Local invertibility of adapted shifts on Wiener space,under finite energy condition

In this paper we study the connection between local invertibility and global invertibility of adapted shifts on Wiener space. First we go from the global to the local and we obtain an explicit formula for the inverse of a stopped adapted shift which is invertible. Then we take the opposite direction and we show that under finite energy condition, a shift which is locally invertible is also invertible. We work in a general framework which applies both to the classical Wiener space and to the Brownian sheet.