A note on stochastic Schr\"odinger equations with fractional multiplicative noise

TL;DR

This paper studies nonlinear stochastic Schr"odinger equations driven by fractional Brownian motion, establishing local existence of solutions using a novel phase change technique due to the weak smoothing properties of the Schr"odinger semigroup.

Contribution

It introduces a new method involving a phase change to handle fractional noise in Schr"odinger equations, overcoming limitations of standard fixed point approaches.

Findings

Proves local existence and uniqueness of solutions under specified conditions.

Develops a phase change method to handle non-differentiable magnetic potentials.

Addresses challenges posed by weak smoothing in Schr"odinger semigroups.

Abstract

This work is devoted to non-linear stochastic Schr\"odinger equations with multiplicative fractional noise, where the stochastic integral is defined following the Riemann-Stieljes approach of Z\"ahle. Under the assumptions that the initial condition is in the Sobolev space $H^q(\Rm^n)$ for a dimension $n$ less than three and $q$ an integer greater or equal to zero, that the noise is a $Q-$fractional Brownian motion with Hurst index $H\in(1/2,1)$ and spatial regularity $H^{q+4}(\Rm^n)$, as well as appropriate hypotheses on the non-linearity, we obtain the local existence of a unique pathwise solution in $\calC^0(0,T,H^q(\Rm^n)) \cap \calC^{0,\gamma}(0,T,H^{q-2}(\Rm^n))$, for any $\gamma \in [0,H)$. Contrary to the parabolic case, standard fixed point techniques based on the mild formulation of the SPDE cannot be directly used because of the weak smoothing in time properties of the Schr\"odinger semigroup. We follow here a different route and our proof relies on a change of phase that removes the noise and leads to a Schr\"odinger equation with a magnetic potential that is not differentiable in time.