The Gaussian Blog
launched November 2023
Gaussian measures, distributions, random variables, vectors and processes etc., are ubiquitous in mathematics. It is the purpose of this blog 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 expected to be 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, waiting for more suggestions:
Some basics on Gaussian measures and variables
Some basic properties and characterizations of Gaussian measures and variables
Historical notice on Gaussian distributions and variables
Gaussian distributions in Statistics
Gaussian distributions in Physics
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 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
…