A stochastic differential equation arising from the vortex problem

The author studies the system of stochastic differential equations dz i =∑ j≠i α j K(z i −z j )dt+σdW i ,i=1,⋯,n , where W i are 2-dimensional independent Brownian motions, K(z)≡K(x,y)=(G y ,−G x ) , and G(z)=−(2π) −1 log(|z|) . The 2-dimensional variables z i represent the positions of n vortices in a viscous and incompressible fluid, with vorticity intensities α i , respectively. The constant σ is related to the viscosity. The drift is singular on a manifold S in R 2n . It is shown that the first-passage time to S is infinite with probability 1.