A note on stochastic Schrodinger equations with fractional multiplicative noise

作者：Pinaud Olivier^*

来源：Journal of Differential Equations, 2014, 256(4): 1467-1491.

DOI：10.1016/j.jde.2013.11.003

This work is devoted to non-linear stochastic Schrodinger equations with multiplicative fractional noise, where the stochastic integral is defined following the Riemann-Stieljes approach of Zahle. Under the assumptions that the initial condition is in the Sobolev space H-q (R-n) for a dimension n less than three and q an integer greater than or equal to zero, that the noise is a Q-fractional Brownian motion with Hurst index H is an element of (1/2, 1) and spatial regularity Hq+4 (R-n), as well as appropriate hypotheses on the non-linearity, we obtain the local existence of a unique pathwise solution in C-0(0, T, H-q (R-n)) boolean AND C-0,C-gamma (0, T, Hq-2(R-n)), for any gamma is an element of [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 Schrodinger semigroup. We follow here a different route and our proof relies on a change of phase that removes the noise and leads to a Schrodinger equation with a magnetic potential that is not differentiable in time.

• 出版日期2014-2-15