On the existence and uniqueness of SDE describing an n-particle system interacting via a singular potential (Q1067302)

The system of the stochastic differential equations describing an interacting n-particle system in \(R^{2d}\) is considered: \[ dZ^ i(t)=\sum_{j:j\neq i}\gamma_ j(\nabla H(Z^ i(t)-Z^ j(t)))^{\perp}dt+d\quad W^ i(t),\quad Z^ i(0)=z^ i,\quad i=1,...,n, \] where \(H(z)=g(| z|)\) \((g\in C^ 2(0,+\infty))\) and \((x,y)^{\perp}=(x,-y)\in R^{2d}\). Under a suitable condition on the singularity of g(z) at \(z=0\) and supposing that all constants \(\{\gamma_ i\}\) have the same sign existence and uniqueness of a solution is proved.