  It is important to clarify where and when rocky and icy planetesimals are formed in a viscously evolving disk. We wish to understand how local runaway pile-up of solids occurs inside or outside the snow line. We assume an icy pebble contains micron-sized silicate grains that are uniformly mixed with ice and are released during the ice sublimation. Using a local one-dimensional code, we solve the radial drift and the turbulent diffusion of solids and the water vapor, taking account of their sublimation/condensation around the snow line. We systematically investigate effects of back-reactions of the solids to gas on the radial drift and diffusion of solids, scale height evolution of the released silicate particles, and possible difference in effective viscous parameters between that for turbulent diffusion ($\alpha_{\rm tur}$) and that for the gas accretion rate onto the central star ($\alpha_{\rm acc}$). We study the dependence on the ratio of the solid mass flux to the gas ($F_{\rm p/g}$). We show that the favorable locations for the pile-up of silicate grains and icy pebbles are the regions in the proximity of the water snow line inside and outside it, respectively. We found that runaway pile-ups occur when both the back-reactions for radial drift and diffusion are included. In the case with only the back-reaction for the radial drift, no runaway pile-up is found except for extremely high pebble flux, while the condition of streaming instability can be satisfied for relatively large $F_{\rm p/g}$ as found in the past literatures. If the back-reactions for radial diffusion is considered, the runaway pile-up occurs for reasonable value of pebble flux. The runaway pile-up of silicate grains that would lead to formation of rocky planetesimals occurs for $\alpha_{\rm tur} \ll \alpha_{\rm acc}$, while the runaway pile-up of icy pebbles is favored for $\alpha_{\rm tur} \sim \alpha_{\rm acc}$. 
Formation of rocky and icy planetesimals inside and outside the snow   line: Effects of diffusion, sublimation and back-reaction