Quasi-invariance of Gaussian measures of negative regularity for fractional nonlinear Schrödinger equations

Abstract

We consider the Cauchy problem for the fractional nonlinear Schrödinger equation (FNLS) on the one-dimensional torus with cubic nonlinearity and high dispersion parameter α > 1, subject to a Gaussian random initial data of negative Sobolev regularity σ < s − 1 2 , for s ≤ 1 2 . We show that for all s * (α) < s ≤ 1 2 , the equation is almost surely globally well-posed. Moreover, the associated Gaussian measure supported on H s (T) is quasi-invariant under the flow of the equation. For α < 1 20 (17 + 3 √ 21) ≈ 1.537, the regularity of the initial data is lower than the one provided by the deterministic wellposedness theory.We obtain this result by following the approach of DiPerna-Lions (1989); first showing global-in-time bounds for the solution of the infinite-dimensional Liouville equation for the transport of the Gaussian measure, and then transferring these bounds to the solution of the equation by adapting Bourgain's invariant measure argument to the quasi-invariance setting. This allows us to bootstrap almost sure global bounds for the solution of (FNLS) from its probabilistic local well-posedness theory.

Unique Ergodicity for a Class of Stochastic Hyperbolic Equations with Additive Space-Time White Noise

“…When an invariant measure is available, often global well-posedness for a.e. initial data sampled according to the invariant measure follows via an application of Bourgain's invariant measure argument [2] (see also [17,Theorem 6.1] for a general formulation). However, this is more-or-less the only globalisation argument that has been shown to work for singular stochastic wave equations.…”