Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - University of Rouen Normandy

This work (see [1]) is concerned with the existence and uniqueness of the solution for the stochastic fast logarithmic equation with Stratonovich multiplicative noise in $\mathbb{R}^d$. It provides an answer to a critical case (corresponding to the porous media operator $\Delta X^m$ for $m = 0$) left as an open problem in the paper Barbu-Röckner-Russo (see [2]). We face several technical difficulties related both to the degeneracy properties of the logarithm and to the fact that the problem is treated in an unbounded domain. Firstly, the order in which the approximations are considered is very important and different from previous methods. Secondly, the energy estimates needed in the last step can only be achieved with a well-chosen Stratonovich-type rectification of the noise.

[1] I. Ciotir, D. Goreac, and R. Fukuizumi, The stochastic fast logarithmic equation in $\mathbb{R}^d$ with multiplicative Stratonovich noise, submitted, preprint here.
[2] V. Barbu, M. Röckner, and F. Russo, Stochastic porous media equations in $\mathbb{R}^d$. Journal de Mathématiques Pures et Appliquées, 103(4):1024-1052, 2015.