Logarithmic Schrödinger Equations in Infinite Dimensions

Abstract

We study the logarithmic Schrödinger equation with finite range potential on R Z d . Through a ground-state representation, we associate and construct a global Gibbs measure and show that it satisfies a logarithmic Sobolev inequality. We find estimates on the solutions in arbitrary dimension and prove the existence of weak solutions to the infinite-dimensional Cauchy problem.