We are concerned with the three-dimensional incompressible Navier–Stokes equations driven by an additive stochastic forcing of trace class. First, for every divergence free initial condition in $L^{2}$ we establish existence of infinitely many global-in-time probabilistically strong and analytically weak solutions, solving one of the open problems in the field. This result, in particular, implies nonuniqueness in law. Second, we prove nonuniqueness of the associated Markov processes in a suitably chosen class of analytically weak solutions satisfying a relaxed form of an energy inequality. Translated to the deterministic setting, we obtain nonuniqueness of the associated semiflows. 

　　Publication: 

　　The Annals of Probability, 51(2): 524-579 (March 2023). DOI: 10.1214/22-AOP1607 

　　Author: 

　　Martina Hofmanová 

　　Fakult？t für Mathematik, Universit？t Bielefeld 

　　Rongchan Zhu 

　　Department of Mathematics, Beijing Institute of Technology 

　　XiangchanZhu 

　　Academy of Mathematics and Systems Science, Chinese Academy of Sciences 

　　Email: zhuxiangchan@126.com