Stochastic phenomena in magnetic nanowires based on domain wall (DW) motion is scientifically important thus to understand and control such behaviors are very meaningful. Here we report on the investigation of pinning and depinning of DWs in permalloy nanowires with six types of longitudinally asymmetric notches using focused magneto-optic Kerr effect (FMOKE) magnetometer and magnetic force microscopy (MFM). The hysteresis loops obtained by FMOKE indicate the generation of one or two distinct depinning fields by creating one notch close to the edge of the nanowires, in comparison multiple depinning processes occur in the nanowires with two identical notches symmetrically placed along the transverse direction, indicating more remarkable stochastic DW depinning phenomena. The MFM images verify the existence of DW in each type of nanowires and the DW sizes in the latter kind of nanowires are generally larger than those in the former ones. These observations can be explained by considering the thermal perturbation and edge or surface roughness effects in nanowires.