Martingale solutions of stochastic fractional nonlinear Schrödinger equation on a bounded interval*

Abstract

This paper is concerned with stochastic fractional nonlinear Schrödinger equation, which plays a very important role in fractional nonrelativistic quantum mechanics. Due to disturbing and interacting of the fractional Laplacian operator on a bounded interval with white noise, the stochastic fractional nonlinear Schrödinger equation is too complicated to be understood. This paper would explore and analyze this stochastic fractional system. Using a suitable weighted space with some fractional operator skills, it overcame the difficulties coming from the fractional Laplacian operator on a bounded interval. Applying the tightness instead of the common compactness, and combining Prokhorov theorem with Skorokhod embedding theorem, it solved the convergence problem in the case of white noise. It finally established the existence of martingale solutions for the stochastic fractional nonlinear Schrödinger equation on a bounded interval.

Keywords:

• Stochastic Schrödinger equation
• the fractional Laplacian operator
• a bounded interval
• martingale solutions

Notes

No potential conflict of interest was reported by the authors.

This paper was partly done while the second author visited the Center for Mathematical Sciences, at Huazhong University of Science and Technology, Wuhan, China.

Additional information

Funding

This work is supported by the National Natural Science Foundation of China [grant number 11571245], [grant number 11401409]; Scientific Research Project of Education Department in Sichuan Province [grant number 15ZA0031].