ABSTRACT: From the Matsumoto Yor observation to stationary measures for a discrete KdV model Let F(x,y)=(u,v) from R2 into itself such that F∘F(x,y)=(x,y). The discrete Korteweg-de Vries model associated to F considers two dimensional vectors (xtn,ytn) indexed by (t,n)∈Z2 submitted to (xtn,ytn)=F(xt−1n,ytn−1). In the particular case F(x,y)=(y−1−(x+y)−1,(x+y)−1), by considerations on Brownian motion, Matsumoto-Yor in 1999 have found laws of independent random variables X,Y such that U,V defined by F(X,Y)=(U,V) are independent, obtaining stationary measures on the KdV model. In an extraordinary paper, Croydon and Sasada (2021) have generalized this result to the surprizing F(x,y)=(yβxy+1αxy+1,xαxy+1βxy+1). The lecture will prove that if X,Y are independent and such that U,V=F(X,Y) are also independent, then the laws of X and Y are indeed the generalized Gaussian distributions discovered by Croydon and Sasada. This is joint work with Jacek Wesolowski (arXiv 2203.05404).