### Gaussian measures and processes (geometric approach)

#### ideas noted

The most natural measure in a finite-dimensional linear space is, of course, Lebesgue measure. However, it is not a probability measure (the measure of the whole space is infinity, not 1), and it fails to exist in infinite dimension.

The most natural probability measure in a linear space (of finite or infinite dimension) is Gaussian measure.

The modern theory of Gaussian measures lies at the intersection of the theory of random processes, functional analysis, and mathematical physics and is closely connected with diverse applications in quantum field theory, statistical physics, financial mathematics, and other areas of sciences. The study of Gaussian measures combines ideas and methods from probability theory, nonlinear analysis, geometry, linear operators, and topological vector spaces in a beautiful and nontrivial way. (Preface, p. xi.)
V.I. Bogachev, "Gaussian measures", AMS 1998.

#### Isoperimetry

The main geometric property of both measures (Lebesgue and Gaussian) is an isoperimetric inequality. For Lebesgue measure it is classical (J. Steiner 1842, H. Schwarz 1884). Among all bodies of a given volume, a ball minimizes the surface area. For Gaussian measure, an isoperimetric inequality has appeared in the work

V.N. Sudakov, B.S. Tsirelson, "Extremal properties of half-spaces for spherically invariant measures", Zapiski LOMI 41 (1974), 14-24 (Russian); Journal of Soviet Mathematics 9:1 (1978), 9-18 (English). [MR51#1932]

It was found independently in

C. Borell, "The Brunn-Minkowski inequality in Gauss space", Invent. Math. 30:2 (1975), 207-216.

Much better proofs have appeared later (A. Ehrhard 1983; M. Ledoux 1994; S. Bobkov 1997).

Among all sets of a given probability, a half-space minimizes the probability of a neighborhood.

The Gaussian isoperimetry implies highly general theorems about the probability distribution of the norm of a Gaussian random vector, as well as the maximum of a Gaussian random process. Such a distribution must have a density (except for a possible atom at the lower end). Moreover, the density is continuous except, maybe, a finite or countable set, where it jumps down. These facts were found initially in the work

B.S. Tsirelson, "The density of the distribution of the maximum of a Gaussian process", Theory Probab. Appl. 20:4 (1975), 847-856 (transl. from Russian). [MR52#15633]

The proof was unnecessarily complicated. Much better approach, based on beautiful geometric results of A. Ehrhard, was found later, see Corollary 4.4.2 in the book by V.I. Bogachev, "Gaussian measures", AMS 1998.

B.S. Tsirelson, I.A. Ibragimov, V.N. Sudakov, "Norms of Gaussian sample functions."
Lecture Notes in Mathematics (Springer) 550 (1976), 20-41. [MR56#16756]
B.S. Tsirelson, "A geometrical approach to maximum likelihood estimation for infinite-dimensional Gaussian location. I." Theory Probab. Appl. 27:2 (1982), 411-418. [MR83i:62150] [download]
B.S. Tsirelson, "A geometric approach to maximum likelihood estimation for infinite-dimensional Gaussian location. II." Theory Probab. Appl. 30:4 (1985), 820-828. [MR87i:62152] [download]
B.S. Tsirelson, "A geometric approach to maximum likelihood estimation for infinite-dimensional Gaussian location. III." Theory Probab. Appl. 31:3 (1987), 470-483. [MR88c:62140] [download]

#### Citing works (1974-2010)

(including works that do not cite my papers but still use "Borel-TIS inequality" etc.)

Random functions and vectors, measure concentration

```                                                          *
*
*             *   *
*       *     *   *
*                                     *   *   *     * * *
*                           *     * * * * *   *   * * * *
*                         * *     * * * * * * *   * * * *
*           *             * *   * * * * * * * *   * * * *
* *       *   *   * *   *   * *   * * * * * * * * * * * * *
*   * * *     *   *   * *   * * * * * * * * * * * * * * * * * *
*   * *   * * * *   * * * * * *   * * * * * * * * * * * * * * * * * *
4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0
1974        1980                1990                2000                2010
```

