Global Solutions of the Random Vortex Filament Equation

In this article we prove the existence of a global solution for the random vortex filament equation. Our work gives a positive answer to a question left open in recent publications: Berselli and Gubinelli [5] showed the existence of global solution for a smooth initial condition while Bessaih, Gubinelli, Russo [6] proved the existence of a local solution for a general initial condition. In this article we prove the existence of a global solution for the following random vortex filament equation dγ dt = u(γ(t)), t ∈ [0,∞) (0.1) γ(0) = γ0, (0.2) where the initial condition γ0 : [0, 1]→ R3 is a geometric ν-rough path (for some ν ∈ (13 , 1)), see Assumption 2.7. Here γ : [0,∞) → Dγ0 ⊂ C is some trajectory in the subset Dγ0 of C of continuous closed curves in R3, uY , Y ∈ Dγ0 ⊂ C is a vector field given by