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The Gaussian Blog

 

Gaussian measures, distributions, random variables, vectors and processes etc., are ubiquitous in mathematics. The purpose of this blog is to collect, present and disseminate some of the main aspects, results, properties, of Gaussian distributions in short texts (posts), describing both their developments and relevance to the current mathematical interests.

The blog is collaborative and interactive. Everybody with an interest in it can suggest potential topics, propose short texts (5 to 10 pages at most), share and host the blog (on a more visible platform), and participate actively to its diffusion, in particular towards young researchers. 

The posts may be authored, anonymous, invited etc. They should be of interest to a large audience, present some main statements with elements or full proofs, comment on their applications and developments in the current research, and include a few pointers to the bibliography, and some possible historical background. Texts already online may be updated, corrected, completed, and improved. Some repetitions and overlaps between the notes are welcome and useful. (The posts already available mainly reflect the interests and expertise of one of the contributors.) 

Topics to be possibly covered, more suggestions can be proposed:

Some basics on Gaussian measures and variables

Some basic properties and characterizations of Gaussian measures and variables

Historical notice on Gaussian distributions and variables

The Central Limit Theorem

Gaussian distributions in Statistics

Gaussian distributions in Physics

Hermite polynomials

Mehler kernel, and the Ornstein-Uhlenbeck operator

Brownian motion and Wiener measure

Admissible shift, reproducing kernel Hilbert space, and abstract Wiener space

Integrability of norms of Gaussian vectors and processes

Gaussian concentration inequalities

Large deviations of Gaussian vectors

Malliavin calculus

Gaussian Riesz transforms

Gaussian comparison inequalities

Boundedness and continuity of Gaussian processes

Stationary Gaussian processes and spectral measures

Fractional Brownian motions

Gaussian rough paths

Gaussian Free Field

Gaussian Multiplicative Cascade

Gaussian random matrix ensembles

Probability distances between Gaussians

The Gaussian isoperimetric inequality

Some geometric inequalities for Gaussian measures

The Gaussian Poincaré inequality

Logarithmic Sobolev and transportation inequalities

Stein’s method for normal approximation

Superconcentration of some Gaussian models

Gaussian kernels

The Gaussian product conjecture

Wick products

Tensor product and Fock space

Gaussian quantization

Gaussian analytic functions

Random Energy Model

 
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