Adler, Aida, Ajiev, Ambrosio, Arcones, Azais, Bakry, Ball, Barthe, Baudoin, Bayle, Bentkus, Blower, Bobkov, Bogachev, Borell, Brandolini, Byczkowski, Canete, Canzani, Carmona, Cattiaux, Chatterjee, Davydov, Deheuvels, Del Barrio, Diebolt, Dudley, Ehrhard, Fatalov, Gao, Gardner, Gentil, Giannopoulos, Gine, Gluskin, Goldman, Gotze, Gourcy, Gozlan, Guerra, Guillin, Hairer, Hoffman-Jorgensen, Horfelt, Houdre, Hu, Jakobson, Konakov, Kratz, Latala, Le, Ledoux, Lewandowski, Li, Lifshits, Linde, Makarova, Maniglia, Marchal, Martin, Massart, Mathieu, Matran, Maurey, Meckes, Milman E., Milman M., Milman V., Miranda, Montenegro, Morgan, Nazarov, Oleszkiewicz, Pallara, Paulauskas, Peres, Piterbarg, Posse, Privault, Rackauskas, Roberto, Rosales, Ryznar, Samotij, Shao, Shepp, Smolyanov, Smorodina, Sodin, Stamatovich, Sudakov, Talagrand, Taylor, Toninelli, Trombetti, Virag, Vitale, Vittone, Wang, Wigman, Wojtaszczyk, Wschebor, Wu, Yurinsky, Zak. (Detail)

Statistical (and other) applications

```                                                                        *   *
*                   * *               *   *
*               * * * *               *   *
*     *     * *       * * * * *         * * * * *
* *   *   * * *     * * * * * *   * *   * * * * *
4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0
1974        1980                1990                2000                2010
```

Addario-Berry, Arcones, Arlot, d'Aspremont, Barabas, Baraud, Baringhaus, Barron, Beran, Bickel, Bigot, Birge, Blanchard, Bobkov, Broutin, Burr, Byambazhav, Chesneau, Chung, Csorgo, Dabrowska, Devroye, Doss, Dossal, Gaenssler, Gadat, van de Geer, Ghaoui, Gill, Gine, Grigoriev, Grubel, Hall, Ho, Horvath, Huet, Johnstone, Kerkyacharian, Kokoszka, Koltchinskii, Krieger, Le Pennec, Li, Lugosi, Mallat, Mammen, Mason, Massart, Millar, Milstein, Mogulskii, Molnar, Neumann, Nussbaum, Pensky, Picard, Pitts, Politis, Polzehl, Reynaud-Bouret, Romano, Roquain, Rost, Sapatinas, Tillich, Tribouley, Vakulenco, Van Keilegom, Veraverbeke, Wolf, Yandell, Zemor, Zhang, Zhou, Zilberburg, Zitikis. (Detail)

Partial differential equations

```                                                                      * *   *
4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0
1974        1980                1990                2000                2010
```

Betta, Brock, Chiacchio, Feo, Ferone, Mercaldo. (Detail)

#### Structure, isomorphism, convergence of series

It is worth noting here that one of the fundamental ideas in the theory of Gaussian measures is that the various centered Radon Gaussian measures are realizations of one and the same "canonical" Gaussian measure: the countable product of the standard normal Gaussian distributions on the line. (Preface, p. xi.)
The main tool of transferring the classical results to the general locally convex spaces setting are theorems 3.4.1, 3.4.4, and 3.5.1 which are essentially due to Tsirelson [774], [775]. ...
Convergence of series and sequences of Gaussian vectors was investigated... The works of Ito and Nisio [371] and Tsirelson [774], [775] were of particular importance for this direction. (Bibl. comments to Chapter 3, p. 383-384.)
V.I. Bogachev, "Gaussian measures", AMS 1998.

The results have appeared in

B.S. Tsirelson, "A natural modification of a random process and its application to stochastic functional series and Gaussian measures", Zapiski LOMI 55 (1976), 35-63 (Russian); Journal of Soviet Mathematics 16:2 (1981), 940-956 (English). [MR53#11727]

B.S. Tsirel'son, "Addendum to the article on natural modification." Zapiski LOMI 72 (1977), 202-211 (Russian); Journal of Soviet Mathematics 23:3 (1983), 2363-2369 (English). [MR58#24499]

#### Citing works (1976-2010)

```                                                *
*                           *     *
* * *                       *     *
* * *       *   * *   *     * *   *                     *
6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0
1976    1980                1990                2000                2010
```

Bobkov, Bogachev, Chevet, Chuprunov, Gardner, Kobanenko, Krylov, Lifshits, Rockner, Sato, Smolyanov, Sudakov, Talagrand. (Detail)