Global solutions of the random vortex filament equation

We prove the existence of a global solution for the filament equation with the inital condition given by a geometric rough path in the sense of Lyons (1998 Rev. Mat. Iberoamericana 14 215-310). Our work gives a positive answer to a question left open in recent publications: Berselli and Gubinelli (2007 Commun. Math. Phys. 269 693-713) showed the existence of a global solution for a smooth initial condition while Bessaih et al (2005 Ann. Probab. 33 1825-55) proved the existence of a local solution for a general initial condition given by a rough path.