Journal of Mathematical Sciences, Vol. 79, No. 2, 1996 GAUSSIAN MEASURES ON LINEAR SPACES V. I. Bogachev UDC 517.987.5 Introduction Gaussian distributions play a fundamental role in all natural sciences. The growing importance of infinite-dimensional Gaussian distributions has become clear over the past decades. The aim of this survey is to present a systematic exposition of the theory of Gaussian measures on general locally convex spaces. Various results on this topic are scattered in a very extensive literature. There are several books devoted to some special aspects of the theory of Gaussian measures such as [420, 395, 478, 199, 407, 429, 289, 230, 118, 480, 80, 359, 304, 114]; however, only [420, 395] and [289] (written more than 20 years ago) present a general theory. The locally convex space setting provides a natural framework for most of the results in this theory. It should be noted that overall this generality does not bring new technical difficulties; moreover, it makes basic constructions more natural and clearer. There are very few cases where the proofs in the Hilbert space setting are really simpler. In such cases we shall mention these shorter proofs. There are several main directions in the study of Gaussian measures. The first one might be called a "linear-topological theory." We discuss the most important results in this direction in Chapter 1 and Chapter 5. Among them: classical theorems on equivalence/singularity, zero-one laws, the reproducing kernel Hilbert spaces, measurable linear functionals and operators, and topological properties of supports. The second direction is connected with convexity and isoperimetry. In Chapter 2 we discuss several fundamental inequalities and estimates, such as exponential integrability, tail behavior, and measures of small balls. This direction is closely related to the theory of Ganssian processes, in particular, to the study of the sample path properties. However, we do not discuss these important applications here. Finally, we make several introductory remarks about the central limit theorem and related questions. A good account of the whole direction is given in a recent monograph by Ledoux and Talagrand [304]. The third direction we discuss might be called a "nonlinear theory" or "analysis on Wiener spaces." This is a very actively developing part of infinite-dimensional stochastic analysis. Nonlinear transformations of Gaussian measures, including the study of absolute continuity, and the Malliavin calculus are the heart of this direction. These problems are briefly discussed in Chapter 4. They will be the subject of a separate survey of the author. Some basic information about Sobolev classes over Ganssian measures is presented in Chapter 3, where, in addition, we discuss Gaussian capacities. In Chapter 5 we introduce Wiener processes in infinite dimensions and study several related problems. This survey is based on the lectures of the author at Moscow State University. Functional-analytic aspects are emphasized, though we do not avoid probabilistic constructions. The results presented below have been discussed with many experts in this field. I am especially grateful to L. Accardi, S. Albeverio, G. Ben Arous, V. Bentkus, A. B. Cruzeiro, G. Da Prato, Yu. L. Daletskii, Yu. A. Davydov, D. Elworthy~ H. Fhllmer, F. Ghtze, N. V. Krylov, M. Lifshits, P. Malliavin, E. Mayer- Wolf, P. Meyer, S. A. Molchanov, D. Nualart, E. Pardoux, D. Preiss, Yu. V. Prokhorov, M. Rhckner, V. V. Sazonov, B. Schmuland, A. N. Shiryaev, A. V. Skorohod, O. G. Smolyanov, V. N. Sudakov, A. V. Uglanov, H. Weizss M. Zakai, and O. Zeitouni. The support of the Russian Fundamental Research Foundation (Grant N 94-01-01556) and the Interna- tional Science Foundation (Grant N M38000)is gratefully acknowledged. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika iee Prilozheniya. Tematicheskiye Obzory. Vol. 16, Analiz-8, 1994 1072-3324/~6/7902-0933515.00 9 Plenum Publishing Corporation 933