener space, De Gruyter Studies in Mathematics, vol. 14, Walter de Gruyter & Co., Berlin, 1991. MR 1133391, DOI 10.1515/9783110858389

C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13

Albert Eagle, Series for all the roots of the equation $(z-a)^m=k(z-b)^n$, Amer. Math. Monthly 46 (1939), 425–428. MR 6, DOI 10.2307/2303037

P. Erdös and T. Grünwald, On polynomials with only real roots, Ann. of Math. (2) 40 (1939), 537–548. MR 7, DOI 10.2307/1968938

Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944

Bruce K. Driver, Towards calculus and geometry on path spaces, Stochastic analysis (Ithaca, NY, 1993) Proc. Sympos. Pure Math., vol. 57, Amer. Math. Soc., Providence, RI, 1995, pp. 405–422. MR 1335485, DOI 10.1090/pspum/057/1335485

I. V. Girsanov, On transforming a class of stochastic processes by absolutely continuous substitution of measures, Teor. Verojatnost. i Primenen. 5 (1960), 314–330 (Russian, with English summary). MR 0133152

James Glimm and Arthur Jaffe, Quantum physics, 2nd ed., Springer-Verlag, New York, 1987. A functional integral point of view. MR 887102, DOI 10.1007/978-1-4612-4728-9

12.

L. Gross, Abstract Wiener Spaces, Proc. 5th. Berkeley Symposium Math. Stat. Prob. 2, (1965), 31 - 42.

Leonard Gross, Potential theory on Hilbert space, J. Functional Analysis 1 (1967), 123–181. MR 0227747, DOI 10.1016/0022-1236(67)90030-4

Nobuyuki Ikeda and Shinzo Watanabe, Stochastic differential equations and diffusion processes, 2nd ed., North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. MR 1011252

Hui Hsiung Kuo, Gaussian measures in Banach spaces, Lecture Notes in Mathematics, Vol. 463, Springer-Verlag, Berlin-New York, 1975. MR 0461643

Shigeo Kusuoka and Daniel Stroock, Applications of the Malliavin calculus. I, Stochastic analysis (Katata/Kyoto, 1982) North-Holland Math. Library, vol. 32, North-Holland, Amsterdam, 1984, pp. 271–306. MR 780762, DOI 10.1016/S0924-6509(08)70397-0

Paul Malliavin, Géométrie différentielle stochastique, Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 64, Presses de l’Université de Montréal, Montreal, Que., 1978 (French). Notes prepared by Danièle Dehen and Dominique Michel. MR 540035

Paul Malliavin, Stochastic calculus of variation and hypoelliptic operators, Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976) Wiley, New York-Chichester-Brisbane, 1978, pp. 195–263. MR 536013

Paul Malliavin, $C^{k}$-hypoellipticity with degeneracy, Stochastic analysis (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1978) Academic Press, New York-London, 1978, pp. 199–214. MR 517243

Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951

James Norris, Simplified Malliavin calculus, Séminaire de Probabilités, XX, 1984/85, Lecture Notes in Math., vol. 1204, Springer, Berlin, 1986, pp. 101–130. MR 942019, DOI 10.1007/BFb0075716

David Nualart, The Malliavin calculus and related topics, Probability and its Applications (New York), Springer-Verlag, New York, 1995. MR 1344217, DOI 10.1007/978-1-4757-2437-0

Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3

I. E. Segal, Distributions in Hilbert space and canonical systems of operators, Trans. Amer. Math. Soc. 88 (1958), 12–41. MR 102759, DOI 10.1090/S0002-9947-1958-0102759-X

Ichiro Shigekawa, Absolute continuity of probability laws of Wiener functionals, Proc. Japan Acad. Ser. A Math. Sci. 54 (1978), no. 8, 230–233. MR 517327

Ichiro Shigekawa, Derivatives of Wiener functionals and absolute continuity of induced measures, J. Math. Kyoto Univ. 20 (1980), no. 2, 263–289. MR 582167, DOI 10.1215/kjm/1250522278

Daniel W. Stroock, The Malliavin calculus and its application to second order parabolic differential equations. I, Math. Systems Theory 14 (1981), no. 1, 25–65. MR 603973, DOI 10.1007/BF01752389

Daniel W. Stroock, The Malliavin calculus, a functional analytic approach, J. Functional Analysis 44 (1981), no. 2, 212–257. MR 642917, DOI 10.1016/0022-1236(81)90011-2

S. Watanabe, Lectures on stochastic differential equations and Malliavin calculus, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 73, Published for the Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin, 1984. Notes by M. Gopalan Nair and B. Rajeev. MR 742628

31.

N. Wiener, Differential space, J. Math. Phys. 2, 131-174 (1923).

Review Information:

Reviewer: Bruce K. Driver
Affiliation: University of California, San Diego
Email: driver@euclid.ucsd.edu