Quasi-sure non-self-intersection for rough differential equations driven by fractional Brownian motion

Abstract

In this paper we study the self-intersection of paths solving elliptic stochastic differential equations driven by fractional Brownian motion. We show that such a path has no self-intersection – except for paths forming a set of zero $(r,q)$-capacity in the sample space – provided the dimension d of the space and the Hurst parameter H satisfy the inequality $d>rq+2∕H$. This inequality is sharp in the case of brownian motion and fractional brownian motion according to existing results. Various results exist for the critical case where $d=rq+4$ for Brownian motion.